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UNIVERSITY OF CALIFORNIA.

UNIVERSITY OF CALIFORNIA

LIBRARY

OF THE

DEPARTMENT OF PHYSICS

NOY1419H

Received

Accessions No. (p $ fy Book No. y .

SMITHSONIAN MISCELLANEOUS COLLECTIONS

VOLUME 58, NUMBER 1

SMITHSONIAN

PHYSICAL TABLES

FIFTH REVISED EDITION

PREPARED BY

F. E. FOWLE

AID. SMITHSONIAN ASTROPHYSICAL OBSERVATORY

sosr

(PUBLICATION 1944),

CITY OF WASHINGTON

PUBLISHED BY THE SMITHSONIAN INSTITUTION 1910

StT 110

ADVERTISEMENT.

In connection with the system of meteorological observations established by the Smithsonian Institution about 1850, a series of meteorological tables was compiled by Dr. Arnold Guyot, at the request of Secretary Henry, and the first edition was published in 1852. Though primarily designed for meteorological observers reporting to the Smithsonian Institution, the tables were so widely used by physicists that it seemed desirable to recast the work entirely. It was decided to publish three sets of tables, each representative of the latest knowledge in its field, and independent of one another, but forming a homogeneous series. The first of the new series, Meteorological Tables, was published in 1893, tne second, Geographical Tables, in 1894, and the third, Physical Tables, in 1896. In 1909 yet another volume was added, so that the series now comprises : Smithsonian Meteorological Tables, Smithsonian Geographical Tables, Smithsonian Physical Tables, and Smithsonian Mathematical Tables.

The fourteen years which have elapsed since the publication of the first edition of the Physical Tables, prepared by Professor Thomas Gray, have brought such changes in the material upon which the tables must be based that it became necessary to prepare this almost wholly new set of tables for the present edition.

CHARLES D. WALCOTT, Secretary -, Smithsonian Institution.

June, 1910.

222536

PREFACE.

The present Smithsonian Physical Tables are the outcome of a radical revision of the set of tables compiled by Professor Thomas Gray in 1896. Recent data and many new tables have been added for which the references to the sources have been made more complete ; and several mathematical tables have been added, — some of them especially computed for this work. The inclusion of these mathematical tables seems warranted by the demand for them. In order to pre- serve a uniform change of argument and to facilitate comparison, many of the numbers given in some tables have been obtained by interpolation in the data actually given in the papers quoted.

Our gratitude is expressed for many suggestions and for help in the improve- ment of the present edition : to the U. S. Bureau of Standards for the revision of the electrical, magnetic, and metrological tables and other suggestions \ to the U. S. Coast and Geodetic Survey for the revision of the magnetic and geodetic tables ; to the U. S. Geological Survey for various data ; to Mr. Van Orstrand for several of the mathematical tables ; to Mr. Wead for the data on the musical scales ; to Mr. Sosman for the new physical-chemistry data ; to Messrs. Abbot, Becker, Lanza, Rosa, and Wood ; to the U. S. Bureau of Forestry and to others. We are also under obligation to the authors and publishers of Landolt-Bornstein- Meyerhoffer's Physikalisch-chemische Tabellen (1905) and B. O. Peirce's Mathe- matical Tables for the use of certain tables.

It is hardly possible that any series of tables involving so much transcribing, interpolation, and calculation should be entirely free from errors, and the Smith- sonian Institution will be grateful, not only for notice of whatever errors may be found, but also for suggestions as to other changes which may seem advisable for later editions.

F. E. FOWLE.

ASTROPHYSICAL OBSERVATORY

OF THE SMITHSONIAN INSTITUTION,

June, 1910

TABLE OF CONTENTS

PACK

Introduction on units of measurement and conversion factors . . xv

Units of measurement : general discussion xv

Dimension formulae for dynamic units . . ... . . . xvii

" " " heat units xxiii

of electric and magnetic units •' general discussion . . xxv

" formulae in electrostatic system ...... xxvi

" electromagnetic system . . . . xxix

Practical units of electricity, legalization of . . . . . . xxxiii

TABLE

1. Formulae for conversion factors : (a) Fundamental units ... 2

(b) Derived units ... 2

I. Geometric and dynamic units 2

II. Heat units .... 3

III. Magnetic and electric units 3

2. Tables for converting U. S. weights and measures :

(1) Customary to metric ........ 5

(2) Metric to customary ........ 6

3. Equivalents of metric and British imperial weights and measures :

(1) Metric to imperial ........ 7

(2) Multiples, metric to imperial ...... 8

(3) Imperial to metric ........ 9

(4) Multiples, imperial to metric . . . . . .10

4. Volume of a glass vessel from weight of its volume of water or mercury 1 1

5. Elementary differential coefficients and integrals . . . .12

6. Reciprocals, squares, cubes and square roots of natural numbers . 13

7. Logarithms, 1000-2000 . . . . . . . .22

8. Logarithms ........... 24

9. Antilogarithms . . . . . . . . . . . 26

10. Antilogarithms, .9000-1.0000 ........ 28

11. Circular (trigonometric) functions, argument (° ,'.) . . . . 30

12. " " " argument (radians) . . . 35 i2a. Factorials, n!, n = i to 100 ......... 38

13. Values of - - (hyperbolic sines), for values of x from o to 5 . 39

14. Logarithms of - (hyperbolic sines), for values of x from o to 5 . 40

15. Values of - ^ — (hyperbolic cosines), for values of x from o to 5 . 41

Vi CONTENTS.

20.

1 6. Logarithms of — -^ — (hyperbolic cosines) for values of x from o to 5 42

17. Values of c* and e~* and their logarithms for x from o to 10

18. " " log. ? for values of x from i o to 30 . . .

19. " " e* and e'3* and their logarithms . . . .

" "4" " "

21. " " e4 and* * « .....

22. " " ** and <r-* and " " for fractional values of x

23. Probability of errors of observations : probability integral . .

n A << « << " « « «<

24. . .

25. Values of 0.6745 ~—^

26. " " °'6745 ^-^ri

29. Inverse of probability integral. Diffusion ......

30. Logarithms of the gamma function T(n) for values of n between i and 2

31. Values for the first seven zonal harmonics from 0 = o° to # = 90° .

32. " " log. M/q.Tr^/aa1 for facilitating the calculation of the mutual inductance between two coaxial circles . . . . . .

33. Value fory 2(i — sm*0sin?<£)±}d<l> for different values of 0 ; also the

corresponding logarithms .........

34. Moments of inertia, radii of gyration, corresponding weights

35. British standard wire gauge : diameters, sections .....

36. Birmingham wire gauge " " ..... (For Brown and Sharp gauge, see tables 40 and 41)

37. Cross section and weights of wires (copper, iron, brass), British units .

38. " " " " " " " " " metric units .

39. " " " " " aluminum wire: British and metric units .

40. Size, weight and electrical constants of copper wire, Brown and Sharp

gauge : common units .........

41. Same as table 40, but in metric measure ......

42. Weight in grammes per square metre of sheet metal ....

43. " " various common units of sheet metal .....

44. Strength of materials : (a) metals ......

(b} stones . . .....

(f) brick ........

(d} concretes .......

45. " " " timber tests .......

Af. a « a ti (i

40. . • • • • ...

47. Moduli of rigidity . ..... . • . .

CONTENTS. Vll

470. Variation of the moduli of rigidity with the temperature . . . 74

48. Young's modulus .......... 75

49. Compressibility of the more important solid elements . . . -76

50. Hardness ............ 76

51. Relative hardness of the elements 76

52. Poisson's ratio . . . . . . . . . . 76

53. Elastic moduli of crystals, formulae 77

54. " " " " numerical results 78

55. Compressibility of O, air, N, H at different pressures and temperatures 79

56. " " ethylene " " " " " . 79

-„ it n « « « it (6 it j~

58. " " carbon dioxide at " " " " . 80

59. " gases, values of a . . . . . . .80

60. " " air and oxygen between 18° and 22°C . . .80

61. Relation between pressure, temperature and volume of sulphur dioxide 81

62. " " " " " " " ammonia . 81

63. Compressibility of liquids 82

64. " " " 83

65. Specific gravities corresponding to the Beaume' scale . . . .84

66. Densities of the solid and liquid elements 85

67. " various woods . 87

68. " " " solids 88

69. " " " alloys 89

70. " " " liquids 90

71. " " " gases 91

72. " " " aqueous solutions of salts, bases and acids . . 92

73. Density of water between o° and 36° C ...... 94

74. Volume of water at temperatures between o° and 36° C in terms of its

volume at the temperature of maximum density . . . -95

75. Density and volume of water at different temperatures from -10 to 25o°C 96

76. " " " " mercury at " " " -10 " 36o°C 97

77. Specific gravity of aqueous ethyl alcohol ...... 98

78. Density of aqueous methyl alcohol ....... 99

79. Variation of the density of alcohol with the temperature . . .100

80. Velocity of sound in solids ......... 101

8 1. " " " " liquids and gases 102

82. Musical scales 103

83. " " 103

84. Force of gravity at sea level and different latitudes . . . .104

85. Results of some of the more recent gravity determinations . . . 105

86. Value of gravity at some of the U. S. C. and G. Survey stations . .106

87. Length of seconds pendulum for sea level and different latitudes . 107

88. Determinations of the length of the seconds pendulum . . . 107

89. Miscellaneous data as to the earth and planets . . . . .108

90. Terrestrial magnetism : secular change of declination . . .no

91. " " dip or inclination 112

92. " secular change of dip 112

93. " " horizontal intensity 113

Vlll CONTENTS.

94. Terrestrial magnetism : secular change of horizontal intensity . -113

95. " " total intensity 114

96. " " secular change of total intensity . , .114

97. " " agonic line . . . * ;, . 115

98. Pressure of mercury and water columns . . . . • .116

99. Reduction of barometer to standard temperature . . . . . 117

100. " " " " " gravity, inch and metric scales . 118

101. " " " " latitude 45° : inch scale . . v .119

102. " " " " " " metric scale . . ,. . 120

103. Correction of barometer for capillarity : inch and metric scale . .121

104. Aerodynamics: data for wind pressures . . . . • .122

105. " " " the soaring of planes 123

1 06. Coefficients of friction . . . . . . . . . .124

107. Viscosity of water at different temperatures . . . . . 125

108. Coefficients of viscosity for solutions of alcohol in water . . .126

109. Specific viscosity of mineral oils . . . . • » . .126

no. " " " various oils > « . 126

in. " " " " liquids .127

112. " " " " " temperature variation . --Y . 128

113. " " solutions: variation with density and temperature. 129

114. " " " " atomic concentrations .... 133

115. " " " gases and vapors . . . . . . . 134

116. " u " " " " temperature variation . . 135

117. Diffusion of an aqueous solution into pure water 136

118. " " vapors 7~ 137

119. " gases and vapors ........ 138

1190. " metals into metals ........ 138

120. Solubility of inorganic salts in water: temperature variation . . 139

121. " " a few organic salts in water : temperature variation . . 140

122. " " gases in water . . 140

123. Absorption of gases by liquids . . . . . . . .141

124. Capillarity and surface tension : water and alcohol in air . .142

125. " " " " miscellaneous liquids in air . . 142

126. " aqueous solutions of salts . . .142

127. Capillarity and surface tension : liquids in contact with air, water or

mercury ...........

128. Capillarity and surface tension : liquids at solidifying point .

129. " " " " thickness of soap films

130. Vapor pressures

131. " " of ethyl alcohol . . . .

132. " " " methyl "

133. and temperatures : (a) carbon disulphide .

(b) chlorobenzine

(d) aniline .... (<?) methyl salicylate .

(f) bromonaphthaline .

(g) mercury

CONTENTS. IX

134. Vapor pressures of solutions of salts in water 149

135. Pressure of aqueous vapor at low temperatures . . . . . 151

136. " " " " o° to 100° C (Broch) 152

137. " " " " 100° to 230° C (Regnault) . . . .153

138. Weight in grains of aqueous vapor in a cubic foot of saturated air . 154 139- " " grammes of " " " " " metre of " " . 154

140. Hygrometry, vapor pressure in the atmosphere 155

141. " dew-points ......... 156

142. Relative humidity .......... 158

143. Values of 0.378*? in the atmospheric pressure equation h=.B — 0.3781?. 159

144. Table for facilitating the calculation of ^760 ..... 160

145. Logarithms of ^760 for values of h between 80 and 800 . . . 160

146. Values of 1-1-0.00367 t\

(a) for values of /between o° and 10° C, by tenths . . . 162

(b) " " " " " —90° " +1990° C, by tens . . 163 (f) logarithms for t " — 49° " +399° c» bv units • • l64 (d) « « " " 400° " 1990° C, by tens . . .166

147. Determination of heights by the barometer ...... 167

148. Barometric pressures corresponding to different temperatures of the

boiling-point of water :

(a) Common measure . . . . . . . .168

(<£) Metric measure ......... 169

149. Standard wave-lengths : Fabry-Buisson's iron arc lines . . . 170

150. " " " red cadmium line 170

151. Stronger lines of some of the elements . . . . .170

152. Rowland's standard solar wave-lengths (also corrections) . . . 171

153. Kayser's standard iron arc lines (also corrections) .... 174

154. Wave-lengths of the Fraunhofer lines . . . . . . .176

155. Photometric standards .......... 177

156. Sensitiveness of the eye to radiation of different wave-lengths: low

(threshold) intensities . . . . . . . . .178

157. Sensitiveness of the eye : greater intensities . ..... 178

158. Sensibility of the eye to small differences of intensity (Fechner) . .178

159. Solar energy and its absorption by the earth's atmosphere . . . 179

1 60. The solar " constant " of radiation and temperature of sun . . . 179

161. Distribution of intensity of radiation over solar disk . . . . 179

162. Relative intensities of sunlight and sky-light . . . . 179

163. Indices of refraction of Jena glasses 180

164. MM « « « « !8o

165. " " " " " " temperature coefficients . . 180

166. " " " " various alums 181

167. " " " " metals and metallic oxides :

(a) Kundt's experiments 182

(b) Du Bois and Rubens' experiments . . . . . 182 (<r) Drude's experiments 182

168. Indices of refraction for rock salt 183

169. " " " " " " temperature coefficients . . .183

170. " " " " sylvine 183

* X CONTENTS.

171. Indices of refraction for fluorite 184

, 172. " " " " " temperature coefficients . . . 184

v 173. " " " " Iceland spar 'V . 184

: 174. " " " " nitroso-dimethyl-aniline . . . . 184

175. " « " " quartz 185

176. " " " " various monorefringents 186

177. " " " " " uniaxial crystals 187

178. " " " " " biaxial crystals 187

179. " " " " solutions of salts and acids :

(a) solutions in water . . . .188

(b) « " alcohol . . . .188

180. " " " " various liquids 189

181. " " " " gases and vapors 190

182. Reflection of light, perpendicular incidence : various values of n . . 191

183. " " " incidence varying : n near unity .... 191

184. " " " " " «=i.55 • • • • .191

185. Reflection from metals 192

1 86. Transmission of Jena glasses . . . . . . . 193

187. " " " 193

188. " " " ultra-violet glasses 193

189. " " alum, rock salt, sylvine, fluorite, Iceland spar, quartz 194

190. Color screens (Landolt) 195

191. " " (Wood) 195

192. " " (Jena glasses) ........ 196

193. Rotation of the plane of polarized light by solutions .... 197

194. " " " " " " " " sodium chlorate and quartz 197

195. Colors of thin films, Newton's rings 198

196. Thermal conductivity of metals and alloys 199

197. " " " various substances ...... 200

198. " " " water and salt solutions . • . . .200

199. " " " organic liquids 200

200. " " " gases 200

201. Heat of combustion . . . . . . . . . . 201

202. Heat values and analyses of various fuels : (a) coals .... 202

(fr) peats . . . .202 (c) liquid fuels . . .202

203. Chemical and physical properties of explosives . . . . . 203

204. Heat of combination .......... 204

205. Latent heat of vaporization 206

206. " " " fusion 208

207. Melting-points of the chemical elements 209

208. Boiling-points " " " " 210

209. Melting-points of various inorganic compounds . • • . .211

210. Boiling-points " " " " 213

211. Melting points of various mixtures of metals . • • . .-214

212. " " " " " " " 214

213. Low-melting-point alloys .. .214

CONTENTS. XI

214. Densities, melting-points, boiling-points of organic compounds:

(a) Paraffin series .215

(£) Olefine series 215

(f) Acetylene series 216

(d) Monatomic alcohols 216

(e) Alcoholic ethers 216

(/) Ethyl ethers . . . . . . . . .216

215. Lowering of freezing-points by salts in solution 217

216. Raising of boiling-points by salts in solution 219

217. Freezing mixtures .......... 220

218. Critical temperatures, pressure, volumes and densities of gases . .221

219. Coefficients of linear expansion of the chemical elements . . .222

220. " " " " " miscellaneous substances . . 223

221. " " cubical " " crystalline and other solids . . 224

222. " " " " " liquids 225

223. " " thermal expansion of gases 226

224. Mechanical equivalent of heat: various data ..... 227

225. " " " " adopted values (Ames) . . . 227

226. " " " " conversion values . . . .227

227. Specific heats of the chemical elements 228

228. " " " water and mercury 229

229. " " " various solids 230

230. " " " " liquids 230

231. " " " " minerals and rocks 231

232. " " " " gases and vapors ...... 232

233. Gas and mercury thermometers : formulae 233

234. Comparison of hydrogen and i6m thermometers : o° to 100° C. . . 233

235. " " " " 59IU " o°toioo°C. . .233

236. " " " " 16'" and 59™ thermometers: -5° to -35° C. 233

237. Comparison of air and i6m glass thermometers : o° to 300° C. . . 234

238. " " " " 59m " " 100° to 200° C. . 234

239. " " hydrogen and various mercury thermometers . . 235

240. " " air and high temperature (59m) mercury thermometer . 235

241. " " H., toluol, alcohol, petrol ether, pentane thermome- ters 235

242. Stem correction for thermometers 236

243- " " " 237

244. " " 237

245. Radiation formulae and constants for perfect radiator . . . .238

246. " in calories for perfect radiators at various temperatures . 238

247. " distribution in spectrum at various temperatures . . . 238

248. Cooling by radiation and convection ; ordinary pressures . . . 239

249. " " " " " different pressures . . . 239

250. " " " " " very small pressures . . . 240

251. Cooling by radiation and convection : temperature and pressure effects 240

252. Properties and constants of saturated steam : metric measure . • 241

253. " " " " " " common measure . . 242

254. Ratio of the electrostatic to the electromagnetic unit of electricity . 247

Xl CONTENTS.

255. Dielectric strength ; steady potential for spark in air . . . .248

256. " " alternating potential for spark in air . . . 248

257. " potentials for longer sparks in air ... 249

258. " " effect of (air) pressure ...... 249

259. " " of various materials . . , . . . 250

260. " " u kerosene . ...... 250

261. Electromotive force of standard cells : absolute current measures . 251

262. Data for voltaic cells : (a) double fluid cells 252

(b) single fluid cells ..... 253

(d) secondary (storage) cells .... 253

263. Contact differences of potential, solids with liquids and liquids with

liquids in air 254

264. Contact differences of potential, solids with solids in air . . 256

265. Potential difference between metals in various salt solutions . . 257

266. Thermoelectric powers ......... 258

267. " " with platinum 259

268. Peltier effect 260

269. Various determinations of the ohm 261

270. Specific resistance of metallic wires ....... 262

271. " " " metals 263

272. Resistance of metals and alloys at low temperatures . . . .264

273. Conductivity of three-metal and miscellaneous alloys .... 266

274. Conducting power of alloys 267

275. Electric resistance with alternating currents (straight wires) . . 269

276. International atomic weights and electrochemical equivalents . . 270

277. Conductivity of a few dilute solutions 272

278. Electrochemical equivalents and densities of nearly normal solutions . 272

279. Specific molecular conductivity of solutions . . . . . . 273

280. " " " " " limiting values . . . 274

281. " " " " " temperature coefficients . 274

282. Equivalent conductivity of salts, acids, bases in solution . . . 275

283. " " " some additional salts in solution . -277

284. " conductance of the separate ions 278

285. Hydrolysis of ammonium acetate : ionization of water . . . .278

286. Dielectric constants (specific inductive capacity) of gases . . -279

287. " " " " " " " temperature coefficient 279

288. Dielectric constants (specific inductive capacity) of gases : pressure co-

efficient ............ 279

289. Dielectric constants of liquids ^ . 280

290. " " " " temperature coefficient .... 282

291. " " liquefied gases 282

292. " " standard solutions for calibrations . . . 283

293. Dielectric constants of solids 283

294. " " " crystals 284

295. Temperature variation of electrical resistance of glass, porcelain . . 285

296. Permeability of iron rings and wire, various inductions . . . 286

CONTENTS. xiii

297. Permeability of transformer iron :

(a) specimen of Westinghouse No. 8 transformer . . . 286 (*) " " " 6 287

(V) " " " 4 " . . 287

(d) " " Thomson-Houston 1 5oo-watt transformer . . 287

298. Magnetic properties of iron and steel ....... 288

299. " " " cast iron in intense fields 288

300. " corrections for ring specimens ...... 288

301. Demagnetizing factors for rods ........ 289

302. " " Shuddemagen's values ..... 289

303. Composition and magnetic properties of iron and steel . . . 290

304. Permeability of some of the specimens in Table 303 .... 292

305. Magnetic properties of soft iron at o° and 100° C 292

306. " " " steel at o° and 100° C 292

307. " " " cobalt at 100° C 293

308. " " " nickel " " " 293

309. " " " magnetite ....... 293

310. " " " Lowmoor wrought iron ..... 293

311. " " " Vicker's tool steel 293

312. " " " Hadfield's manganese steel .... 293

313. Saturation values for different steels 293

314. Magnetic properties of iron in very weak fields 294

315. Dissipation of energy in cyclic magnetization of magnetic substances . 294

316. " " " " " m " " cable transformers . 294

317. " " " " " " " various substances .295

318. " " " " " " " transformer steels .296

319. Magneto-optic rotation, formulae : Verdet's constant .... 297

320. " " " in solids 298

321. " " " " liquids 299

322. " " " " solutions of salts and acids in water . . 301

323. " " " " " " " in alcohol . . . .303

324. " " " " " " " " hydrochloric acid . . 303

325. " « " " gases .... ... 304

326. Verdet's and Kundt's constants 304

327. Magnetic susceptibility of liquids and gases ...... 305

328. Values of Kerr's constant ......... 305

329. Variation of the resistance of bismuth in magnetic field . . . 306

330. « " " " " nickel " " " . 306

331. " " " " " various metals in a magnetic field . .306

332. Transverse galvanomagnetic and thermomagnetic effects . . . 307

333. Variation of the Hall constant with the temperature .... 307

334. Appendix : Mean specific heat of iron at high temperatures . . . 308

335. Total heat of iron to high temperatures .... 308 — " Definitions of units 309

Index 313

INTRODUCTION.

UNITS OF MEASUREMENT AND CONVERSION FORMULA.

Units. — The quantitative measure of anything is a number which expresses the ratio of the magnitude of the thing to the magnitude of some other thing of the same kind. In order that the number expressing the measure may be intelligi- ble, the magnitude of the thing used for comparison must be known. This leads to the conventional choice of certain magnitudes as units of measurement, and any other magnitude is then simply expressed by a number which tells how many magnitudes equal to the unit of the same kind of magnitude it contains. For example, the distance between two places may be stated as a certain number of miles or of yards or of feet. In the first case, the mile is assumed as a known distance ; in the second, the yard, and in the third, the foot. What is sought for in the statement is to convey an idea of the distance by describing it in terms of distances which are either familiar or easily referred to for comparison. Similarly quantities of matter are referred to as so many tons or pounds or grains and so forth, and intervals of time as a number of hours or minutes or seconds. Gen- erally in ordinary affairs such statements appeal to experience j but, whether this be so or not, the statement must involve some magnitude as a fundamental quan- tity, and this must be of such a character that, if it is not known, it can be readily referred to. We become familiar with the length of a mile by walking over dis- tances expressed in miles, with the length of a yard or a foot by examining a yard or a foot measure and comparing it with something easily referred to, — say our own height, the length of our foot or step, — and similarly for quantities of other kinds. This leads us to be able to form a mental picture of such magnitudes when the numbers expressing them are stated, and hence to follow intelligently descriptions of the results of scientific work. The possession of copies of the units enables us by proper comparisons to find the magnitude-numbers express- ing physical quantities for ourselves. The numbers descriptive of any quan- tity must depend on the intrinsic magnitude of the unit in terms of which it is described. Thus a mile is 1760 yards, or 5280 feet, and hence when a mile is taken as the unit the magnitude-number for the distance is i, when a yard is taken as the unit the magnitude-number is 1760, and when afoot is taken it is 5280. Thus, to obtain the magnitude-number for a quantity in terms of a new unit when it is already known in terms of another we have to multiply the old magnitude- number by the ratio of the intrinsic values of the old and new units ; that is, by the number of the new units required to make one of the old.

XVi INTRODUCTION.

Fundamental Units of Length and Mass. — It is desirable that as few dif- ferent kinds of unit quantities as possible should be introduced into our measure- ments, and since it has been found possible and convenient to express a large number of physical quantities in terms of length or mass or time units and com- binations of these they have been very generally adopted as fundamental units. Two systems of such units are used in this country for scientific measurements, namely, the British, and the French or metric, systems. Tables of conversion factors are given in the book for facilitating comparisons between quantities ex- pressed in terms of one system with similar quantities expressed in the other. In the customary system the standard unit of length is the yard and is now defined as 3600/3937 metre. The unit of mass is the avoirdupois pound and is denned as 1/2.20462 kilogramme.

The British yard is defined as the " straight line or distance (at 62° F.) between the transverse lines in the two gold plugs in the bronze bar deposited in the office of the exchequer." The British standard of mass is the pound avoirdupois and is the mass of a piece of platinum marked "P. S. 1844, i lb.," preserved in the exchequer office.

In the metric system the standard of length is defined as the distance between the ends of a certain platinum bar (the metre des Archives) when the whole bar is at the temperature o° Centigrade. The bar was made by Borda, and is preserved in the national archives of France. A line-standard metre has been constructed by the International Bureau of Weights and Measures, and is known as the Inter- national Prototype Metre. A number of standard-metre bars which have been carefully compared with the International Prototype have lately been made by the International Bureau of Weights and Measures and furnished to the various gov- ernments who have contributed to the support of that bureau. These copies are called National Prototypes.

Borda, Delambre, Laplace, and others, acting as a committee of the French Academy, recommended that the standard unit of length should be the ten mil- lionth part of the length, from the equator to the pole, of the meridian passing through Paris. In 1795 the French Republic passed a decree making this the legal standard of length, and an arc of the meridian extending from Dunkirk to Barcelona was measured by Delambre and Mechain for the purpose of realizing the standard. From the results of that measurement the metre bar was made by Borda. The metre is not now defined as stated above, but as the length of Borda's rod, and hence subsequent measurements of the length of the meridian have not affected the length of the metre.

The French, or metric, standard of mass, the kilogramme, is the mass of a piece of platinum also made by Borda in accordance with the same decree' of the Republic. It was connected with the standard of length by being made as nearly as possible of the same mass as that of a cubic decimetre of distilled water at the temperature of 4° C., or nearly the temperature of maximum density.

As in the case of the metre, the International Bureau of Weights and Measures has made copies of the kilogramme. One of these is taken as a standard, and

INTRODUCTION. XV11

is called the International Prototype Kilogramme. The others were distrib- uted in the same manner as the metre standards, and are called National Proto- types.

Comparisons of the French and customary standards are given in tabular form in Table 2 ; and similarly Table 3, differing slightly, compares the British and French systems. In the metric system the decimal subdivision is used, and thus we have the decimetre, the centimetre, and the millimetre as subdivisions, and the dekametre, hektometre, and kilometre as multiples. The centimetre is most commonly used in scientific work.

Time. — The unit of time in both the systems here referred to is the mean solar second, or the 86,4ooth part of the mean solar day. The unit of time is thus founded on the average time required for the earth to make one revolution on its axis relatively to the sun as a fixed point of reference.

Derived Units. — Units of quantities depending on powers greater than unity of the fundamental length, mass, and time units, or on combinations of different powers of these units, are called " derived units." Thus, the unit of area and of volume are respectively the area of a square whose side is the unit of length and the volume of a cube whose edge is the unit of length. Suppose that the area of a surface is expressed in terms of the foot as fundamental unit, and we wish to find the area-number when the yard is taken as fundamental unit. The yard is 3 times as long as the foot, and therefore the area of a square whose side is a yard is 3 X 3 times as great as that whose side is a foot. Thus, the surface will only make one ninth as many units of area when the yard is the unit of length as it will make when the foot is that unit. To transform, then, from the foot as old unit to the yard as new unit, we have to multiply the old area-number by 1/9, or by the ratio of the magnitude of the old to that of the new unit of area. This is the same rule as that given above, but it is usually more convenient to express the transformations in terms of the fundamental units directly. In the above case, since on the method of measurement here adopted an area-number is the product of a length-number by a length-number the ratio of two units is the square of the ratio of the intrinsic values of the two units of length. Hence, if / be the ratio of the magnitude of the old to that of the new unit of length, the ratio of the cor- responding units of area is /2. Similarly the ratio of two units of volume will be /*, and so on for other quantities.

Dimensional Formulae. — It is convenient to adopt symbols for the ratios of length units, mass units, and time units, and adhere to their use throughout ; and in what follows, the small letters, /, m, t, will be used for these ratios. These letters will always represent simple numbers, but the magnitude of the number will depend on the relative magnitudes of the units the ratios of which they repre- sent. When the values of the numbers represented by /, m, t are known, and the powers of /, m, and / involved in any particular unit are also known, the factor for transformation is at once obtained. Thus, in the above example, the value of / was 1/3 and the power of /involved in the expression for area is /*; hence, the factor for transforming from square feet to square yards is 1/9. These factors

XV111 INTRODUCTION.

have been called by Prof. James Thomson "change ratios," which seems an appropriate term. The term " conversion factor " is perhaps more generally known, and has been used throughout this book.

Conversion Factor. — In order to determine the symbolic expression for the conversion factor for any physical quantity, it is sufficient to determine the degree to which the quantities length, mass, and time are involved in the quantity. Thus, a velocity is expressed by the ratio of the number representing a length to that representing an interval of time, or L/T, an acceleration by a velocity-number divided by an interval of time-number, or L/T2, and so on, and the correspond- ing ratios of units must therefore enter to precisely the same degree. The fac- tors would thus be for the above cases, /// and ///2. Equations of the form above given for velocity and acceleration which show the dimensions of the quantity in terms of the fundamental units are called " dimensional equations." Thus

is the dimensional equation for energy, and MLaT~2 is the dimensional formula for energy.

In general, if we have an equation for a physical quantity

Q=CLaM6Tc,

where C is a constant and LMT represents length, mass, and time in terms of one set of units, and we wish to transform to another set of units in terms of which

T TiyT T*

the length, mass, and time are LyMyTy, we have to find the value of _ ',— J ', which

J_/ JYl 1

in accordance with the convention adopted above will be / m t, or the ratios of the magnitudes of the old to those of the new units.

Thus Ly = L/, My = Mm, Ty = T/, and if Qy be the new quantity-number

Q, = CL/-M,T/'

= CLataMbmbTctc =

or the conversion factor is PnPf, a quantity of precisely the same form as the dimension formula LaM6Tc.

We now proceed to form the dimensional and conversion factor formulae for the more commonly occurring derived units.

1. Area. — The unit of area is the square the side of which is measured by the unit of length. The area of a surface is therefore expressed as

S = CL2,

where C is a constant depending on the shape of the boundary of the surface and L a linear dimension. For example, if the surface be square and L be the length of a side C is unity. If the boundary be a circle and L be a diameter C = ir/4, and so on. The dimensional formula is thus L2, and the conversion factor /*.

2. Volume. — The unit of volume is the volume of a cube the edge of which is measured by the unit of length. The volume of a body is therefore expressed as

INTRODUCTION. XIX

V = CL8,

where as before C is a constant depending on the shape of the boundary. The dimensional formula is L8 and the conversion factor /8.

3. Density. — The density of a substance is the quantity of matter in the unit of volume. The dimension formula is therefore M/V or ML~8, and conversion factor ml~*.

Example. — The density of a body is 150 in pounds per cubic foot: required the density in grains per cubic inch.

Here m is the number of grains in a pound = 7000, and / is the number of inches in a foot = 12 ; /. ml~B = 7000/1 2s = 4.051. Hence the density is 150 X 4.051 =607.6 in grains per cubic inch.

NOTE. — The specific gravity of a body is the ratio of its density to the density of a standard substance. The dimension formula and conversion factor are therefore both unity.

4. Velocity. — The velocity of a body at any instant is given by the equation v = -p, or velocity is the ratio of a length-number to a time-number. The di-

d r

mension formula is LT"1, and the conversion factor lt~\

Example. — A train has a velocity of 60 miles an hour : what is its velocity in feet per second ?

Here 7=5280 and / = 36oo ; .'. trl = = — — 1-467. Hence the velo- city =60 X 1-467 = 88.0 in feet per second.

5. Angle. — An angle is measured by the ratio of the length of an arc to the length of the radius of the arc. The dimension formula and the conversion factor are therefore both unity.

6. Angular Velocity. — Angular velocity is the ratio of the magnitude of the angle described in an interval of time to the length of the interval. The dimen- sion formula is therefore T"1, and the conversion factor is t~\

7. Linear Acceleration. — Acceleration is the rate of change of velocity or

a = -?• The dimension formula is therefore VT"1 or LT~a, and the conversion at

factor is /r2.

Example? — A body acquires velocity at a uniform rate, and at the end of one minute is moving at the rate of 20 kilometres per hour : what is the acceleration in centimetres per second per second ?

Since the velocity gained was 20 kilometres per hour in one minute, the accel- eration was 1 200 kilometres per hour per hour.

Here/=iooooo and /=36oo; /. //~2= 100000/3600* = .00771, and there- fore acceleration =^.007 7 1 X 1200 = 9.26 centimetres per second.

8. Angular Acceleration. — Angular acceleration is rate of change of angu-

XX INTRODUCTION.

lar velocity. The dimensional formula is thus angulayelocity or T~2, and the conversion factor /~2.

9. Solid Angle. — A solid angle is measured by the ratio of the surface of the portion of a sphere enclosed by the conical surface forming the angle to the square of radius of the spherical surface, the centre of the sphere being at the

vertex of the cone. The dimensional formula is therefore — ^ or i, and hence

l_i

the conversion factor is also i.

10. Curvature. — Curvature is measured by the rate of change of direction of the curve with reference to distance measured along the curve as independent

variable. The dimension formula is therefore .ang G. or Lr1, and the conversion

length

factor is l~\

11. Tortuosity. — Tortuosity is measured by the rate of rotation of the tan- gent plane round the tangent to the curve of reference when length along the

curve is independent variable. The dimension formula is therefore - — ^—? or

length

Lr1, and the conversion factor is l~l.

12. Specific Curvature of a Surface. — This was defined by Gauss to be» at any point of the surface, the ratio of the solid angle enclosed by a surface formed by moving a normal to the surface round the periphery of a small area containing the point, to the magnitude of the area. The dimensional formula is

therefore solld angle Or L~2, and the conversion factor is thus /-* surface

13. Momentum. — This is quantity of motion in the Newtonian sense, and is, at any instant, measured by the product of the mass-number and the velocity- number for the body.

Thus the dimension formula is MV or MLT"1, and the conversion factor mlf~\ Example. — A mass of 10 pounds is moving with a velocity of 30 feet per sec- ond : what is its momentum when the centimetre, the gramme, and the second are fundamental units ?

Here m = 453-59, /= 30.48, and /= i ; .*. mtrl = 453-59 X 30.48 = 13825. The momentum is thus 13825 X 10 X 30 = 4 147 500.

14. Moment of Momentum. — The moment of momentum of a body with reference to a point is the product of its momentum-number and the number expressing the distance of its line of motion from the point. The dimensional formula is thus ML^T"1, and hence the conversion factor is mPr1.

15. Moment of Inertia. — The moment of inertia of a body round any axis is expressed by the formula ^mr*, where m is the mass of any particle of the body

INTRODUCTION. Xxi

and r its distance from the axis. The dimension formula for the sum is clearly the same as for each element, and hence is ML2. The conversion factor is there- fore mt*.

16. Angular Momentum. — The angular momentum of a body round any axis is the product of the numbers expressing the moment of inertia and the angular velocity of the body. The dimensional formula and the conversion fac- tor are therefore the same as for moment of momentum given above.

17. Force. — A force is measured by the rate of change of momentum it is capable of producing. The dimension formulae for force and "time rate of change of momentum " are therefore the same, and are expressed by the ratio of momentum-number to time-number or MLT~2. The conversion factor is thus

NOTE. — When mass is expressed in pounds, length in feet, and time in seconds, the unit force is called the poundal. When grammes, centimetres, and seconds are the corresponding units the unit of force is called the dyne.

Example. Find the number of dynes in 25 poundals.

Here m = 453-59> l = 3°-48, and t= i ; .-. m/r*= 453-59 X 30.48 — 13825 nearly. The number of dynes is thus 13825 X 25 =345625 approximately.

18. Moment of a Couple, Torque, or Twisting Motive. — These are dif- ferent names for a quantity which can be expressed as the product of two numbers representing a force and a length. The dimension formula is therefore FL or ML2T~2, and the conversion factor is ml2*-*.

19. Intensity of a Stress. — The intensity of a stress is the ratio of the num- ber expressing the total stress to the number expressing the area over which the stress is distributed. The dimensional formula is thus FLr2 or ML"1'!"2, and the conversion factor is ml~lt~*.

20. Intensity of Attraction, or " Force at a Point." — This is the force of attraction per unit mass on a body placed at the point, and the dimensional for- mula is therefore FM"1 or LT~2, the same as acceleration. The conversion fac- tors for acceleration therefore apply.

21. Absolute Force of a Centre of Attraction, or " Strength of a Cen- tre." — This is the intensity of force at unit distance from the centre, and is there- fore the force per unit mass at any point multiplied by the square of the distance from the centre. The dimensional formula thus becomes FL2M~J or L8T~2. The conversion factor is therefore T8/"2.

22. Modulus of Elasticity. — A modulus of elasticity is the ratio of stress intensity to percentage strain. The dimension of percentage strain is a length divided by a length, and is therefore unity. Hence, the dimensional formula of a modulus of elasticity is the same as that of stress intensity, or ML-1T~2, and the conversion factor is thus also ml~lt~*.

Xxii INTRODUCTION.

23. "Work and Energy. — When the point of application of a force, acting on a body, moves in the direction of the force, work is done by the force, and the amount is measured by the product of the force and displacement numbers. The dimensional formula is therefore FL or ML2T~2.

The work done by the force either produces a change in the velocity of the body or a change of shape or configuration of the body, or both. In the first case it produces a change of kinetic energy, in the second a change of potential energy. The dimension formulae of energy and work, representing quantities of the same kind, are identical, and the conversion factor for both is #z/2/~2.

24. Resilience. — This is the work done per unit volume of a body in distort- ing it to the elastic limit or in producing rupture. The dimension formula is there- fore ML2T-2L~8 or MI/^T-2, and the conversion factor

25. Power, or Activity. — Power — or, as it is now very commonly called, ac- tivity — is defined as the time rate of doing work, or if W represent work and P power

P = — . The dimensional formula is therefore WT"1 or ML'T-8, and the con- dt

version factor mPr*, or for problems in gravitation units more conveniently./?/"1, where /stands for the force factor.

Examples, (a) Find the number of gramme centimetres in one foot pound. Here the units of force are the attraction of the earth on the pound* and the gramme of matter, and the conversion factor is./7, where/ is 453.59 and /is

30.48-

Hence the number is 453.59 X 30.48 = 13825.

(ft) Find the number of foot poundals in i oooooo centimetre dynes. Here m = i/453-59> '= 1/30.48, and / = i ; .-. mt*r2 = 1/453-59 X 30.48', and io6»i/»/-*= 107453.59 X 3°-482= 2.373.

(c) If gravity produces an acceleration of 32.2 feet per second per second, how many watts are required to make one horse-power ?

One horse-power is 550 foot pounds per second, or 550X32.2 = 17710 foot poundals per second. One watt is io7 ergs per second, that is, io7 dyne centi- metres per second. The conversion factor is mf*t~s, where m = 453-59> ^= 3°-48, and /= i, and the result has to be divided by io7, the number of dyne centime- tres per second in the watt.

Hence, 17710 mZ*r*/iol = 17710 X 453-59 X 30.487 io7 = 746.3.

(//) How many gramme centimetres per second correspond to 33000 foot pounds per minute ?

The conversion factor suitable for this case is./?/""1, where/ is 453-59> ' is 30.48, and / is 60.

Hence, 33000 //~1= 33000 X 453-59 X 30.48/60= 7604000 nearly.

* It is important to remember that in problems like that here given the term "pound" or " gramme " refers to force and not to mass.

INTRODUCTION. XX111

HEAT UNITS.

i. If heat be measured in dynamical units its dimensions are the same as those of energy, namely ML2T~2. The most common measurements, however, are made in thermal units, that is, in terms of the amount of heat required to raise the temperature of unit mass of water one degree of temperature at some stated temperature. This method of measurement involves the unit of mass and some unit of temperature ; and hence, if we denote temperature-numbers by ® and their conversion factors by 0, the dimensional formula and conversion factor for quan- tity of heat will be M© and mO respectively. The relative amount of heat com- pared with water as standard substance required to raise unit mass of different substances one degree in temperature is called their specific heat, and is a simple number.

Unit volume is sometimes used instead of unit mass in the measurement of heat, the units being then called thermometric units. The dimensional formula is in that case changed by the substitution of volume for mass, and becomes L8@, and hence the conversion factor is to be calculated from the formula 1*6.

For other physical quantities involving heat we have : —

2. Coefficient of Expansion. — The coefficient of expansion of a substance is equal to the ratio of the change of length per unit length (linear), or change of volume per unit volume (voluminal) to the change of temperature. These ratios are simple numbers, and the change of temperature is inversely as the mag- nitude of the unit of temperature. Hence the dimensional and conversion-factor formulae are ®-1 and 6~1.

3. Conductivity, or Specific Conductance. — This is the quantity of heat transmitted per unit of time per unit of surface per unit of temperature gradient. The equation for conductivity is therefore, with H as quantity of heat,

and the dimensional formula 7^r^ = ^-^, which gives ml~lf~l for conversion factor.

In thermometric units the formula becomes L^T"1, which properly represents diffusivity. In dynamical units H becomes ML2T~2, and the formula changes to MLT-8®-1. The conversion factors obtained from these are 72/"1 and respectively.

XXIV INTRODUCTION.

4. Thermal Capacity. — This is the product of the number for mass and the specific heat, and hence the dimensional formula and conversion factor are simply M and m.

5. Latent Heat. — Latent heat is the ratio of the number representing the quantity of heat required to change the state of a body to the number represent- ing the quantity of matter in the body. The dimensional formula is therefore M®/M or 0, and hence the conversion factor is simply the ratio of the tempera- ture units or 0. In dynamical units the factor is /2/~2.*

6. Joule's Equivalent. — Joule's dynamical equivalent is connected with quantity of heat by the equation

ML2T-2 = JHorJM®.

This gives for the dimensional formula of J the expression U*T~*&~1. The conver- sion factor is thus represented by /V"8^"1. When heat is measured in dynamical units J is a simple number.

7. Entropy. — The entropy of a body is directly proportional to the quantity of heat it contains and inversely proportional to its temperature. The dimen- sional formula is thus M®/® or M, and the conversion factor is m. When heat is measured in dynamical units the factor is mlzt~^6~l.

Examples, (a) Find the relation between the British thermal unit, the calorie, and the therm.

Neglecting the variation of the specific heat of water with temperature, or de- fining all the units for the same temperature of the standard substance, we have the following definitions. The British thermal unit is the quantity of heat required to raise the temperature of one pound of water i° F. The calorie is the quan- tity of heat required to raise the temperature of one kilogramme of water i° C. The therm is the quantity of heat required to raise the temperature of one gramme of water i° C. Hence : —

(1) To find the number of calories in one British thermal unit, we have »*— 45399 and 0 = f ; .'• w<9 = . 45399 X 5/9—25199.

(2) To find the number of therms in one calorie, m=iooo and 6=1; .*. mO= 1000.

It follows at once that the number of therms in one British thermal unit is 1000 X .25199 = 251.99.

(£) What is the relation between the foot grain second Fahrenheit-degree and the centimetre gramme second Centigrade-degree units of conductivity ?

The number of the latter units in one of the former is given by the for-

* It will be noticed that when <=> is given the dimension formula L2T~2 the formulae in^ thermal and dynamical units are always identical. The thermometric units practically suppress mass.

INTRODUCTION. XXV

mula ml~lt~l6°j where m — . 064 799, /= 30.48, and /= i, and is therefore = .064799/30.48 = 2.126 X io~8.

(c) Find the relation between the units stated in (ft) for emissivity. In this case the conversion formula is w/"2/""1, where ml and / have the same value as before. Hence the number of the latter units in the former is 2 = 6.975 X io~6.

(d) Find the number of centimetre gramme second units in the inch grain hour unit of emissivity.

Here the formula is ml~*t~l, where m — 0.064 799» ^=2.54, and ^ = 3600. Therefore the required number is 0.064 799/2-542 X 3600 = 2.790 X io~*.

(e) If Joule's equivalent be 776 foot pounds per pound of water per degree Fahrenheit, what will be its value in gravitation units when the metre, the kilogramme, aud the degree Centigrade are units ?

The conversion factor in this case is ,,_a or I0~l, where / = .3048 and ff-l = i.S-, .'. 776 X .3048 X 1.8 = 425.7.

(/) If Joule's equivalent be 24832 foot poundals when the degree Fahren- heit is unit of temperature, what will be its value when kilogramme metre second and degree-Centigrade units are used ?

The conversion factor is Pr*0~l, where /= .3048, t = i, and 0~l = 1.8 ;

.-. 24832 x r-r2d~l = 24832 x .3048' x 1.8 = 4152.5.

In gravitation units this would give 4152.5/9.81 = 423.3.

ELECTRIC AND MAGNETIC UNITS.

There are two systems of these units, the electrostatic and the electromagnetic systems, which differ from each other because of the different fundamental suppo- sitions on which they are based. In the electrostatic system the repulsive force between two quantities of static electricity is made the basis. This connects force,

quantity of electricity, and length by the equation /=a 22l, where / is force, a a

quantity depending on the units employed and on the nature of the medium, q and ql quantities of electricity, and / the distance between q and qt. The magnitude of the force / for any particular values of q, qt and / depends on a property of the medium across which the force takes place called its inductive capacity. The in- ductive capacity of air has generally been assumed as unity, and the inductive capacity of other media expressed as a number representing the ratio of the induc- tive capacity of the medium to that of air. These numbers are known as the spe- cific inductive capacities of the media. According to the ordinary assumption, then, of air as the standard medium, we obtain unit quantity of electricity when in the above equation y = ?{, and/, a, and / are each unity. A formal definition is given below.

In the electromagnetic system the repulsion between two magnetic poles or

XXvi INTRODUCTION.

quantities of magnetism is taken as the basis. In this system the quantities force, quantity of magnetism, and length are connected by an equation of the form

where m and mt are in this case quantities of magnetism, and the other symbols have the same meaning as before. In this case it has been usual to assume the magnetic inductive capacity of air to be unity, and to express the magnetic induc- tive capacity of other media as a simple number representing the ratio of the in- ductive capacity of the medium to that of air. These numbers, by analogy with specific inductive capacity for electricity, might be called specific inductive capac- ities for magnetism. They are usually called permeabilities. {Vide Thomson, " Papers on Electrostatics and Magnetism," p. 484.) In this case, also, like that for electricity, the unit quantity of magnetism is obtained by making m = mt, and /, a, and / each unity.

In both these cases the intrinsic inductive capacity of the standard medium is suppressed, and hence also that of all other media. Whether this be done or not, direct experiment has to be resorted to for the determination of the absolute val- ues of the units and the relations of the units in the one system to those in the other. The character of this relation can be directly inferred from the dimen- sional formulae of the different quantities, but these can give no information as to the relative absolute values of the units in the two systems. Prof. Riicker has suggested (Phil. Mag. vol. 27) the advisability of at least indicating the exist- ence of the suppressed properties by putting symbols for them in the dimensional formulae. This has the advantage of showing how the magnitudes of the different units would be affected by a change in the standard medium, or by making the standard medium different for the two systems. In accordance with this idea, the symbols K and P have been introduced into the formulae given below to represent inductive capacity in the electrostatic and the electromagnetic systems respectively. In the conversion formulae k and/ are the ordinary specific inductive capacities and permeabilities of the media when air is taken as the standard, or generally those with reference to the first medium taken as standard. The ordinary for- mulae may be obtained by putting K and P equal to unity.

ELECTROSTATIC UNITS.

i. Quantity of Electricity. — The unit quantity of electricity is defined as that quantity which if concentrated at a point and placed at unit distance from an equal and similarly concentrated quantity repels it, or is repelled by it, with unit force. The medium or dielectric is usually taken as air, and the other units in ac- cordance with the centimetre gramme second system.

In this case we have the force of repulsion proportional directly to the square of the quantity of electricity and inversely to the square of the distance between the quantities and to the inductive capacity. The dimensional formula is there- fore the same as that for [force X length2 X inductive capacity]* or and the conversion factor is

INTRODUCTION. XXVii

2. Electric Surface Density and Electric Displacement. — The density of an electric distribution at any point on a surface is measured by the quantity per unit of area, and the electric displacement at any point in a dielectric is mea- sured by the quantity displaced per unit of area. These quantities have therefore the same dimensional formula, namely, the ratio of the formulae for quantity of electricity and for area or M^Lr^T^K*, and the conversion factor m*l~lt-l$.

3. Electric Force at a Point, or Intensity of Electric Field. — This is measured by the ratio of the magnitude of the force on a quantity of electricity at a point to the magnitude of the quantity of electricity. The dimensional formula is therefore the ratio of the formulae for force and electric quantity, or

which gives the conversion factor

4. Electric Potential and Electromotive Force. — Change of potential is proportional to the work done per unit of electricity in producing the change. The dimensional formula is therefore the ratio of the formulae for work and elec- tric quantity, or

which gives the conversion factor

5. Capacity of a Conductor. — The capacity of an insulated conductor is proportional to the ratio of the numbers representing the quantity of electricity in a charge and the potential of the charge. The dimensional formula is thus the ratio of the two formulae for electric quantity and potential, or

* _ T K

which gives Ik for conversion factor. When K is taken as unity, as in the ordinary units, the capacity of an insulated conductor is simply a length.

6. Specific Inductive Capacity. — This is the ratio of the inductive capac- ity of the substance to that of a standard substance, and hence the dimensional formula is K/K or i.*

7. Electric Current. — Current is quantity flowing past a point per unit of time. The dimensional formula is thus the ratio of the formulae for electric quan- tity and for time, or

and the conversion factor

* According to the ordinary definition referred to air as standard medium, the specific inductive capacity of a substance is K, or is identical in dimensions with what is here taken as inductive ca- pacity. Hence in that case the conversion factor must be taken as i on the electrostatic and as on the electromagnetic system.

XXV111 INTRODUCTION.

8. Conductivity, or Specific* Conductance. — This, like the corresponding term for heat, is quantity per unit area per unit potential gradient per unit of time. The dimensional formula is therefore

__ ,p_1K or "" *

electric quantity

_ jj-, area X potential gradient X time

The conversion factor is

9. Specific * Resistance. — This is the reciprocal of conductivity as above defined, and hence the dimensional formula and conversion factor are respec- tively TK.-1 and tk~\

10. Conductance. — The conductance of any part of an electric circuit, not containing a source of electromotive force, is the ratio of the numbers represent- ing the current flowing through it and the difference of potential between its ends. The dimensional formula is thus the ratio of the formulae for current and poten- tial, or

from which we get the conversion factor

n. Resistance. — This is the reciprocal of conductance, and therefore the dimensional formula and the conversion factor are respectively L^TK^1 and

EXAMPLES OF CONVERSION IN ELECTROSTATIC UNITS.

(a) Pind the factor for converting quantity of electricity expressed in foot grain second units to the same expressed in c. g. s. units.

By (i) the formula is wV3/"1^, in which in this case m = 0.0648, /= 30.48, / = i, and k = i ; .*. the factor is 0.0648* X 30.48* = 4.2836.

(£) Find the factor required to convert electric potential from millimetre milli- gramme second units to c. g. s. units.

By (4) the formula is »/i/i/~1^"~J, and in this case m = o.ooi, /= o.i, /= i, and £=i; .*. the factor = o.ooi1 X o.ij=o.oi.

(<:) Find the factor required to convert from foot grain second and specific in- ductive capacity 6 units to c. g. s. units.

By (5) the formula is /£, and in this case 7=30.48 and £ = 6; .*. the factor = 30.48 X 6 = 182.88.

* The term " specific/' as used here and in 9, refers conductance and resistance to that between the ends of a bar of unit section and unit length, and hence is different from the same term in specific heat, specific inductivity, capacity, etc., which refer to a standard substance.

INTRODUCTION. XXIX

ELECTROMAGNETIC UNITS.

As stated above, these units bear the same relation to unit quantity of magne- tism that the electric units do to quantity of electricity. Thus, when inductive capacity is suppressed, the dimensional formula for magnetic quantity on this sys- tem is the same as that for electric quantity on the electrostatic system. All quan- tities in this system which only differ from corresponding quantities defined above by the substitution of magnetic for electric quantity may have their dimensional formulae derived from those of the corresponding quantity by substituting P forK.

i. Magnetic Pole, or Quantity of Magnetism. — Two unit quantities of magnetism concentrated at points unit distance apart repel each other with unit force. The dimensional formula is thus the same as for [force X length2 X in- ductive capacity] or M^UT"1?1, and the conversion factor is

2. Density of Surface Distribution of Magnetism. — This is measured by quantity of magnetism per unit area, and the dimension formula is therefore the ratio of the expressions for magnetic quantity and for area, or MiLriT~1P}, which gives the conversion factor

3. Magnetic Force at a Point, or Intensity of Magnetic Field. — The number for this is the ratio of the numbers representing the magnitudes of the force on a magnetic pole placed at the point and the magnitude of the magnetic pole.

The dimensional formula is therefore the ratio of the expressions for force and magnetic quantity, or

MiJJT-lpi

and the conversion factor

4. Magnetic Potential. — The magnetic potential at a point is measured by the work which is required to bring unit quantity of positive magnetism from zero potential to the point. The dimensional formula is thus the ratio of the formula for work and magnetic quantity, or

which gives the conversion factor

5. Magnetic Moment. — This is the product of the numbers for pole strength and length of a magnet. The dimensional formula is therefore the pro- duct of the formulae for magnetic quantity and length, or M^T"1?*, and the con- version factor

6. Intensity of Magnetization. — The intensity of magnetization of any por- tion of a magnetized body is the ratio of the numbers representing the magni-

XXX INTRODUCTION.

tude of the magnetic moment of that portion and its volume. The dimensional formula is therefore the ratio of the formulae for magnetic moment and volume, or

The conversion factor is therefore

7. Magnetic Permeability,* or Specific Magnetic Inductive Capacity. — This is the analogue in magnetism to specific inductive capacity in electricity. It is the ratio of the magnetic induction in the substance to the magnetic induc- tion in the field which produces the magnetization, and therefore its dimensional formula and conversion factor are unity.

8. Magnetic Susceptibility. — This is the ratio of the numbers which repre- sent the values of the intensity of magnetization produced and the intensity of the magnetic field producing it. The dimensional formula is therefore the ratio of the formulae for intensity of magnetization and magnetic field or

* *

The conversion factor is therefore /, and both the dimensional formula and con- version factor are unity in the ordinary system.

9. Current Strength. — A current of strength c flowing round a circle of radius r produces a magnetic field at the centre of intensity 2Trcjr. The dimen- sional formula is therefore the product of the formulae for magnetic field intensity and length, or M^T"1?"*, which gives the conversion factor

10. Current Density, or Strength of Current at a Point. — This is the ratio of the numbers for current strength and area. The dimensional formula and the conversion factor are therefore M^L^T-1?-1 and

ii. Quantity of Electricity. — This is the product of the numbers for cur- rent and time. The dimensional formula is therefore WL*T~lp-* X T= MJL*P~*, and the conversion factor

12. Electric Potential, or Electromotive Force. — As in the electrostatic system, this is the ratio of the numbers for work and quantity of electricity. The dimensional formula is therefore

and the conversion factor

* Permeability, as ordinarily taken with the standard medium as unity, has the same dimension formula and conversion factor as that which is here taken as magnetic inductive capacity. Hence for ordinary transformations the conversion factor should be taken as I in the electromagnetic and j~2t2 in the electrostatic systems.

INTRODUCTION. XXXI

13. Electrostatic Capacity. — This is the ratio of the numbers for quantity of electricity and difference of potential. The dimensional formula is therefore

and the conversion factor

14. Resistance of a Conductor. — The resistance of a conductor or elec- trode is the ratio of the numbers for difference of potential between its ends and the constant current it is capable of producing. The dimensional formula is therefore the ratio of those for potential and current or

The conversion factor thus becomes #-1/, and in the ordinary system resistance has the same conversion factor as velocity.

15. Conductance. — This is the reciprocal of resistance, and hence the dimen- sional formula and conversion factor are respectively Lr^TP"1 and

16. Conductivity, or Specific Conductance. — This is quantity of electric- ity transmitted per unit of area per unit of potential gradient per unit of time. The dimensional formula is therefore derived from those of the quantities men- tioned as follows : —

The conversion factor is therefore

17. Specific Resistance. — This is the reciprocal of conductivity as defined in 1 6, and hence the dimensional formula and conversion factor are respectively and

18. Coefficient of Self-induction, or Inductance, or Electro-kinetic In- ertia. — These are for any circuit the electromotive force produced in it by unit rate of variation of the current through it. The dimensional formula is therefore the product of the formulae for electromotive force and time divided by that for current or

•» rl-r arn_o-r»l

X T = LP.

The conversion factor is therefore lp, and in the ordinary system is the same as that for length.

19. Coefficient of Mutual Induction. — The mutual induction of two cir- cuits is the electromotive force produced in one per unit rate of variation of the current in the other. The dimensional formula and the conversion factor are therefore the same as those for self-induction.

XXX11 INTRODUCTION.

20. Electro-kinetic Momentum. — The number for this is the product of the numbers for current and for electro-kinetic inertia. The dimensional formula is therefore the product of the formulae for these quantities, or M^T"1?"* X LP = M*UT-1P*, and the conversion factor is

21. Electromotive Force at a Point. — The number for this quantity is the ratio of the numbers for electric potential or electromotive force as given in 12, and for length. The dimensional formula is therefore MiLiT~2PJ, and the conversion factor

22. Vector Potential. — This is time integral of electromotive force at a point, or the electro-kinetic momentum at a point. The dimensional formula may therefore be derived from 21 by multiplying by T, or from 20 by dividing by L. It is therefore M*!,*!""1?*, and the conversion factor

23. Thermoelectric Height. — This is measured by the ratio of the num- bers for electromotive force and for temperature. The dimensional formula is therefore the ratio of the formulae for these two quantities, or MiLiT~2Pi®~1, and the conversion factor

24. Specific Heat of Electricity. — This quantity is measured in the same way as 23, and hence has the same formulas.

25. Coefficient of Peltier Effect. — This is measured by the ratio of the numbers for quantity of heat and for quantity of electricity. The dimensional formula is therefore

and the conversion factor

EXAMPLES OF CONVERSION IN ELECTROMAGNETIC UNITS.

(a) Find the factor required to convert intensity of magnetic field from foot grain minute units to c. g. s. units.

By (3) the formula is w*/"*/"1/"*, and in this case m = 0.0648, /= 30.48, / = 60, and/ = i ; .*. the factors = 0.0648* X 30.48"* X 6o~1 = 0.00076847.

Similarly to convert from foot grain second units to c. g. s. units the factor is 0.0648* X 30.48"* = 0.046 1 08.

(£) How many c. g. s. units of magnetic moment make one foot grain second unit of the same quantity ?

By (5) the formula is #z*/*/~~^*, and the values for this problem are m = 0.0648, /= 30.48, t= i, and/ = i ; .'. the number = 0.0648* X 30.48*= 1305.6.

(c) If the intensity of magnetization of a steel bar be 700 in c. g. s. units, what will it be in millimetre milligramme second units ?

INTRODUCTION. XXX111

By (6) the formula is wW"1/*, and in this case m = 1000, /= 10, /== i, and p = i j /.the intensity = 700 X 1000* X 10* = 70000.

(d) Find the factor required to convert current strength from c. g. s. units to earth quadrant io~u gramme and second units.

By (9) the formula is mll}rlp~*, and the values of these quantities are here m = lo11, /= io~9, / = i, and/ = i ; /. the factor = ioH x io~J = 10.

(e) Find the factor required to convert resistance expressed in c. g. s. units into the same expressed in earth-quadrant io~u grammes and second units.

By (14) the formula is #~^, and for this case /= io~', /= i, and / = i ; /. the factor = io~9.

(/) Find the factor required to convert electromotive force from earth-quadrant io~n gramme and second units to c. g. s. units.

By (12) the formula is f^*/8/"^*, and for this case m = io~u, /== io9, /= i, and/ = i ; .*. the factor = io8.

PRACTICAL UNITS.

In practical electrical measurements the units adopted are either multiples or submultiples of the units founded on the centimetre, the gramme, and the second as fundamental units, and air is taken as the standard medium, for which K and P are assumed.unity. The following, quoted from the report to the Honorable the Secretary of State, under date of November 6th, 1893, by the delegates repre- senting the United States, gives the ordinary units with their names and values as defined by the International Congress at Chicago in 1893 : —

" Resolved, That the several governments represented by the delegates of this International Congress of Electricians be, and they are hereby, recommended to formally adopt as legal units of electrical measure the following : As a unit of re- sistance, the international ohm, which is based upon the ohm equal to io9 units of resistance of the C. G. S. system of electro-magnetic units, and is represented by the resistance offered to an unvarying electric current by a column of mercury at the temperature of melting ice 14.4521 grammes in mass, of a constant cross- sectional area and of the length of 106.3 centimetres.

" As a unit of current, the international ampere, which is one tenth of the unit of current of the C. G. S. system of electro-magnetic units, and which is represented sufficiently well for practical use by the unvarying current which, when passed through a solution of nitrate of silver in water, and in accordance with accom- panying specifications,* deposits silver at the rate of 0.001118 of a gramme per second.

* " In the following specification the term ' silver voltameter ' means the arrangement of appara- tus by means of which an electric current is passed through a solution of nitrate of silver in water. The silver voltameter measures the total electrical quantity which has passed during the time of the experiment, and by noting this time the time average of the current, or, if the current has been kept constant, the current itself can be deduced.

" In employing the silver voltameter to measure currents of about one ampere, the following arrangements should be adopted : —

XXXIV INTRODUCTION.

" As a unit of electromotive force, the international volt, which is the electro- motive force that, steadily applied to a conductor whose resistance is one interna- tional ohm, will produce a current of one international ampere, and which is rep- resented sufficiently well for practical use by T$§£ of the electromotive force between the poles or electrodes of the voltaic cell known as Clark's cell, at a tem- perature of 15° C., and prepared in the manner described in the accompanying specification.*

" As a unit of quantity, the international coulomb, which is the quantity of elec- tricity transferred by a current of one international ampere in one second.

"As a unit of capacity, the international farad, which is the capacity of a con- denser charged to a potential of one international volt by one international cou- lomb of electricity. t

" As a unit of work, the joule, which is equal to io7 units of work in the c. g. s. system, and which is represented sufficiently well for practical use by the energy expended in one second by an international ampere in an international ohm.

"As a unit of power, the watt, which is equal to io7 units of power in the c. g. s. system, and which is represented sufficiently well for practical use by the work done at the rate of one joule per second.

" As the unit of induction, the henry, which is the induction in a circuit when the electromotive force induced in this circuit is one international volt, while the inducing current varies at the rate of one ampere per second.

" The Chamber also voted that it was not wise to adopt or recommend a stand- ard of light at the present time."

By an Act of Congress approved July i2th, 1894, the units recommended by the Chicago Congress were adopted in this country with only some unimportant verbal changes in the definitions.

By an Order in Council of date August 23d, 1894, the British Board of Trade adopted the ohm, the ampere, and the volt, substantially as recommended by the Chicago Congress. The other units were not legalized in Great Britain. They are, however, in general use in that country and all over the world.

" The kathode on which the silver is to be deposited should take the form of a platinum bowl not less than io centimetres in diameter and from 4 to 5 centimetres in depth.

" The anode should be a plate of pure silver some 30 square centimetres in area and 2 or 3 millimetres in thickness.

" This is supported horizontally in the liquid near the top of the solution by a platinum wire passed through holes in the plate at opposite corners. To prevent the disintegrated silver which is formed on the anode from falling on to the kathode, the anode should be wrapped round with pure filter paper, secured at the back with sealing wax.

"The liquid should consist of a neutral solution of pure silver nitrate, containing about 15 parts by weight of the nitrate to 85 parts of water.

" The resistance of the voltameter changes somewhat as the current passes. To prevent these changes having too great an effect on the current, some resistance besides that of the voltameter should be inserted in the circuit. The total metallic resistance of the circuit should not be less than io ohms."

* A committee, consisting of Messrs. Helmholtz, Ayrton, and Carhart, was appointed to pre- pare specifications for the Clark's cell, but no report was made, on account of Helmholtz's death.

t The one millionth part of the farad is more commonly used in practical measurements, and is called the microfarad.

PHYSICAL TABLES

T ABLE 1 .

FUNDAMENTAL AND DERIVED UNITS,

To change a quantity from one system of units to another : substitute in the correspond- ing conversion factor from the following table the ratio of the magnitudes of the old units to the new and multiply the old quantity by the resulting number. For example : to reduce velocity in miles per hour to feet per second, the conversion factor is //—1; /=528o/i, /=36oo/i, therefore the factor=528o/36oo=i.467.

(a) FUNDAMENTAL UNITS.

Name of Unit.

Symbol.

Conversion Factor.

Length.

Mass.

Time.

Temperature.

Electric Inductive Capacity.

Magnetic Inductive Capacity.

M T © K P

(£) DERIVED UNITS. I. Geometric and Dynamic Units.

Name of Unit.

Conversion Factor.

Area. Volume. Angle. Solid Angle. Curvature. Tortuosity.

Specific curvature of a surface. Angular velocity. Angular acceleration. Linear velocity. Linear acceleration. Density.

Moment of inertia.

Intensity of attraction, or " force at a point." Absolute force of a centre of attraction, or " strength ") of a centre." )

Momentum.

Moment of momentum, or angular momentum. Force.

Moment of a couple, or torque. Intensity of stress. Modulus of elasticity. Work and energy. Resilience. Power or activity.

//~2

w/2 //~2

mtr1 mtr*

m /-1 /-'

SMITHSONIAN TABLES.

TABLE 1 . FUNDAMENTAL AND DERIVED UNITS.

//. Heat Units.

Name of Unit.

Conversion Factor.

Quantity of heat (thermal units).

" (thermometric units). " " (dynamical units).

Coefficient of thermal expansion. Conductivity (thermal units).

f thermometric units), or diffusivity. " (dynamical units).

Thermal capacity. Latent heat (thermal units).

" " (dynamical units). Joule's equivalent.

Entropy (heat measured in thermal units). " ( " " " dynamical units).

mO 1*0

III. Magnetic and Electric Units.

Name of Unit.

Conversion factor for electrostatic system.

Conversion factor for electromag- netic system.

Magnetic pole, or quantity of mag- netism.

Density of surface distribution of magnetism.

Intensity of magnetic field.

Magnetic potential.

Magnetic moment.

Intensity of magnetisation.

Magnetic permeability.

Magnetic susceptibility and mag-) netic inductive capacity. j

Quantity of electricity.

Electric surface density and electric )

/> r1/1

«*/'

m*ll

displacement. Intensity of electric field. Electric potential and e. m. f. Capacity of a condenser. Inductive capacity. Specific inductive capacity. Electric current.

m*l*

t-^k-*

r*#

nj> /-I/ *»/*

SMITHSONIAN TABLES.

TABLE 1. FUNDAMENTAL AND DERIVED UNITS.

///. Magnetic and Electric Units.

Conversion factor

Name of Unit.

for electrostatic

for electromag-

system.

netic system.

Conductivity. Specific resistance.

jj*

Conductance.

1 1~^ k

f~i t p~i

Resistance.

t*tK+

i t~i p

Coefficient of self induction and)

^ . 2 t-i

7 -A

coefficient of mutual induction, j

rrr k

Electrokinetic momentum.

m\ l\ £-*

m* I* rlp*

Electromotive force at a point.

m\ /-* /-I £-i

m* /l /~2/*

Vector potential.

fffi /~i k~*

m* /* r"1/*

Thermoelectric height and specific) heat of electricity. j"

*flr*ir***

*>> /' rv» «-*

Coefficient of Peltier effect.

m* f* t IT* 6

SMITHSONIAN TABLES.

TABLE 2. TABLES FOR CONVERTING U. S. WEIGHTS AND MEASURES.*

(1) CUSTOMARY TO METRIC.

, LINEAR.

CAPACITY.

Inches to millimetres.

Feet to

metres.

Yards to metres.

Miles to kilometres.

Fluid drams to millilitres or cubic

Fluid ounces to

Liquid quarts to litres.

Gallons to litres.

centimetres.

•ft

25.4001

0.304801

0.914402

1.60935

3-70

29-57

0.94636

378543

50.8001

0.60960!

1.828804

3.21869

7-39

59- * 5

1.89272

7.57087

76.2002

0.914402

2.743205

4.82804

11.09

88.72

2.83908

11.35630

IOI.6002

1.219202

3.657607

6-43739

14.79

118.29

378543

15.14174

127.0003

1.524003

4.572009

8.04674

18.48

147.87

18.92717

152.4003 177.8004 203.2004 228.6005

1.828804 2.133604 2.438405 2.743205

5.48641 1 6.400813

7-3I52I5 8.229616

9.65608 11.26543 12.87478 14.48412

22.18 25.88 29.57 33.27

177-44 207.02

236.59 266.16

5.67815 6.62451 7.57087 8.51723

22.71261 26.49804 30-28348 34.06891

SQUARE.

WEIGHT.

Square inches to square cen- timetres.

Square feet to square decimetres.

Square yards to square metres.

Acres to hectares.

Grains to

milli- grammes.

Avoirdu- pois ounces to grammes.

Avoirdu- pois pounds to kilo- grammes.

Troy ounces to grammes.

6.452 12.903

ifc??

0.836 1.672

0.4047 0.8094

64.7989 129.5978

28.3495 $6.6991

0-45359 0.90718

• 31.10348 62.20696

J9-355 25.807

32.258

27.871 37.161 46.452

2.508

3-345 4.181

1.2141 1.6187 2.0234

194.3968

2 59- ! 957

323.9946

85.0486 113.3981 141.7476

1.36078

I.8I437 2.26796

93-3 ! 044 124.41392

38.710 45.161

55-742 65.032

5-o 1 7 5-853

2.4281 2.8328

388.7935 453-5924

170.0972 198.4467

2.72I55

186.62088 217.72437

51.613 58.065

74-323 83-613

6.689

7.525

3-2375 3.6422

8 9

583-1903

226.7962 255-H57

3-62874 4.08233

248.82785

279.93 * 33

CUBIC.

Cubic inches to cubic cen- timetres.

Cubic feet to cubic metres.

Cubic yards to cubic metres.

Bushels to hectolitres.

i Gunter's chain = 20.1168 metres, i sq. statute mile = 259.000 hectares.

i fathom = 1.829 metres.

16.387

0.02832

0.765

0.35239

i nautical mile = 1853.25 metres.

32.774

0.05663

1-529

0.70479

i foot = 0.304801 metre.

49.161 65.549

0.08495 0.11327

2.294 3.058

1.05718 1.40957

i avoir, pound = 453.5924277 grammes.

81.936

0.14159

3.823

1.76196

1 5432.35639 grains = i.ooo kilogramme.

98.323

0.16990

4.587

2.11436

II47IO

0.19822

5-352

2.46675

131.097

0.22654

6.II6

2.81914

147.484

0.25485

6.881

3.I7I54

According to an executive order dated April 15, 1893, the United States yard is defined as 3600/3937 metre, and the avoirdupois pound as 1/2.20462 kilogramme.

The only authorized material standard of customary weight is the Troy pound of the Mint. It is of brass of un- known density, and therefore not suitable for a standard of mass. It was derived from the British standard Troy pound of 1758 by direct comparison.

The British gallon = 4.5459631 litres.

The British bushel = 36.3477 litres.

The length of the nautical mile given above and adopted by the U. S. Coast and Geodetic Survey many years ago, is defined as that of a minute of arc of a great circle of a sphere whose surface equals that of the earth (Clarke's Sphe- roid of 1866).

* Quoted from sheets issued by the United States Bureau of Standards. SMITHSONIAN TABLES.

TABLE 2. TABLES FOR CONVERTING U. S. WEIGHTS AND MEASURES.

(2) METRIC TO CUSTOMARY.

LINEAR.

CAPACITY.

Millilitres

or cubic

Centi-

Deca

Hecto-

Metres to

Kilometres

centi-

litres to

litres

inches.

feet.

yards.

to miles.

metres

fluid

to fluid

ounces.

gallons.

bushels.

drams.

I 2

39-3700 78.7400

6.56167

1.093611 2.187222

0.62137 1.24274

0.27

0.676

1.0567 2.1134

2.6417

2.8377 5-6755

3 4 5

118.1100 157.4800 196.8500

9.84250

I3-I2333 16.40417

3-280833 4.374444 5.468056

1.86411 2.48548 3-I0685

3 4

?!o8

1.014

1-353 1.691

3.1700 4.2267

7-9251 10.5668 13.2085

8.5132 H.35'0

14.1887

236.2200 275.5900 314.9600

19.68500 22.96583 26.24667

6.561667 7-655278 8.748889

3.72822

4-34959 4.97096

6 8

1.62 1.89 2.l6

2.029 2.367 2.705

6.3401 7-3968

8-4535

1 5.8502 18.4919 21.1336

17.0265 19.8642 22.7019

354-3300

29.52750

9.842500

5-59233

2-43

3-043

9.5101

23-7753

25'5397

SQUARE.

WEIGHT.

Square

Milli-

Kilo-

Hecto-

Kilo-

centimetres

metres to

Hectares

grammes

gra

mmes

grammes

to square

square

to acres.

to c

unces

•

o pounds

inches.

feet.

yards.

grains.

avoirdupois.

0.1550

10.764

1.196

2.471

0.01543

15432-36

3-5274

2.20462 '

0.3100

21.528

2.392

4.942

0.03086

30864.71

7.0548

4.40924

3 4

0.4650 0.6200

32.292 43-055

3.588 4.784

7-4I3 9.884

0.04630 0.06173

46297.07 61729.43

10 14

5822 1096

6.61 387 8.81849

0.7750

53-8I9

5-980

I2-355

0.07716

77161.78

17.6370

11.02311

6 9

0.9300 1.0850 1.2400

I-395°

64.583 75-347 86.1 1 1 96.875

7.176 8.372 9.568 10.764

14.826 17.297 19.768 22.239

8 9

0.09259 0.10803 0.12346 0.13889

92594.14 108026.49 123458.85 138891.21

21.1644 24.6918 28.2192 31.7466

13.22773 I5-43236 17.63698 19.84160

CUBIC.

WEIGHT.

Cubic centimetres to cubic

Cubic decimetres to cubic

Cubic metres to cubic

Quintals to pounds av.

Milliers or tonnes to pounds

Kilogrammes to ounces

inches.

feet.

yards.

0.06 10

61.023

35-3H

1-308

220.46

2204.6

32.1507

0.1220

122.047

70.629

2.616

440.92

4409.2

64.3015

0.1831

183.070

105.943

3.924

661.39

661-

•9

96.4522

0.2441

244.094

141.258

5-232

881.85

881*

•5

128.6030

0.3051

3°5-II7

176.572

6.540

1102.31

11023.1

160.7537

0.3661

366.140

211.887

7.848

1322.77

13227.7

192.9045

0.4272

427.164

247.201

9.156

I543-24

15432.4

225.0552

0.4882

488.187

282.516

10.464

1763.70

17637.0

57.2059

0.5492

549.210

317.830

11.771

1984.16

19841

59.3567

By the concurrent action of the principal governments of the world an International Bureau of Weights and Measures has been established near Paris. Under the direction of the International Committee, two ingots were cast of pure platinum-iridium in the proportion of 9 parts of the former to i of the latter metal. From one of these a certain number of kilogrammes were prepared, from the other a definite number of metre bars. These standards of weight and length were intercompared, without preference, and certain ones were selected as Internationarproto- type standards. The others were distributed by lot, in September, 1889, to tne different governments, and are called National prototype standards. Those apportioned to the United States were received in 1890, and are kept at the Bureau of Standards in Washington, D. C.

The metric system was legalized in the United States in 1866.

The International Standard Metre is derived from the Metre des Archives, and its length is defined by the distance between two lines at o° Centigrade, on a platinum-iridium bar deposited at the International Bureau of Weights and Measures.

The International Standard Kilogramme is a mass of platinum-iridium deposited at the same place, and its weight in vacuo is the same as that of the Kilogramme des Archives.

The litre is equal to a cubic decimetre, and it is measured by the quantity of distilled water which, at its maxi- mum density, will counterpoise the standard kilogramme in a vacuum, the volume of such a quantity of water being, as nearly as has been ascertained, equal to a cubic decimetre.

SMITHSONIAN TABLES.

TABLE 3.

EQUIVALENTS OF METRIC AND BRITISH IMPERIAL WEIGHTS AND MEASURES.*

(1) METRIC TO IMPERIAL.

LINEAR MEASURE.

MEASURE OF CAPACITY.

zmim.ne.re (mm.) |

= 0.03937 in.

, rnimmre (ml., (.00, | = ^

I centimetre (.01 m.) i decimetre (.1 m.)

= 0.39370 " = 3-93701 (39.370113 "

i centilitre (.01 litre) = j °*oi£24in" i decilitre (.1 litre) . . = 0.176 pint.

I METRE (m.) . . .

= \ 3.280843 ^

i LITRE (1,000 cub. )

I dekametre

( i. 09361 425 yds.

centimetres or i j- = 1.75980 pints, cub. decimetre) )

(10 m.) i " * *

— J O-936 1 4

i dekalitre (10 litres) . = 2.200 gallons.

I hectometre

i hectolitre (ioo " ) . = 2.75 bushels.

I09'36l42S

i kilolitre (1,000 " ) . = 3.437 quarters.

I myriametre ) ( 1 0,000 m.) j * '

= 6.21372 miles.

APOTHECARIES' MEASURE.

= o.ooi mm.

i cubic centi- ) ( 0.03520 fluid ounce, metre (i > = } 0.28157 fluid drachm, gramme w't) ) ( 15.43236 grains weight, i cub. millimetre = 0.01693 minim.

SQUARE MEASURE.

AVOIRDUPOIS WEIGHT.

I sq. centimetre . . I sq. decimetre ) (ioo sq. centm.) f I sq. metre or centi- j are (loosq. dcm.) j i ARE (ioo sq. m.) i hectare (ioo ares or 10,000 sq. m.)

= 0.1550 sq. in.

_ i 10.7639 sq. ft. ( 1.1960 sq. yds. = 119.60 sq. yds.

= 2.4711 acres.

i milligramme (mgr.) . . = o.oi 543 grain, i centigramme (.01 gram.) = 0.15432 " i decigramme (.1 " ) = 1.54324 grains.

i dekagramme (10 gram.) = 5.64383 drams, i hectogramme (ioo " ) = 3.52739 oz. {2.2046223 Ibs. 15432.3564

grains.

I myriagramme (iokilog.)= 22.04622 Ibs.

i quintal (ioo " )= 1.96841 cwt.

CUBIC MEASURE.

i millier or tonne | ~nQM~ (1,000 kilog.) } • •- 0-9842 ton.

I cub. centimetre

(c.c.) (1,000 cubic

= 0.0610 cub. in.

TROY WEIGHT.

millimetres)

I cub. decimetre

( 0.03215 oz. Troy.

(c.d.) (1,000 cubic

= 61.024 " "

i GRAMME . . = 1 0.64301 pennyweight.

centimetres)

( 15.43236 grains.

1 CorB"stefeTRE I .

_ J 35-3 ^8 cub. ft. i i. 307954 cub. yds.

APOTHECARIES' WEIGHT.

( 0.25721 drachm.

I GRAMME . . . . = < 0.77162 scruple.

( 1 5.43236 grains.

NOTE. — The METRE is the length, at the temperature of o° C., of the platinum-iridium bar deposited at the International Bureau of Weights and Measures at Sevres, near Paris, France.

The present legal equivalent of the metre is 39.370113 inches, as above stated.

The KILOGRAMME is the mass of a platinum-iridium weight deposited at the same place.

The LITRE contains one kilogramme weight of distilled water at its maximum density (4° C.), the barometer being at 760 millimetres.

*In accordance with the schedule adopted under the Weights and Measures (metric system) Act, 1897. SMITHSONIAN TABLES.

8 TABLES.

EQUIVALENTS OF METRIC AND BRITISH IMPERIAL WEIGHTS AND MEASURES.

(2) METRIC TO IMPERIAL

LINEAR MEASURE.

MEASURE OF CAPACITY.

3 4

Millimetres to inches.

Metres to feet.

Metres to yards.

Kilo- metres to miles.

Litres to pints.

Dekalitres to gallons.

Hectolitres to bushels.

Kilolitres to quarters.

0-039370II 0.07874023 0.11811034 0.15748045 0.19685056

3.28084 6.56169 9.84253 I3-I2337 16.40421

1.09361

2.18723 3.28084 4-37446 546807

0.62137 1.24274 1.86412 2.48549 3.10686

4 5

1.75980 3.51961 5.27941 7.03921 8.79902

2.19975

4-39951 6.59926 8.79902 10.99877

2.74969 5-49938 8.24908 10.99877 13.74846

343712 6.87423 10.31135 13.74846 17.18558

0.23622068 0.27559079 0.31496090 0.35433102

19.68506 22.96590 26.24674 29.52758

6.56169 7.65530 8.74891

9-84253

3.72823 4.34960 4.97097 5-59235

8 9

10.55882 12.31862 14.07842 I5-83823

13.19852 15.39828 17.59803 19.79778

16.49815 19.24785 21.99754 2474723

20.62269 24.05981 27.49692 30.93404

SQUARE MEASURE.

WEIGHT (AVOIRDUPOIS).

I 2

4 5

Square centimetres to square inches.

Square metres to

Square metres to square yards.

Hectares to acres.

I 2

3 4 5

Milli- grammes to grains.

Kilogrammes to grains.

Kilo- grammes to pounds,

Quintals to hundred- weights.

0.15500

0.31000 0.46500 0.62000 0.77500

10.76393 21.52786 32.29179 43.05572 53-8I965

I.I9599 2.39198 3.58798 4-78397 5.97996

2.4711 4.9421

74132 9.8842

12.3553

0.01543 0.03086 0.04630 0.06173 0.07716

15432.356 30864.713 46297.069 61729.426 77161.782

2.20462 4.40924 6.61387 8.81849 II.023II

1.96841 3.93683 5-90524 7.87365 9.84206

0.93000 1.08500 I-24OOO L3950I

64.58357 75-34750 86.11143 96.87536

7-17595 8.37194

9.56794 10.76393

14.8263 17.2974 19.7685 22.2395

0.09259 0.10803 0.12346 0.13889

92594.138 108026.495 123458.851 138891.208

13.22773 1543236 17.63698 19.84160

11.81048 13.77889

1574730 17.71572

CUBIC MEASURE.

APOTHE- CARIES' MEASURE.

AVOIRDUPOIS (cont.)

TROY WEIGHT.

APOTHE- CARIES' WEIGHT.

Cubic

decimetres to cubic inches.

Cubic metres to cubic feet.

Cubic metres to cubic yards.

Cub. cen- timetres to fluid drachms.

Milliers or tonnes to tons.

Grammes to ounces Troy.

, Grammes to penny- weights.

Grammes to scruples.

I 2

3 4

61.02390 122.04781 183.07171 244.09561 305.H9S2

35-3I476 70.62952 105.94428 141.25904 176.57379

1.30795 2.61591 3.92386 5.23182

6-53977

0.28157 0.56314 0.84471 1.12627 1.40784

I 2

4 5

0.98421 1.96841 2.95262

3.93683 4.92103

0.03215 0.06430 0.09645 0.12860 0.16075

0.64301 1.28603 1.92904 2.57206 3.21507

0.77162 1.54324 2.31485 3.08647 3.85809

366.14342 427.16732 488.19123 549-2I5I3

211.88855 247.20331 282.51807 317.83283

7.84772 9.15568 10.46363 11.77159

1.68941 1.97098 2-25255 2.53412

8 9

5-90524 6.88944

7.87365 8.85786

0.19290 0.22506 0.25721 0.28936

3.85809 4.50110 5.14412 578713

4.62971 5.40132 6.17294 6.94456

SMITHSONIAN TABLES.

TABLE 3.

EQUIVALENTS OF BRITISH IMPERIAL AND METRIC WEIGHTS AND MEASURES.

(3) IMPERIAL TO METRIC.

LINEAR MEASURE.

MEASURE OF CAPACITY.

f 25.400 milli-

i gill . . . — 1.42 decilitres.

i inch = \ metres, i foot (12 in.) . .= 0.30480 metre.

i pint (4 gills) . . . =0.568 litre, i quart (2 pints) . . = 1.136 litres.

i YARD (3 ft.) . . = 0.914399 i pole (si yd.) . .= 5.0292 metres.

i GALLON (4 quarts) =4.5459631 " i peck (2 galls.) . . = 9.092

i chain (22 yd. or) _ gg „ 100 links) ) i furlong (220 yd.) = 201.168 "

i bushel (8 galls.) . = 3.637 dekalitres, i quarter (8 bushels) = 2.909 hectolitres.

AVOIRDUPOIS WEIGHT.

SQUARE MEASURE.

(64.8 milli-

i Grain • • — <

(6.4516 sq. cen-

i square inch . = \ timetres.

dram — T-772 grammes.

f 9.2903 sq. deci-

ounce (16 dr.) . .= 28.350 "

i sq.ft. (144 sq. in.) = ) metres, f 0.836126 sq. i SQ. YARD (9 sq. ft.) = \ metres.

P°7"olo grlins)°r \ = °-45359243 kilogr. stone (14 lb.) . .= 6.350 "

f 21? 201 sa me-

quarter (28 lb.) .= 12.70 "

i rood (40 perches) = 10.117 ares.

hundredweight 1 j 50.80 " (ii2lb.) J I 0.5080 quintal.

i ACRE (4840 sq. yd.) = 0.40468 hectare.

{i. 0160 tonnes or

i sq. mile (640 acres) = J259.oo hectares.

1016 kilo- grammes.

TROY WEIGHT.

CUBIC MEASURE.

i cub. inch— 16.387 cub. centimetres, i cub. foot (1728 I (0.028317 cub me- cub.in.) ' ]— \ tre, or 28.317 I cub. decimetres.

i Troy OUNCE (480 ) s==3I.Io35 grammes, grains avoir.) ) i pennyweight (24 1 _ _ ,« grains) f

i CUB. YARD (27 f __ 0.76455 cub. metre.

NOTE. — The Troy grain is of the same weight as the Avoirdupois grain.

APOTHECARIES' MEASURE.

•

APOTHECARIES' WEIGHT.

i gallon (8 pints or ) 4-5459631 litres. 1 60 fluid ounces) J I fluid ounce, f 3 ) (28.4123 cubic (8 drachms) f } centimetres. I fluid drachm, f 3 I __ f 3-55 1 5 cubic (60 minims) f ~= \ centimetres, i minim, n\ (0.91146 ) ( 0.05919 cubic grain weight) ) " \ centimetres.

i ounce (8 drachms) = 31.1035 grammes, i drachm, 31 (3 scru- ) ggg <« pies) f.. -••* i scruple, £i (20 } g grains) f

NOTE. — The Apothecaries' ounce is of the same weight as the Troy ounce. The Apothecaries'

NOTE. — The Apothecaries' gallon is of the same

grain is also of the same weight as the Avoir dupois

capacity as the Imperial gallon.

grain.

NOTE. —The YARD is the length at 62° Fahr., marked on a bronze bar deposited with the Board of Trade.

The POUND is the weight of a piece of platinum weighed in vacuo at the temperature of o° C., and which is also deposited with the Board of Trade.

The GALLON contains 10 lb. weight of distilled water at the temperature of 62° Fahr., the barometer being at 30 inches.

SMITHSONIAN TABLES.

IO TABLE 3.

EQUIVALENTS OF BRITISH IMPERIAL AND METRIC WEIGHTS AND MEASURES.

(4) IMPERIAL TO METRIC.

LINEAR MEASURE.

MEASURE OF CAPACITY.

Inches to centimetres.

Feet to metres.

Yards to metres.

Miles to kilo- metres.

Quarts to litres.

Gallons to litres.

Bushels to dekalitres.

Quarters to hectolitres.

2-539998 5.079996 7.619993 10.159991 12.699989

0.30480 0.60960 0.91440 1.21920 1.52400

0.91440 1.82880 2.74320 3.65760 4.57200

1.60934 3.21869 4.82803

6-43737 8.04671

4 5

1.13649 2.27298 340947 4.54596 5.68245

4.54596 9.09193

I3-63789 18.18385 22.72982

3-63677 7-27354 10.91031 14.54708 18.18385

2.90942 5.81883 8.72825 11.63767 14.54708

1 5-239987 17.779984 20.319982 22.859980

1.82880 2.13360 2.43840 2.74320

5.48640 6.40080

7.3I5I9

8.22959

9.65606 11.26540 12.87474 14.48408

6 9

6.81894

7-95544 9.09193 10.22842

27.27578 31.82174 36.36770 40.91367

21.82062

25-45739 29.09416

32.73093

17.45650 20.36591

23-27533 26.18475

SQUARE MEASURE.

WEIGHT (AVOIRDUPOIS).

Square inches to square centimetres.

Square feet to square decimetres.

Square yards to square metres.

Acres to hectares.

Grains to milli- grammes.

Ounces to grammes.

Pounds to kilo- grammes.

Hundred- weights to 'quintals.

4 5

6.45*59

12.90318

19.35477 25.80636

32'25794

9.29029 18.58058 27.87086 37.16115 46.45144

0.83613 1.67225 2.50838

3-3445°

4.18063

0.40468 0.80937 1.21405 1.61874 2.02342

64.79892 129.59784

I94-39675 259- i 9567 323-99459

28.34953 56.69905 85.04858 113.39811 141.74763

0-45359 0.90718 1.36078 1.81437 2.26796

0.50802 1.01605 1.52407 2.03209 2.54012

6 9

38.70953 45.16112 51.61271 58.06430

55-74I73 65.03201 74.32230 83.61259

5.01676

5.85288 6.68901 7.52513

2.42811 2.83279 3.23748 3.64216

6 9

388.79351 453-59243 5*8.39135 583.19026

170.09716 198.44669 226.79621 255.14574

2.72155

3.I75I5 3.62874 4.08233

3.04814 3-556l6 4.06419 4.57221

CUBIC MEASURE.

APOTHE- CARIES' MEASURE.

AVOIRDUPOIS (font.-).

TROY WEIGHT.

APOTHE- CARIES' WEIGHT.

Cubic inches to cubic centimetres.

Cubic feet to cubic metres.

Cubic yards to cubic metres

Fluid drachms to cubic centi- metres.

Tons to milliers or tonnes.

Ounces to grammes.

Penny- weights to grammes.

Scruples to grammes.

3 4 5

16.38702 32.77404 49.16106 65.54808 81.93511

0.02832 0.05663 0.08495 0.11327 0.14158

0.76455 I.529II 2.29366 3.05821 3.82276

3-55I53 7.10307 10.65460 14.20613 17.75767

3 4 5

1.01605 2.03209 3.04814 4.06419 5.08024

31.10348 62.20696

93-3 * 044 124.41392

I55.5I740

I.555I7 3-II035 4.66552 6.22070 7.77587

1.29598 2.59196 3.88794

5-l839i 6.47989

98.32213 114.70915 131.09617 147.48319

0.16990 0.19822 0.22653

0.25485

4-58732 5.35187 6.11642 6.88098

21.30920 24.86074 28.41227 31.96380

6.09628

7-iJ233 8.12838 9.14442

186.62088

217-72437 248.82785

279-93I33

9.33104 10.88622 12.44139 I3-99657

7.77587 9.07185 10.36783 11.66381

SMITHSONIAN TABLES.

TABLE 4.

VOLUME OF A CLASS VESSEL FROM THE WEIGHT OF ITS EQUIVALENT VOLUME OF MERCURY OR WATER.

If a glass vessel contains at f> C, P grammes of mercury, weighted with brass weights in air at 760 mm. pressure, then its volume in c. cm.

at the same temperature, t, \ V-= PR = P^f

at another temperature, /i, : V = PR\ = Ppjd \ I + y (t\ — t) \

p = the weight, reduced to vacuum, of the mass of mercury or water which, weighed with brass weights, equals i gramme ;

d ' = the density of mercury or water at /°C,

and 7 = o.ooo 025, is the cubical expansion coefficient of glass.

Temper- ature t

WATER.

MERCURY.

Rlt ti — 10°.

J?lf /j = 20°.

Rit /j = 10°.

/?!, /! = 20°.

0°

I.OOII92

1.001443

I.OOI693

0.0735499

0.0735683

0.0735867

1133

1358

l6O9

5633

5798

5982

IO92

1292

1542

5766

59H

6898

3 4 5

1068 I060 1068

1243

I2IO "93

H93 I46O

1443

5900

6029 6144 6259

6213 6328 6443

I.OOI092

I.OOII92

I.OOI442

0.0736301

0.0736374

0.0736558

II3I II84

I2O6 1234

1456

1485

6434 6568

6490 6605

6674

I252

1277

1527

6702

6720

6904

1333

1584

6835

7020

I.OOI428

I.OOI4O3

001653

0.0736969

0.0736951

0.0737135

1536

1486

1736

7103

7066

7250

1657

1582

I832

7236

7181

7365

1790

1690

1940

7370

7297

7481

J935

1810

2O6O

75°4

7412

750

1.002092

1.001942

I.002I93

0.0737637

0.0737527

0.07377II

2261

2086

2337

7771

7642

7826

18 19

2441 2633

2241 2407

2491 2658

7905 8039

7757 7872

7941 8057

2835

2584

2835

8172

7988

1.003048

1.002772

I.OO3O23

0.0738306

0.0738103

0.0738288

3271

2970

3220

8440

8218

8403

35°4

3178

3429

8573

8333

8518

3748

3396

3647

8707

8449

8633

4001

3624

3875

8841

8564

8748

1.004264

1.003862

I.OO4II3

0.0738974

0.0738679

0.0738864

4537

4110

4361

9108

8794

8979

4818

4366

4616

9242

8910

9094

5110

4632

4884

9376

9025

9210

54io

4908

S1S9

9510

9140

9325

Taken from Landolt, Bornstein, and Meyerhofifer's Physikalisch-Chemische Tabellen. SMITHSONIAN TABLES.

TABLE 5. DIFFERENTIAL COEFFICIENTS.

INTEGRALS.

DIFFERENTIAL

COEFFICIENTS.

INTEGRALS.

a* ex

loge*

sin.* cos. x tan. x cot.*

sec.* cosec. * sin.-1 * cos.—1 * tan.-1 * cot.-1 * sec.—1 * cosec.-1 * vers.-1 * covers.—1 x

=nx»~l

ax loge a e*

i *

COS.*

—sin. * sec.2 * —cosec.2 * sin. *

fx*dx

faxdx

fe*dx fdx J~x

fcos.ax-dx

/sin. ax • dx /sec.2 ax • dx

/cosec.2 ax - dx fs[n- x dx

_*n+i

W+I

loge <*

ex loge* sin. ax

—cos. ax

a tan. ax

cos.2 *

COS.*

~~ sin.2 *

VC-*2)

a —cot. ax

a sec. *

—cosec. *

sin-^ a

' -cos-** a

I tan-1 2 a a

i , , *

J cos.2 * fCOS' Xdx

1 ' • t ax J sm.2*

r dx

JV(a2-*?)

rdx

Ja*+x*

r dx

Vd-*") I

I+*2

*V/(*2-l)

cot. * a a

( i , *

*V(*2-i)

- sec. l - a a

I _, _ __ ! *

J x\/(x2 a2)

X/(2 *-*2)

r dx

cosec. l — a a

{vers.—1 * —covers.—1 *

V/(2 OC-X>)

J*S(2X-X*)

Taylor's series :

u=f(x+h)=f(x) +f'(x)h+f"(x) ^ +/'"(

The remainder after the first w terms is expressed by

•f0nfn+l(x+h-z)z*.dz.

I.2.3

Maclaurin's series :

u=f(x)=f(o)+f'(o)x+f"(o) ~

^=3.14159265359 i =0.3 1 83098861 8

^=9.86960440109 ^=2.71828182846

=0.497 1 4987 269

^=0.88622692546

loge 10=2.30258509299

log^(number) =loge (number) • logs e _ logff(number) logeJ5

SMITHSONIAN TABLES.

TABLE 6. 13

VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, OF NATURAL NUMBERS.

.000.1

lOOO.Jl

«*

tf*

100.000

100

IOOO

3.1623

15-3846

4225

274625

8.0623

90.9091

121

J331

3.3166

15.1515

4356

287496

8.1240

83-3333

144

1728

3.4641

14.9254

4489

300763

8.1854

76.9231

I69

2197

3.6056

68 •

14.7059

4624

3 * 4432

8.2462

71.4286

196

2744

3-7417

14.4928

4761

328509

8,3066

66.6667

225

3375

3-8730

14.2857

4900

343ooo

8.3666

62.5000

256

4096

4.0000

14.0845

5041

3579"

8.4261

58.8235

289

4.1231

13.8889

5^4

373248

8.4853

55-5556

324

§32

4.2426

13.6986

5329

389017

8.5440

52.6316

361

4.3589

13-5135

5476

405224

8.6023

50.0000

400

8000

4.4721

I3-3333

5625

421875

8.6603

47.6190

441

9261

4.5826

I3-I579

5776

438976

8.7178

45-4545

484

10648

4.6904

12.9870

5929

456533

8.7750

434783

529

12167

4.7958

12.8205

6084

474552

8.8318

41.6667

576

13824

4.8990

12.6582

6241

493039

8.8882

40.0000

625

15625

5.0000

12.5000

6400

512000

8.9443

38.4615 37-0370

676 729

17576 19683

5.0990 5.1962

81 82

12.3457 12.1951

6561 6724

53 '44i 551368

9.0000

9.0554

28 29

35-7I43 34.4828

784 84I

21952 24389

5-2915 5-3852

83 84

1 2.0482 11.9048

6889 7056

571787 592704

9.1104

9.1652

33-3333

900

27000

5-4772

11.7647

7225

614125

9.2195

32.2581

961

29791

5-5678

11.6279

7396

636056

9.2736

31.2500

1024

32768

5-6569

11.4943

7569

658503

9.3274

30-3030

I089

35937

5-7446

11.3636

7744

681472

9.3808

29.4118

1156

39304

5-8310

11.2360

7921

704969

9.4340

28.5714

1225

42875

5.9161

n. mi

8100

729000

9.4868

27.7778

1296

46656

6.0000

10.9890

8281

753571

9-5394

27.0270

1369

50653

6.0828

10.8696

8464

778688

9-59I7

38 39

26.3158 25.6410

1444 1521

54872 59319

6.1644 6.2450

93 94

10.7527 10.6383

8649 8836

8043^7 830584

9.6437 9.6954

25.0000 24.3902

I6OO

1681

64000 68921

6.3246 6.4031

10.5263 10.4167

9025 9216

857375 884736

9.7468 9.7980

23.8095

1764

74088

6.4807

10.3093

9409

912673

9.8489

23.2558

1849

79507

6-5574

10.2041

9604

941192

9.8995

22.7273

1936

85184

6.6332

10.1010

9801

970299

9.9499

22.2222

2025

91125

6.7082

100

10.0000

IOOOO

IOOOOOO

10.0000

21.2766

2116 2209

97336 103823

6.7823 6-8557

101 IO2

9.90099 9.80392

IO2OI 10404

1030301

1061208

10.0499

10.0995

48 49

20.8333 20.4082

2304 2401

110592 117649

6.9282 7.0000

103 104

9.70874 9.61538

10609 I08I6

1092727 1124864

10.1489 10.1980

2O.OOOO 19.6078

2500 2601

125000 132651

7.0711 7.1414

105

106

9.52381 9.43396

II025 11236

1157625

1191016

10.2470 10.2956

52 53

19.2308 18.8679

2704 2809

140608 148877

7.2111

7.2801

107 108

9-34579 9.25926

II449 11664

1225043 1259712

10.3441 10.3923

18.5185

2916

157464

7.3485

109

9.I743I

Il88l

1295029

10.4403

18.1818

17-8571

3025

166375 175616

7.4162 7-4833

110

9.09091 9.00901

1 2 100 I232I

1331000 1367631

10.4881

10.5357

17-5439

3249

185193

7-5498

112

8.92857

12544

1404928

10.5830

17.2414

3364

195112

7-6158

8.84956

12769

1442897

10.6301

16.9492

348i

205379

7.6811

114

8.77193

12096

1481544

10.6771

16.6667

3600

216000

7.7460

115

8.69565

13225

1520875

10.7238

16.3934

3721

226981

7.8102

116

8.62069

13456

1560896

10.7703

16.1290

3844

238328

7.8740

117

8.54701

13689

1601613

10.8167

63 64

15-8730 15.6250

3969 4096

250047 9 262144

7.9373 8.0000

118 119

8.47458 8.40336

13924 I4l6l

1643032 1685159

10.8628 10.9087

SMITHSONIAN TABLES.

14 TABLE 6 (continued'}.

VALUES OF RECIPROCALS, SQUARES, CUBES, SQUARE ROOTS, OF NATURAL NUMBERS.

1000.*

„•

1000.1

*•

120

8.33333

14400

1728000

10.9545

175

5.71429

30625

5359375

13.2288

121

8.26446

14641

I77I56I

1 1 .0000

176

5.68182

30976

545r776

13.2665

122 I23

8.19672 8.13008

14884 15129

1815848 1860867

11.0454 11.0905

177

178

5.64972 5.61798

31329 31684

5545233

13-3041

U-34I7

124

8.06452

15376

1906624

179

5.58659

32041

5735339

I3-379I

125

126

8.00000

7-93651 7.87402 7.81250

15625 15876 16129 16384

I953I25

2000376 2048383 2097IW

11.1803 11.2250 11.2694 H.3I37

180

181 182 183

5-55556 5.52486

5-49451 5.46448

32400 32761 33I24 33489

5832000

5929741 6028568 6128487

13.4164 U-4536 13.4907

!3-5277

129

7W94

16641

2146689

n.3578

184

5-43478

33856

6229504

13-5647

130

7.69231

16900

2197000

11.4018

185

5-40541

34225

6331625

13.6015

I3I

7.63359

17161

2248091

11-4455

186

5-37634

34596

6434856

13.6382

132 133

7-57576 7.51880

17424 17689

2352637

11.4891 11.5326

187 188

5-34759

34969 35344

6539203 6644672

13-6748 i3-7"3

134

7.46269

17956

2406104

II.5758

189

5.29101

35721

6751269

l3-7477

135

136

137 138

7.40741

7.35294 7.29927

7-24638

18225 18496 18769 19044

2460375 2515456 2571353 2628072

11.6190 11.6619 11.7047 n-7473

190

191 192

5.26316

5.23560 5-20833 S-^^S

36100 36481 36864 37249

6859000 6967871 7077888 7189057

13.7840 13.8203 13.8564 13.8924

139

7.19424

19321

2685619

11.7898

194

5.15464

37636

7301384

13.9284

140

7.14286

19600

2744000

11.8322

195

5.12821

38025

74M875

13.9642

141

142

7.09220 7.04225

19881 20164

2803221 2863288

11.8743 11.9164

196 197

5.10204

5.07614

38416 38809

7529536 7645373

14.0000 1 4-03 57

143

6.99301

20449

2924207

n-9583

198

5-05051

39204

7762392

14.0712

144

6,94444

20736

2985984

I2.OOOO

199

S-02S13

39601

7880599

14.1067

145

6.89655

21025

3048625

I2.O4I6

200

5.00000

40000

8000000

14.1421

146

6.84932

21316

3II2I36

12.0830

201

4.97512

40401

8120601

14.1774

148

6.80272 6.75676

21609 21904

3176523 3241792

12.1244 I2.I655

2O2 203

4.95050 4.92611

40804 41209

8242408 8365427

14.2127 14.2478

149

6.71141

22201

3307949

12.2066

2O4

4.90196

41616

8489664

14.2829

150

6.66667

22500

3375000

12.2474

205

4-87805

42025

8615125

14.3178

151

6.62252

22501

3442951

12.2882

206

4-85437

8741816

I4-3527

152

6.57895

23104

3511808

12.3288

207

4.83092

42849

8869743

XQ« ,• I4'3w5

153

6-53595

23409

3581577

12.3693

208

4.80769

43264

8998912

14.4222

6.49351

23716

3652264

12.4097

209

4-78469

43681

9129329

14.4568

155

6.45161

24025

3723875

12.4499

210

4.76190

44100

9261000

14.4914

156

6.41026

24336

3796416

12.4900

211

4-73934

44521

939393 i

14.5258

'57

6.36943

24649

3869893

12.5300

212

4.71698

44944

9528128

14.5602

158 159

6.32911 6.28931

24964 25281

3944312 4019679

12.5698 12.6095

2I3

214

4.69484 4.67290

45369 45796

9663597 9800344

14-5945 14.6287

160

6.25000

25600

4096000

12.6491

215

4.65116

46225

9938375

14.6629

161

6.21118

25921

4173281

12.6886

216

4.62963

46656

10077696

14.6969

162

6.17284

26244

4251528

12.7279

217

4.60829

47089

10218313

14.7309

163

6.13497

26569

4330747

12.7671

218

4.58716

47524

10360232

14.7648

164

6.09756

26896

4410944

12.8062

219

4.56621

4796i

10503459

14.7986

165

6.06061

27225

4492125

12.8452

220

4-54545

48400

10648000

14.8324

166

6.02410

27556

4574296

12.8841

221

4.52489

48841

10793861

14.8661

167

5.98802

27889

4657463

12.9228

222

4-50450

49284

10941048

14.8997

168

5-95238

28224

4741632

12.9615

223

4.48431

49729

11089567

I4-9332

169

5.91716

28561

4826809

13.0000

224

4.46429

50176

11239424

14.9666

170

5.88235

28900

4913000

13.0384

225

4.44444

50625

11390625

15.0000

171 172

5-84795

29241 29584

5000211 5088448

13.0767 13.1149

226 227

4.42478 440529

51076

11543176 11697083

1 5.0665

173 174

5*78035 5-747I3

29929 30276

5268024

13-1529 13.1909

228 229

$85

5^4 52441

"852352 12008989

15.0997 15-1327

SMITHSONIAN TABLES.

TABLE 6 (continued). l$

VALUES OF RECIPROCALS, SQUARES, CUBES, AND SQUARE ROOTS, OF

NATURAL NUMBERS.

IOOO.J

„«

<•

1000.1

V»

230

23I

232

4.34783 4.32900

4-3I034

52900 5336i 53824

12167000 12326391 12487168

15.1658 15-1987 15-2315

285

286 287

3.50877 3-49650 3-48432

81225 81796 82369

23149125 23393656 23639903

16.8819 16.9115 16.9411

233

4.29185

54289

12649337

15.2643

288

3.47222

82944

23887872

16.9706

234

4.27350

54756

12812904

15.2971

289

3.46021

83521

24137569

17.0000

235

4.25532

55225

12977875

15-3297

290

3.44828

84100

24389000

17.0294

236

4.23729

55696

13144256

15.3623

291

3-43643

84681

24642171

17.0587

237

4.21941

56169

13312053

I5-3948

292

3.42466

85264

24897088

17.0880

238

4.20168

56644

13481272

15-4272

293

3.41297

85849

25153757

17.1172

239

4.18410

57121

13651919

154596

294

3.40136

86436

25412184

17.1464

240

4.16667

57600

13824000

15.4919

295

3-38983

87025

25672375

17.1756

241

4.14938

58081

I399752I

15.5242

296

3.37838

87616

25934336

17.2047

242 244

4.13223 4-11523 4-09836

58564 59049 59536

14172488 14348907 14526784

1 5- 5 563 15.5885 15.6205

297 298

299

3.36700 3-35570 3-34448

88209 88804 89401

26198073 26463592 26730899

17-2337 17.2627 17.2916

245

246

4.08163 4.06504

60025 60516

14706125 14886936

15.6525 15.6844

300

301

3-33333 3.32226

90000 90601

27000000 27270901

17.3205 17.3494

247

4.04858

61009

15069223

15.7162

302

3.31126

91204

27543608

17.3781

248 249

4.03226 4.01606

61504 62001

15252992 15438249

15.7480 I57797

3°3 304

3-3J033 3.28947

91809 92416

27818127 28094464

17.4069 J7-4356

250

251

252

4.00000 3.98406 3-96825

62500 63001 63504

15625000

15813251 16003008

15.8114 15.8430 15-8745

305

306

3°7

3.27869 3.26797 3-25733

93025 93636

28372625 28652616 28934443

17.4642 17.4929 17.5214

253

3-95257

64009

16194277

15.9060

308

3-24675

94864

29218112

17.5499

254

3-93701

64516

16387064

15-9374

309

3.23625

95481

29503629

17.5784

255

3-92I57

65025

16581375

15.9687

310

3.22581

96100

29791000

17.6068

256

3.90625

65536

16777216

16.0000

3-21543

96721

30080231

17.6352

257 258

3.89105 3-87597

66049 66564

16974593 I7I73512

16.0312 16.0624

312

3-20513 3.19489

97344 97969

30371328 30664297

17-6635 17.6918

259

3.86100

67081

17373979

16.0935

3*4

3.18471

98596

30959144

17.7200

260

3-84615

67600

17576000

16.1245

315

3.17460

99225

31255875

17.7482

261 262

3-83M2 3.81679

68121 68644

17779581 17984728

16.1555 16.1864

3i7

3.16456

3- i 5457

99856 100489

31554496 3l855OI3

17.7764 17.8045

263 264

3.80228 3-78788

69169 69696

18191447 18399744

16.2173 16.2481

319

3- i 4465 3.13480

101124 101761

32157432 32461759

17.8326 17.8606

265

3.77358

70225

18609625

16.2788

320

3.12500

102400

32768000

17.8885

266

3-75940

70756

18821096

16.3095

321

103041

33076161

17.9165

267

3.74532

71289

19034163

16.3401

322

3-I0559

103684

33386248

17.9444

268

3.73134

71824

19248832

16.3707

323

3-09598

104329

33698267

17.9722

269

37I747

72361

19465109

16.4012

324

3.08042

104976

34012224

18.0000

270

271

272

3-70370 3.69004 3.67647

72900 73441 73984

19683000 19902511 20123648

16.4317 16.4621 16.4924

325

326

3.07692 3.06748 3.05810

106276 106929

34328125 34645976 34965783

18.0278 18.0355 18.0831

273 274

3.66300 3.64964

74529 75076

20346417 20570824

16.5227 16.5529

329

3.04878 3-0395 i

107584 108241

35287552 35611289

18.1108 18.1384

275

276

3-63636 3.62319

76176

20796875 21024576

16.5831 16.6132

330

3.03030 3-02115

108900 109561

35937000 36264691

18.1659 18.1934

277

3.61011

76729

21253933

16.6433

332

3.01205

110224

36594368

18.2209

278

3.59712

77284

21484952

16.6733

333

3.00300

110889

36926037

18.2483

279

3-58423

77841

21717639

16.7033

334

2.99401

111556

37259704

18.2757

280

281

3.57U3 3.55872

78400 78961

21952000 22188041

16.7332 16.7631

335

336

2.98507 2.97619

112225 112896

37595375 37933056

18.3030 18.3303

282 283

3-53357

79524 80089

22425768 22665187

16.7929 16.8226

337 338

2-96736 2.958^8

H3569 114244

38272753 38614472

18.3576 18.3848

284

80656

22906304

16.8523

339

2.94985

114921

38958219

18.4120

SMITHSONIAN TABLES.

TABLE 6

VALUES OF RECIPROCALS, SQUARES, CUBES, AND SQUARE ROOTS OF NATURAL NUMBERS.

lOOO.i

v«

lOOO.i

340

2.94118

115600

39304000

18.4391

395

2.53165

156025

61629875

19.8746

341

2.93255

116281

39651821

18.4662

396

2.52525

156816

62099136

19.8997

342

2.92398

116964

40001688

18.4932

2.51889

157609

62570773

19.9249

343

2.91545

117649

40353607

18.5203

398

2.51256

158404

63044792

19.9499

344

2.90698

118336

40707584

18.5472

399

2.50627

159201

63521199

19.9750

345

2.89855

119025

41063625

18.5742

400

2.50000

160000

64000000

20.0000

346

2.89017

119716

41421736

18.6011

401

2-49377

160801

64481201

20.0250

2.88184

2.87356

120409 121104

41781923 42144192

18.6279 18.6548

402 403

2.48756 2.48139

161604 162409

64964808 65450827

20.0499 20.0749

349

2.86533

121801

42508549

18.6815

404

2.47525

163216

65939264

20.0998

350

2.85714

122500

42875000

18.7083

405

2.46914

164025

66430125

20.1246

352

2.84900 2.84091

123201 123904

43243551 43614208

18.7350 18.7617

406 407

2.46305 2.45700

164836 165649

66923416 67419143

20.1494 20.1742

353

2.83286

124609

43986977

18.7883

408

2.45098

166464

67917312

20.1990

354

2.82486

125316

44361864

18.8149

409

2-44499

167281

68417929

20.2237

355

2.81690

126025

44738875

18.8414

410

2.43902

168100

68921000

20.2485

356

2.80899

126736

45118016

18.8680

411

2.43309

168921

69426531

20.2731

357

2.80112

127449

45499293

18.8944

412

2.42718

169744

69934528

20.2978

358

2.79330

128164

45882712

18.9209

413

2.42131

170569

70444997

20.3224

359

2.78552

128881

46268279

18.9473

414

2.41546

171396

70957944

20.3470

360

2.77778

129600

46656000

18.9737

415

2.40964

172225

7M73375

20.3715

361

2.77008

130321

47045881

19.0000

416

2.40385

173056

71991296

20.3961

362

2.76243

131044

47437928

19.0263

417

2.39808

173889

72511713

20.4206

363 364

2.75482 2.74725

131769 132496

47832147 48228544

19.0526 19.0788

418 419

174724

73034632 73560059

20.4450 20.4695

365

2.73973

133225

48627125

19.1050

420

2.38095

176400

74088000

20.4939

366

2.73224 2.72480

133956 134689

49027896 49430863

19.1311 19.1372

421 422

2.37530 2.36967

177241 178084

74618461 75I5I448

20.5183 20.5426

368

2.71739

135424

49836032

«9-l833

423

2.36407

178929

75686967

20.5670

369'

2.71003

136161

50243409

19.2094

424

2-35849

179776

76225024

20.5913

370

2.70270

136900

50653000

19.2354

425

2.35294

180625

76765625

20.6155

371

2.69542

137641

51064811

19.2614

426

2-34742

181476

77308776

20.6398

372

2.68817

138384

51478848

19.2873

427

2.34192

182329

77854483

20.6640

373

2.68097

139129

5l895II7

19.3132

428

2-33645

183184

78402752

20.6882

374

2.67380

139876

52313624

429

2.33100

184041

789535»9

20.7123

375

2.66667

140625

52734375

19.3649

430

2.32558

184900

79507000

20.7364

376 377

2.65957 2.65252

141376

142129

53157376 53582633

19.3907 19.4165

432

2.32019 2.31481

185761 186624

80062991 80621568

20.7605 20.7846

378 379

2.64550 2.63852

142884 143641

54010152 54439939

19.4422 19.4679

433 434

2.30947 2.30415

187489 188356

81182737 81746504

20.8087 20.8327

380

2.63158 2.62467

144400 145161

54872000 55306341

19.4936 19.5192

435

436

2.29885 2.29358

189225 190096

82312875 82881856

20.8567 20.8806

382 383

2.61780

2.61097

145924 146689

55742968 56181887

19.5448 19.5704

438

2.28833 2.28311

190969 191844

83453453 84027672

20.9045 20.9284

384

2.60417

147456

56623104

19-5959

439

2.27790

192721

84604519

20.9523

385

2.59740

148225

57066625

19.6214

440

2.27273

193600

85184000

20.9762

386

148996

57512456

19.6469

441

2.26757

194481

85766121

2I.OOOO

387

2.58398

149769

57960603

19.6723

442

2.26244

195364

86350888

21.0238

388

2.57732

150544

58411072

19.6977

443

2.25734

196249

86938307

21.0476

389

2.57069

151321

58863869

19.7231

444

2.25225

197136

87528384

21.0713

390

2.56410

152100

59319000

19.7484

445

2.24719

198025

88121125

21.0950

392

2.55754 2.55102

152881 153664

59776471 60236288

19.7737 19.7990

446

447

2.24215 2.23714

198916 199809

88716536 89314623

2I.II87 21.1424

393

2.54453

154449

60698457

19.8242

448

2.23214

200704

89915392

21. 1660

394

2.53807

155236

e.^

19.8494

449

2.22717

201601

90518849

21.1896

SMITHSONIAN TABLES.

TABLE 6 (continued). 1 7

VALUES OF RECIPROCALS, SQUARES, CUBES, AND SQUARE ROOTS OF NATURAL NUMBERS.

,000.1

IOOO.J

450

2.22222

2O25OO

9II25OOO

21.2132

505

1.98020

255025

128787625

22.4722

451

2.21730

2O34OI

9I73385I

21.2368

506

1.97628

256036

129554216

22.4944

452

2.21239

204304

92345408

21.2603

5°7

1.97239

257049

130323843

22.5167

453 454

2.20751 2.2O264

205209 2o6ll6

92959677 93576664

21.2838 21.3073

508 509

1.96850

1.96464

258064 259081

131096512 131872229

22.5389 22.5610

455

456

2.19780 2.19298

207025 207936

94818816

21.3307 21.3542

510

511

1.96078 1 -95695

260100 261121

132651000 1 3343283 i

22.5832 22.6053

457

2.l88l8

208849

95443993

21.3776

512

r-95312

262144

134217728

22.6274

458 459

2.18341 2.17865

209764 2I068I

96071912 96702579

21.4009 21.4243

1.94932 1-94553

263169 264196

135005697 135796744

22.6495 22.6716

460

2.I739I

2II6OO

97336000

21.4476

515

I-94I75

265225

136590875

22.6936

461

2.16920

2I252I

97972181

21.4709

516

1.93798

266256

137388096

22.7156

462

2.16450

213444

98611128

21.4942

5*7

1.93424

267289

138188413

22.7376

463

2.15983

214369

99252847

21.5174

518

1.93050

268324

1 38991 832

22.7596

464

215296

99897344

21.5407

5*9

1.92678

269361

^9798359

22.7816

465

2.15054

216225

100544625

21.5639

520

1.92308

270400

140608000

22.8035

466

2.14592

217156

101194696

21.5870

521

1.91939

271441

141420761

22.8254

467

2.I4I33

218089

101847563

21.6102

522

272484

142236648

22.8473

468 469

2.13675 2.13220

219024 219961

102503232 103161709

21.6333 21.6564

523 524

1.91205 1.90840

273529 274576

143055667 143877824

22.8692 22.8910

470

2.12766

22O9OO

103823000

21.6795

525

1.90476

275625

144703125

22.9129

471

2.12314

221841

104487111

21.7025

526

1.90114

276676

I4553I576

22.9347

472

2.II864

222784

105154048

21.7256

527

1 -897 53

277729

146363183

22.9565

473

2.II4I6

223729

105823817

21.7486

528

1.89394

278784

147197952

22.9783

474

2.IO97O

224676

106496424

21.7715

529

1.89036

279841

148035889

23.0000

475

2.10526

225625

107171875

21.7945

530

1.88679

280900

148877000

23.0217

476

2.IOO84

226576

107850176

21.8174

531

1.88324

281961

149721291

23-0434

477

2.09644

227529

108531333

21.8403

532

1.87970

283024

150568768

23.0651 ;

478

2.09205

228484

109215352

21.8632

533

1.87617

284089

I5HI9437

23.0868

479

2.08768

229441

109902239

21.8861

534

1.87266

285156

152273304

23.1084

480

2-08333

230400

110592000

21.9089

535

1.86916

286225

I53I30375

23.1301

481

2.0790C

231361

111284641

21.9317

536

1.86567

287296

153990656

482 483

2.07469 2.07039

232324 233289

111980168 112678587

21-9545 21.9773

537

538

1.86220 1.85874

288369 289444

154854153 155720872

23-1733 23.1948

484

2.06612

234256

"3379904

22.0000

539

1.85529

290521

156590819

23.2164

485

486

2.06186 2.05761

235225 236196

114084125 114791256

22.0227

22.0454

540

54i

1.85185 1.84843

291600 292681

157464000 158340421

23-2379 23.2594

487

2-05339

237169

115501303

22.O68I

542

1.84502

293764

159220088

23.2809

488

2.04918

238144

116214272

22.0907

543

1.84162

294849

160103007

23.3024

489

2.04499

239I2I

116930169

22.1133

544

1.83824

295936

160989184

23-3238

490

2.O4O82

24OIOO

117649000

22.1359

545

1.83486

297025

161878625

23-3452

491

2.03666

241081

118370771

22.1585

546

1.83150

298116

162771336

23.3666

492

2.03252

242064

119095488

22.I8II

547

1.82815

299209

163667323

23.3880

493

2.02840

243049

119823157

22.2036

548

1.82482

300304

164566592

23.4094

494

2.02429

244036

120553784

22.2261

549

1.82149

301401

165469149

23-4307

495

2.O2O2O

245025

121287375

22.2486

550

1.81818

302500

166375000

23.4521

496

2.0l6l3

246016

122023936

22.2711

551

1.81488

303601

167284151

23-4734

497

2.OI2O7

247009

122763473

22.2935

552

1.81159

304704

168196608

23-4947

498 499

2.00803 2.00401

248004 249OOI

123505992 124251499

22.3159 22.3383

553 554

1.80832 1.80505

$&$

169112377 170031464

23.5160 23-5372

500

2.00000 I.9900I

250000 25IOOI

125000000 125751501

22.3607 22.3830

555

556

.80180 .79856

308025 309136

170953875 171879616

23.5584 23.5797

502

1.99203

252004

i 26506008

22.4054

557

•79533

310249

172808693

23.6008

5°3

1.98807

253009

127263527

22.4277

558

.79211

311364

173741112

23.6220

5°4

1.98413

254016

128024064

22.4499

559

.78891

312481

174676879

23.6432

SMITHSONIAN TABLES.

1 8 TABLE 6 (continued).

VALUES OF RECIPROCALS, SQUARES, CUBES, AND SQUARE ROOTS OF NATURAL NUMBERS.

lOOO.i

tfi

«8

IOOO.I

«3

560

J-7857i

313600

175616000

23.6643

615

1.62602

378225

232608375

24.7992

56i 562

563

1-78253 1.77936 1.77620

314721

176558481 177504328 178453547

23.6854 23.7065 23.7276

616 617 618

1.62338

1.62075 1.61812

380689 381924

233744896 234885113 236029032

24.8193

24-8395 24.8596

564

I-77305

318096

179406144

23.7487

619

1.61551

383161

237176659

24.8797

565

1.76991

319225

180362125

23.7697

620

1.61290

384400

238328000

24.8998

566

1.76678

320356

181321496

23.7908

621

1.61031

385641

239483061

24.9199

567

1.76367

321489

182284263

23.8118

622

1.60772

386884

240641848

24-9399

568

-*•** >-

1.76056

322624

183250432

23.8328

623

1.60514

388129

241804367

24.9600

569

1-75747

323761

184220009

23-8537

624

1.60256

389376

242970624

24.9800

570

1 -7 5439

324900

185193000

23.8747

625

1.60000

390625

244140625

25.0000

57i

I-75I3I

326041

186169411

23.8956

626

1-59744

391876

2453 ! 4376

25.0200

572

1.74825

327184

187149248

23.9165

627

1.59490

393 i 29

246491883

25.0400

573

1.74520

328329

188132517

23-9374

628

1.59236

394384

247673132

25.0599

574

1.74216

329476

189119224

23-9583

629

1.58983

395641

248858189

25.0799 1

575

i-739I3

330625

190109375

23.9792

630

1-58730

396900

25OO47OOO

25.0998

576

1.73611

33^76

191102976

24.0000

631

1.58479

398161

25I23959!

25.1197

577

1.73310

332929

192100033

24.0208

632

1.58228

399424

252435968

25.1396

578

1.73010

334084

193100552

24.0416

633

1.57978

400689

253636137

25-1595

579

1.72712

335241

194104539

24.0624

634

1.57729

401956

254840104

25.1794

580

1.72414

336400

195112000

24.0832

635

1.57480

403225

256047875

25.1992

581

1.72117

33756i

196122941

24.1039

636

1-57233

404496

257259456

25.2190

582

1.71821

338724

I97I37368

24.1247

637

1.56986

405769

258474853

25.2389

583

1.71527

339889

198155287

24.1454

638

1.56740

407044

259694072

25.2587

584

1-71233

341056

199176704

24.1661

639

1.56495

408321

260917119

25.2784

585

1.70940

342225

200201625

24.1868

640

1.56250

409600

262144000

25.2982

586

1.70648

343396

201230056

24.2074

641

1.56006

410881

263374721

25.3180

587

1-70358

344569

202262003

24.2281

642

I-55763

412164

264609288

25-3377

588

1.70068

345744

203297472

24.2487

643

L55521

413449

265847707

25-3574

589

1.69779

346921

204336469

24-2693

644

1.55280

4H736

267089984

25-3772

590

1.69492

348100

205379000

24.2899

645

L55039

416025

268336125

25-3969

591

1.69205

349281

206425071

24.3105

646

1-54799

4173*6

269586136

25.4165

592

1.68919

350464

207474688

24.3311

647

1.54560

418609

270840023

25.4362

593

1.68634

35 l 649

208527857

24.3516

648

I-5432I

419904

272097792

25-4558

594

1.68350

352836

209584584

24.3721

649

1.54083

421201

273359449

25.4755

595

1.68067

354025

210644875

24.3926

650

1.53846

422500

274625000

25.4951

596

1.67785

3552i6

211708736

24.4131

651

1.53610

423801

275894451

25-5M7

597

1.67504

356409

212776173

24-4336

652

1-53374

425104

277167808

25-5343

i98

1.67224

357604

213847192

24.4540

653

I-53I39

426409

278445077

25-5539

599

1.66945

3588oi

214921799

24.4745

654

1.52905

427716

279726264

25.5734

600

1.66667

360000

216000000

24.4949

655

1.52672

429025

281011375

25-5930

601

1.66389

361201

217081801

24-5r53

656

i.52439

430336

282300416

25.6125

602

1.66113

362404

218167208

24-5357

657

1.52207

431649

283593393

25.6320

603

1.65837

363609

219256227

24.5561

658

1.51976

432964

284890312

25-651.5

604

1-65563

364816

220348864

24.5764

659

i.5'745

43428i

286191179

25.6710

605

1.65289

366025

221445125

24.5967

660

I-SISIS

435600

287496000

25.6905

606

1.65017

367236

222545016

24.6171

661

1.51286

436921

288804781

25.7099

607

1.64745

368449

223648543

24.6374

662

1.51057

438244

290117528

25-7294

608

1.64474

369664

224755712

24.6577

663

1.50830

439569

291434247

25.7488

609

1.64204

37o88i

225866529

24.6779

664

1.50602

440896

292754944

25.7682 ;

610

1-63934

372100

226981000

24.6982

665

i.50376

442225

294079625

25.7876

611

1.63666

373321

228099131

24.7184

666

1.50150

443556

295408296

25.8070

612

1 -63399

374544

229220928

24-7386

667

1.49925

444889

296740963

25.8263

613

1.63132

375769

230346397

24.7588

668

1.49701

446224

298077632

25-8457

614

1.62866

376996

23H75544

24.7790

669

1.49477

44756i

299418309

25.8650

SMITHSONIAN TABLES.

TABLE 6 (continued). ' 19

VALUES OF RECIPROCALS, SQUARES, CUBES, AND SQUARE ROOTS OF NATURAL NUMBERS.

iooo.i

„.

iooo.i

670

1.49254

448900

300763000

25.8844

725

I-3793I

525625

381078125

26.9258

671

1.49031

450241

302111711

25.9037

726

I-3774I

527076

382657176

26.9444

672

1.48810

45*584

303464448

25.9230

727

I-37552

528529

384240583

26.9629

673

1.48588

452929

304821217

25.9422

728

1-37363

529984

385828352

26.9815

674

1.48368

454276

306182024

25.9615

729

I.37I74

53I44I

387420489

27.0000

675

1.48148

455625

307546875

25.9808

730

1.36986

532900

389017000

27.0185

676

1.47929

456976

308915776

26.0000

731

1-36799

390617891

27.0370

677

1.47710

310288733

26.0192

732

1.36612

535824

392223168

27.0555

678 679

1.47493 1.47275

459684 461041

311665752 313046839

26.0384 26.0576

733

734

1.36426 1.36240

537289 538756

393832837 395446904

27.0740 27.0924

680

1.47059

462400

314432000

26.0768

735

1.36054

540225

397065375

27.1109

68 1

1.46843

463761

315821241

26.0960

736

1-35870

541696

398688256

27.1293

682

1.46628

465124

317214568

26.1151

737

1.35685

543169

400315553

27.1477

683

1.46413

466489

318611987

26.1343

738

1.35501

544644

401947272

27.1662

684

1.46199

467856

320013504

26.1534

739

I-353l8

546121

403583419

27.1846

685

1 4598 5

469225

321419125

26.1725

740

1.35135

5476oo

405224000

27.2029

686

1-45773

470596

322828856

26.1916

741

1-34953

549081

406869021

27.2213

687

1.45560

471969

324242703

26.2107

742

i-3477i

550564

408518488

27.2397

688

145349

473344

325660672

26.2298

743

1-3459°

552049

410172407

27.2580

689

145*38

474721

327082769

26.2488

744

1.34409

553536

411830784

27.2764

690

1.44928

476100

328509000

26.2679

745

1.34228

555025

413493625

27.2947

691

1.44718

477481

329939371

26.2869

746

1.34048

556516

415160936

27.3130

692 693

1.44509 1.44300

478864 480249

331373888 332812557

26.3059 26.3249

747 748

1.33869 1.33690

558009 559504

416832723 418508992

27.3313 27.3496

694

1.44092

481636

334255384

26.3439

749

1-335"

561001

420189749

27.3679

695

1.43885

483025

335702375

26.3629

750

1-33333

562500

421875000

27.3861

696

1.43678

484416

337153536

26.3818

751

I-33I56

564001

423564751

27.4044

697

1.43472

485809

338608873

26.4008

752

1.32979

565504

425259008

27.4226

698

1.43266

487204

340368392

26.4197

753

1.32802

567009

426957777

27.4408

699

1.43062

488601

341532099

26.4386

754

1.32626

568516

428661064

27.4591

700

1.42857

490000

343000000

26.4575

755

1.32450

570025

430368875

274773

701

1.42653

491401

344472101

26.4764

756

1.32275

571536

432081216

27-4955

702

1.42450

492804

345948408

26.4953

757

1.32100

573049

433798093

27-5136

703 704

1.42248 1.42045

494209 495616

347428927 348913664

26.5141 26.5330

758 759

1.31926 1.31752

574564 576081

4355I9512 437245479

27-5318 27.5500

705

1.41844

497025

350402625

26.5518

760

i-3I579

577600

438976000

27.568!

706

1.41643

498436

351895816

26.5707

761

1.31406

579121

440711081

27.5862

707

1.41443

499849

353393243

26.5895

762

i.3I234

580644

442450728

27-6043

708

1.41243

501264

354894912

26.6083

763

1.31062

582169

444194947

27.6225

709

1.41044

502681

356400829

26.6271

764

1.30890

583696

445943744

27.6405

710

1.40845

504100

357911000

26.6458

765

1.30719

585225

447697125

27.6586

711

712

1.40647 1.40449

505521 506944

359425431 360944128

26.6646 26.6833

766

767

1.30548 1.30378

586756 588289

449455096 451217663

27.6767 27.6948

713

1.40252

508369

362467097

26.7021

768

1.30208

589824

452984832

27.7128

1.40056

509796

363994344

26.7208

769

1.30039

59i36i

454756609

27.7308

715

1.39860

5II225

365525875

26.7395

770

1.29870

592900

456533000

27.7489

716

1.39665

512656

367061696

26.7582

771

1.29702

594441

458314011

27.7669

717

1.39470

*JS^I

514089

368601813

26.7769

772

1.29534

595984

460099648

27.7849

718

1.39276

5T5524

370146232

26.7955

773

1.29366

597529

461889917

27.8029

719

1.39082

516961

371694959

26.8142

774

1.29199

599076

463684824

27.8209

720

721

722

1.38889 .1.38696

"1.38504

518400 519841 521284

373248000 374805361 376367048

26.8328 26.8514 26.8701

775

776 777

1.28700

600625 602176 603729

465484375 467288576

469097433

27.8388 27.8568 27.8747

723 724

1.38313

1.38122

522729 524176

377933067 379503424

26.8887 26.9072

778 779

1-28535 1.28370

605284 606841

470910952 472729139

27.8927

27.9106

SMITHSONIAN TABLES.

2O TABLE 6 (continued}.

VALUES OF RECIPROCALS, SQUARES, CUBES, AND SQUARE ROOTS OF NATURAL NUMBERS.

iooo4

«3

jf*

iooo.i

«2

«8

780

1.28205

608400

474552000

27.9285

835

1.19760

697225

582182875

28.8964

781

1.28041

609961

476379541

27.9464

836

1.19617

698896

584277056

28.9137

782

1.27877

611524

478211768

27.9643

837

1.19474

700569

586376253

28.9310

783

1.27714

613089

480048687

27.9821

838

1.19332

702244

588480472

28.9482

784

1.27551

614656

481890304

28.0000

839

1.19190

703921

590589719

28.9655

785

1.27389

6l6225

483736625

28.0179

840

1.19048

705600

592704000

28.9828

786

1.27226

617796

485587656

28.0357

841

1.18906

707281

594823321

29.0000

787

1.27065

619369

487443403

28.0535

842

1.18765

708964

596947688

29.0172

788 789

1.26904 1.26743

620944 622521

489303872 491169069

28.0713 28.0891

843 844

1.18624 1.18483

710649 712336

599077107 601211584

29-0345 29.0517

790

1.26582

624IOO

493039000

28.1069

845

1.18343

714025

603351125

29.0689

791

1.26422

625681

494913671

28.1247

846

1.18203

715716

605495736

29.0861

792

1.26263

627264

496793088

28.1425

847

1.18064

717409

607645423

29.1033

793

1.26103

628849

498677257

28.1603

848

1.17925

719104

609800192

29.1204

794

1.25945

630436

500566184

28.1780

849

1.17786

720801

611960049

29.1376

795

1.25786

632025

502459875

28.1957

850

1.17647

722500

614125000

29.1548

796

1.25628

633616

504358336

28.2135

851

1.17509

724201

616295051

29.1719

798

1.25471 1.25313

635209 636804

506261573 508169592

28.2312 28.2489

852 853

1.17371 1.17233

725904 727609

618470208 620650477

29.1890 29.2062

799

1.25156

638401

510082399

28.2666

854

1.17096

729316

622835864

29.2233

800

1.25000

64OOOO

512000000

28.2843

855

1.16959

731025

625026375

29.2404

801

1.24844

641601

5I392240I

28.3019

856

1.16822

732736

627222016

29-2575

802

1.24688

643204

515849608

28.3196

857

1.16686

734449

629422793

29.2746

803

I-24533

644809

517781627

28.3373

858

1.16550

736164

631628712

29.2916

804

1.24378

646416

519718464

28.3549

859

1.16414

73788i

633839779

29.3087

805

806

1.24224 1.24069

648025 649636

52I660I25 523606616

28.3725 28.3901

860

86 1

1.16279

1.16144

7396oo 741321

636056000 638277381

29.3258 29.3428

807

1.23916

651249

525557943

28.4077

862

1.16009

743044

640503928

29.3598

808

1.23762

652864

527514112

28.4253

863

1.15875

744769

642735647

29.3769

809

1.23609

654481

529475129

28.4429

864

1.15741

746496

644972544

29-3939

810

1.23457

656100

53I44IOOO

28.4605

865

1.15607

748225

647214625

29.4109

811

1 -23305

657721

5334II73I

28.4781

866

1.15473

749956

649461896

29.4279

812

L23I53

659344

535387328

28.4956

867

1.15340

751689

651714363

29.4449

813

I.23OOI

660969

537367797

28.5132

868

1.15207

753424

653972032

29.4618

814

1.22850

662596

539353M4

28.5307

869

1.15075

755l6i

656234909

29.4788

815

1.22699

664225

541343375

28.5482

870

1.14943

756900

658503000

29.4958

816

1.22549

665856

543338496

28.5657

871

1.14811

758641

660776311

295127

817

1.22399

667489

54533851 3

28.5832

872

1.14679

760384

663054848

29.5296

818

1.22249

669124

547343432

28.6007

873

1.14548

762129

665338617

29.5466

819

1. 22100

670761

549353259

28.6182

874

1.14416

763876

667627624

29-5635

820

I.2I95I

672400

551368000

28.6356

875

1.14286

765625

669921875

29.5804

821

1.21803

674041

553387661

28.6531

876

1.14155

767376

672221376

29-5973

822 823

I.2I655

I.2T507

675684 677329

555412248 557441767

28.6705 28.6880

877 878

1.14025 I-I3895

769129 770884

674526133 676836152

29.6142 29.63 1 1

824

L2I359

678976

559476224

28.7054

879

1.13766

772641

679I5I439

29.6479

825

I.2I2I2

680625

561515625

28.7228

880

1.13636

774400

681472000

29.6648

826

I.2IO65

682276

563559976

28.7402

88 1

1.13507

776161

683797841

29.6816

827

I.209I9

683929

565609283

28.7576

882

I.I3379

777924

686128968

29.6985

828

1.20773

685584

567663552

28.7750

883

1.13250

779689

688465387

29-7153

829

1.20627

687241

569722789

28.7924

884

1.13122

781456

690807104

29.7321

830

1.20482

688900

571787000

28.8097

885

1.12994

783225

693!54i25

29-7489

831

1.20337

690561

573856191

28.8271

886

1.12867

784996

695506456

29.7658

832

I.2OI92

692224

575930368

28.8444

887

1.12740

786769

697864103

29.7825

833

1.20048

693889

578009537

28.8617

888

1.12613

788544

700227072

29-7993

834

1.19904

695556

580093704

28.8791

889

1.12486

790321

702595369

29.8161

SMITHSONIAN TABLES.

TABLE 6 (continued). 21

VALUES OF RECIPROCALS, SQUARES, CUBES, AND SQUARE ROOTS OF NATURAL NUMBERS.

lOOO.jj

»3

*•

IOOO.£

»2

«3

v«

890

1.12360

792100

704969000

29.8329

945

1.05820

893025

843908625

30.7409

891

1.12233

793881

707347971

29.8496

946

1.05708

894916

846590536

30.7571

892

i. 12108

795664

709732288

29.8664

947

1.05597

896809

849278123

30-7734

893

1.11982

797449

7I2I2I957

29.8831

948

1.05485

898704

851971392

30.7896

894

1.11857

799236

714516984

29.8998

949

1.05374

900601

854670349

30.8058

895

1.11732

801025

7I69I7375

29.9166

950

1.05263

902500

857375000

30.8221

896

1.11607

802816

719323136

29-9333

951

1.05152

904401

860085351

30.8383

897

1.11483

804609

721734273

29.9500

952

1.05042

906304

862801408

30.8545

898

I-II359

806404

724150792

29.9666

953

1.04932

908209

865523177

30.8707

899

1.11235

808201

726572699

29-9833

954

1.04822

9IOII6

868250664

30.8869

900

i. inn

810000

729000000

30.0000

955

1.04712

9I2O25

870983875

30.9031

901

1.10988

811801

731432701

30.0167

956

1.04603

913936

873722816

30.9192

902

1.10865

813604

733870808

30-0333

957

1.04493

915849

876467493

30.9354

9°3

1.10742

815409

7363*4327

30.0500

958

1.04384

917764

879217912

30.9516

904

1.10619

817216

738763264

30.0666

959

1.04275

919681

881974079

30.9677

905

1.10497

819025

741217625

30.0832

960

1.04167

921600

884736000

30.9839

906

1-10375

820836

743677416

30.0998

961

1 .04058

923521

887503681

31.0000

907

1.10254

822649

746142643

30.1164

962

1.03950

925444

890277128

31.0161

908

1.10132

824464

748613312

30.1330

963

1.03842

927369

893056347

31.0322

909

I.IOOII

826281

751089429

30.1496

964

1.03734

929296

895841344

31.0483

910

1.09890

828100

753571000

30.1662

965

1.03627

931225

898632125

31.0644

911

1.09769

829921

756058031

30.1828

966

1.03520

933^6

901428696

31.0805

912

1.09649

831744

758550528

30-I993

967

1.03413

935089

904231063

31.0966

9*3 914

1.09529 1.09409

833569 835396

761048497 76355^44

30.2159 30.2324

968 969

1.03306 1.03199

937024 938961

907039232 909853209

31.1127 31.1288

915

1.09290

837225

766060875

30.2490

970

1.03093

940900

912673000

31.1448

916

1.09170

839056

768575296

30-2655

971

1.02987

942841

9I54986II

31.1609

917

1.09051

840889

771095213

3O.282O

972

1.02881

944784

918330048

31.1769

918 919

1.08932 1.08814

842724 844561

773620632 776I5I559

30.2985 30.3150

973 974

1.02775

1.02669

946729 948676

92II673I7 924010424

31.1929 31.2090

920

1.08696

846400

778688000

30.3315

975

1.02564

950625

926859375

31.2250

921

1.08578

848241

781229961

30.3480

976

1.02459

952576

929714176

31.2410

922 923

1.08460 1.08342

850084 851929

783777448 786330467

30-3645 30.3809

977 978

1.02354 1.02249

954529 956484

932574833 935441352

31.2570 31.2730

924

1.08225

853776

788889024

30.3974

979

1.02145

958441

9383*3739

31.2890

925

1. 08 1 08

855625

79M53I25

30.4138

980

1.02041

960400

941 192000

31-3050

926

1.07991

857476

794022776

30.4302

981

1.01937

962361

944076141

31.3209

927

1.07875

859329

796597983

30.4467

982

1.01833

964324

946966168

3 * -3369

928 929

1.07759 1.07643

861184 863041

799178752 801765089

30.4631 30.4795

983 984

1.01729 1.01626

966289 968256

949862087 952763904

3I-3528 31.3688

930

1.07527

864900

804357000

30-4959

985

1.01523

970225

955671625

3I-3847

93 l

1.07411

866761

806954491

3°- 5 i 23

986

1.01420

972196

958585256

31.4006

932

1.07296

868624

809557568

30.5287

987

1.01317

974169

961504803

31.4166

933

1.07181

870489

812166237

30.545o

988

1.01215

976144

964430272

3M325

934

1.07066

872356

814780504

30.5614

989

I.OIII2

978121

967361669

31.4484

935

1.06952

874225

817400375

30.5778

990

I.OIOIO

980100

970299000

3 r -4643

936 937

1.06838 1.06724

876096 877969

820025856 822656953

30-594I 30.6105

991 992

1.00908

1.00806

982081 984064

973242271 976191488

31.4802 31.4960

938 939

1.06610 1.06496

879844 881721

825293672 827936019

30.6268 30.6431

993 994

1.00705

1.00604

986049 988036

979146657 982107784

3i-5ii9

3I-5278

940

1.06383

883600

830584000

30.6594

995

1.00503

990025

985074875

3 * -5436

941

1.06270

885481

833237621

30-6757

996

1.00402

992016

988047936

3J-5595

942

1.06157

887364

835896888

30.6920

997

1.00301

994009

991026973

31-5753

943

1.06045

889249

838561807

30.7083

998

1.00200

996004

994011992

31-59"

944

1.05932

891136

841232384

30.7246

999

1. 00 1 00

998001

997002999

31.6070

SMITHSONIAN TABLES.

TABLE 7. LOGARITHMS.

100

oooo

0004

0009

0013

0017

0022

0026

0030

°°35

0039

0043

101 102

0043 0086

0048 0090

0052 0095

0056 0099

0060 0103

0065 OIO7

0069 oin

0073 0116

0077

0120

0082 0124

0086 0128

103

0128

0*33

0*37

0141

0145

0149

0154

0158

Ol62

0166

0170

104

0170

0*75

0179

0183

0187

0191

oi9S

0199

O204

0208

0212

105

106

O2I2 0253

0216

0257

0220 O26l

0224 0265

0228 0269

0233 0273

0237 0278

0241 0282

0245 0286

0249 0290

0253 0294

107

0294

0298

0302

0306

0310

0314

0318

0322

0326

033°

°334

1 08

0334

0338

0342

0346

0350

0354

0358

0362

0366

0370

0374

109

0374

0378

0382

0386

0390

0394

0398

0402

0406

0410

0414

110

0414

0418

0422

0426

0430

0434

0438

0441

0445

0449

°453

0457

0461

0465

0469

0473

0477

0481

0484

0488

0492

112

0492

0496

0500

0504

0508

0512

°5J5

0519

0523

0527

°53I

0531

0535

0538

0542

0546

0550

0554

0558

0561

0565

0569

114

0569

°573

0577

0580

0584

0588

0592

0596

0599

0603

0607

115

0607

0611

0615

0618

0622

0626

0630

0633

0637

0641

0645

116

0645

0648

0652

0656

0660

0663

0667

0671

0674

0678

0682

0686

0689

0693

0697

0700

0704

0708

0711

0715

0719

118

0719

0722

0726

0730

0734

0737

0741

0745

0748

0752

°75S

119

0755

0759

0763

0766

0770

0774

0777

0781

0785

0788

0792

120

0792

0795

0799

0803

0806

0810

0813

0817

0821

0824

0828

121

0828

0831

0835

0839

0842

0846

0849

0853

0856

0860

0864

122

0864

0867

0871

0874

0878

O88l

0885

0888

0892

0896

0899

I23

0899

0903

0906

0910

0913

0917

0920

0924

0927

0931

0934

124

0934

0938

0941

0945

0948

0952

0955

0959

0962

0966

0969

125

0969

0973

0976

0980

0983

0986

0990

0993

0997

IOOO

1004

126

1004

1007

ion

1014

1017

1021

1024

1028

1031

I035

1038

127

1038

1041

1045

1048

1052

IO55

I059

1062

I065

1069

1072

128

1072

1075

1079

1082

1086

1089

1092

1096

1099

1103

1106

129

1106

1109

1113

ni6

1119

1123

1126

1129

"33

1136

"39

130

"39

"43

1146

1149

"53

1156

"59

1163

1166

1169

"73

131

"73

1176

"79

1183

1186

1189

"93

1196

"99

1 202

1206

132-

1206

1209

1212

1216

1219

1222^

1225

1229

1232

I235

1239

133

1239

1242

1245

1248

1232

I2S5

1258

1261

1265

1268

1271

134

1271

1274

I278

1281

1284

1287

1290

1294

1297

1300

J3°3

135

1303

1307

1310

1313

1316

1319

1323

1326

1329

,332

1335

136

1335

1339

1342

U45

1348

13^1

1358

1361

1364

^67

139

1367 1399 1430

1370 1402 H33

1374 1405

1436

1377 1408 1440

1380 1411 1443

1383 1414 1446

1418 1449

1389 1421 MS2

1392 1424

1455

1396 1427

1458

1399 143° 1461

140

1461

1464

1467

1471

H74

1477

1480

1483

1486

1489

1492

141

1492

'495

I498

1501

1504

1508

'5"

15*4

1517

1520

1523

142

1523

1526

1529

1532

1535

1538

i54i

1544

1547

1550

1553

143

'553

1556

1559

1562

1565

*5^9

1572

i|75

'578

1581

1584

144

1584

I5»7

I59°

1593

1596

1599

1602

1605

1608

IOII

1614

145

1614

1617

1620

1623

1626

1629

1632

l635

1638

1641

1644

146

1644

1647

1649

1652

1655

1658

1661

1664

1667

1670

1673

147

1673

1676

1679

1682

1685

l688

1691

1694

1697

1700

-1703

148

1703

1706

1708

1711

1714

1717

1720

1723

1726

1729

1732

149

1732

1735

1738

1741

1744

1746

1749

1752

1755

1758

1761

SMITHSONIAN TABLES.

TABLE 7 (continued).

LOGARITHMS.

150

1761

1764

1767

1770

1772

I77S

1778

1781

1784

1787

1790

J793

1796

1798

1801

1804

1807

1810

1813

1816

1818

1818 1847

1821 1850

1824 1853

1827 1855

1830

1858

1833 1861

1836 1864

1838 1867

1841 1870

1844 1872

1847 1875

154

1875

1878

1881

1884

1886

1889

1892

1895

1898

1901

1903

155

1903

1906

1909

1912

I9T5

1917

1920

1923

1926

1928

'931

156

157

I931 1959

1934 1962

1965

1940 1967

1942 1970

'945 J973

1948 1976

'951 1978

1953 1981

1956 1984

1959 1987

158

1987

1989

1992

1995

1998

2000

2003

2006

2009

2OII

2014

159

2014

2017

2019

2O22

2O25

2028

2030

2033

2036

2038

2041

160

2041

2044

2047

2049

2052

2055

2057

2060

2063

2066

2068

161

2068

2071

2074

2076

2079

2082

2084

2087

2090

2092

2095

L 162

2095

2098

2IOI

2103

2106

2IO9

2III

2114

2117

2119

2122

If l63

2122

2125

2127

2130

2133

2135

2I38

2140

2143

2146

2148

164

2148

2151

2154

2156

2159

2l62

2164

2167

2170

2173'

2175

165

2175

2177

2180

2183

2185

2188

2191

2193

2196

2198

22OI

166

22OI

2204

2206

22O9

2212

2214

2217

2219

2222

2225

2227

167

2227

2230^

2232

2235

2238

224O

2243

2245

2248

2251

2253

168

2253

2256

2258

2261

2263

2266

2269

2271

2274

2276

2279

169

2279

2281

2284

2287

2289

2292

2294

2297

2299

2302

2304

170

2304

2307

23IO

23I2

2315

2317

2320

2322

2325

2327

2330

171

2330

2333

2335

2338

2340

2343

2345

2348

2350

2353

2355

172

2355

2358

2360

2363

2365

2368

2370

2373

2375

2378

2380

173

2380

2383

2385

2388

2390

2393

2395

2398

2400

2403

2405

174

2405

2408

24IO

24U

2415

2418

2420

2423

2425

2428

2430

17?

2430

2433

2435

2438

2440

2443

2445

2448

2450

2453

2455

176

177

2480

2458 2482

2460 2485

2463 2487

2465 2490

2467 2492

2470 2494

2472 2497

2475 2499

2477 2502

2480 2504

178 179

2504 2529

2507 2531

2509 2533

25I2 2536

2514 2538

2516 2541

2519

2543

2521 2545

2524 2548

2526 2550

2529 2553

ia°

2553

2555

2558

2560

2562

2565

2567

2570

2572

2574

2577

III 183 184

2|77 26OI 2625 2648

2579 2603 2627 2651

2582 2605 2629 2653

2632 2655

2586 26lO 2634 2658

2589 2613 2636 2660

2591 2615

2662

2594 2617 2641 2665

2643 2667

2598 2622 2646 2669

2601 2625 2648 2672

185

2672

2674

2676

2679

268l

2683

2686

2688

2690

2693

2695

186

2695

2697

2700

27O2

2704

2707

2709

2711

2714

2716

27l8

187

27l8

2721

2723

2725

2728

2730

2732

2735

2737

2739

2742

188 189

2742 2765

2744 2767

2746 2769

2749 2772

2751 2774

2753 2776

2755 2778

275» 2781

2760 2783

2762 2785

2765 2788

190

191

2788 28lO

2790 2813

2792 28lC

2794 2817

2797 2819

2799

2822

2801 2824

2804 2826

2806

2828

2808 2831

28lO 2833

192 J93

2833 2856

2835 2858

2838

2840 2862

2842 2865

2844 2867

2869

2849 2871

2851 2874

2853 2876

2856 2878

194

2878

2880

2882

2885

2887

2889

2891

2894

2896

2898

2900

195

2900

2903

2905

2907

2909

2911

2914

2916

2918

2920

2923

196

2923

2925

2927

2929

2931

2934

2936

2938

2940

2942

2945

2947

2949

2951

2953

2956

2958

2960

2962

2964

2967

198

2967

2969

2971

2973

297S

2978

2980

2982

2984

2986

2989

199

2991

2993

2995

2997

2999

3002

3004

3006

3008

3010

SMITHSONIAN TABLES.

TABLE 8. LOGARITHMS.

12 13

17 18

21 22

23 24

26 27 28 29

3i 32 33 34

46 47 48

52 53 54

8 9

0000

0414 0792

"39 1461

1761 2041 2304 2553 2788

3010 3222

3424 3617 3802

3979 415° 43H 4472 4624

4771 4914

5315

6721 6812 6902

6990 7076 7160 7243 7324

0043 0086 0128

0453 0492 0531

0828 0864 0899

1173 1206 1239

1492 1523 1553

1790 1818 1847

2068 2095 2122

233° 2355 2380

2577 2601 2625

2810 2833 2856

3032 3OC4 3075

3243 3263 3284

3444 3464 3483

3636 3655 3674

3820 3838 3856

3997 4014 4031

4166 4183 4200

433° 4346 4362

4487 4502 4518

4639 4654 4669

4786 4800 4814

4928 4942 4955

5065 5079 5092

5198 5211 5224

5328 5340 5353

5453 5465 5478

5575 5587 5599

5694 5705 57i7

5809 5821 5832

5922 5933 5944

6031 6042 6053

6138 6149 6160

6243 6253 6263

6345 6355 6365

6444 6454 6464

6542 6551 6561

6637 6646 6656

673° 6739 6749

6821 6830 6839

6911 6920 6928

6998 7007 7016

7084 7093 7101

7168 7177 7185

7251 7259 7267

7332 7340 7348

OI7O O2I2 0253

0569 0607 0645

0934 0969 1004

I27I 1303 1335

1584 IOI4 1644

1875 19°3 I93T

2148 2-175 22OI

2405 2430 2455

2648 2672 2695

2878 29OO 2923

3096 3118 3139

3304 3324 3345

3502 3522 3541

3692 3711 3729

3874 3892 3909

4048 4065 4082

4216 4232 4249

4378 4393 4409

4533 4548 4564

4683 4698 4713

4829 4843 4857

4969 4983 4997

5I05 5IJ9 5*32

5237 5250 5263

5366 5378 5391

5490 5502 5514

5611 5623 563*5

5729 5740 5752

5843 58

5955

5866 5977

6064 6075 6°85

6170 6180 6191

6274 6284 6294

6375 6385 6395

6474 6484 6493

6571 6580 6590

6665 6675 6684

6758 6767 6776

6848 6857 6866

6937 6946 6955

7024 7033 7042

7110 7118 7126

7193 7202 7210

7275 7284 7292

7356 7364 7372

0294 0334 0374

0682 0719 0755

1038 1072 1106

1367 1399 M30

1673 J703 1732

1959 1987 2014

2227 2253 2270

2480 2504 2529

2718 2742 2765

2945 2967 2989

3160 3181 3201

3365 3385 3404

356o 3579 3598

3747 3766 3784

3927 3945 3962

4099 4116 4133

4265 4281 4298

4425 4440 4456

4579 4594 4609

4728 4742 4757

4871 4886 4900

5011 5024 5038

5r45 5J59 5J72

5276 5289 5302

5403 5416 5428

5527 5539 555i

5647 5658 5670

5763 5775 5786

5877 5888 5899

5988 5999 6010

6096 6107 6117

6201 6212 6222

6304 6314 6325

6405 6415 6425

65°3 65J3 6522

6599 6609 6618

6693 6702 6712

6785 6794 6803

6875 6884 6893

6964 6972 6981

7050 7059 7067

721 ^722 7235 7300 7308 7316 738o 7388 7396

P.P.

10 10

SMITHSONIAN TABLES.

TABLE 8 (continued). LOGARITHMS.

0 123 456 789

]

P. F

7404 7412 7419 7427 7435 7443 7451 7459 7466 7474

7482 7490 7497 7505 7513 7520 7528 7535 7543 7551

7559 7566 7574 7582 7589 7597 7604 7612 7619 7627

7634 7642 7649 7657 7664 7672 7679 7686 7694 7701

7709 7716 7723 7731 7738 7745 7752 7760 7767 7774

7782 7789 7796 7803 7810 7818 7825 7832 7839 7846 7853 7860 7868 7875 7882 7889 7896 7903 7910 7917

4 4

7924 793 i 7938 7945 7952 7959 7966 7973 798° 7987

7993 8000 8007 8014 8021 8028 8035 8041 8048 8055

8062 8069 8075 8082 8089 8096 8102 8109 8116 8122

8129 8136 8142 8149 8156 8162 8169 8176 8182 8189

8195 8202 8209 8215 8222 8228 8235 8241 8248 8254

67 68 69

8261 8267 8274 8280 8287 8293 8299 8306 8312 8319 8325 8331 8338 8344 8351 8357 8363 8370 8376 8382 8388 8395 8401 8407 8414 8420 8426 8432 8439 8445

2 2 2

3 3 3

8451 8457 8463 8470 8476 8482 8488 8494 8500 8506

8513 8519 8525 8531 8537 8543 8549 8555 8561 8567

72 73

8573 8579 8585 8591 8597 8603 8609 8615 8621 8627 8633 8639 8645 8651 8657 8663 8669 8675 8681 8686

2 2

3 3

8692 8698 8704 8710 8716 8722 8727 8733 8739 8745

8751 8756 8762 8768 8774 8779 8785 8791 8797 8802

8808 8814 8820 8825 8831 8837 8842 8848 8854 8859

8865 8871 8876 8882 8887 8893 8899 8904 8910 8915

8921 8927 8932 8938 8943 8949 8954 8960 8965 8971

8976 8982 8987 8993 8998 9004 9009 9015 9020 9025

9031 9036 9042 9047 9053 9058 9063 9069 9074 9079

9085 9090 9096 9101 9106 9112 9117 9122 9128 9133

9138 9143 9149 9154 9159 9165 9170 9175 9180 9186

9191 9196 9201 9206 9212 9217 9222 9227 9232 9238

9243 9248 9253 9258 9263 9269 9274 9279 9284 9289 (

9294 9299 9304 9309 9315 9320 9325 9330 9335 9340

9345 9350 9355 936° 93^5 937° 9375 938° 9385 939°

9395 9400 9405 94io 9415 9420 9425 9430 9435 9440

9445 9450 9455 9460 9465 9469 9474 9479 9484 9489

9494 9499 9504 9509 9513 9518 9523 9528 9533 9538

9542 9547 9552 9557 95^2 9566 957i 9576 9581 95^6

959° 9595 9600 9605 9609 9614 9619 9624 9628 9633

9638 9643 9647 9652 9657 9661 9666 9671 9675 9680

9685 9689 9694 9699 9703 9708 9713 9717 9722 9727

9731 9736 974i 9745 9750 9754 9759 9763 9768 9773

9777 9782 9786 9791 9795 9800 9805 9809 9814 9818

9823 9827 9832 9836 9841 9845 9850 9854 9859 9863

9868 9872 9877 9881 9886 9890 9894 • 9899 9903 9908

9912 9917 9921 9926 9930 9934 9939 9943 9948 9952

9956 9961 9965 9969 9974 9978 9983 9987 9991 9996

SMITHSONIAN TABLES.

TABLE 9. ANTILOGARITHMS.

Onoo 456 7 ft Q

]

3. p

JL « O Tt *J W / O *r

.00

IOOO IOO2 IOO5 IOO7 IOO9 IOI2 IOI4 IOl6 IOI9 IO2I

.01

1023 1026 1028 1030 1033 1035 IO38 IO4° IO42 IO45

.02

1047 I05° I052 I054 I057 I059 Io62 Io64 Io67 Io69

•°3

1072 1074 1076 1079 1081 1084 1086 1089 1091 1094

.04

1096 IO99 IIO2 IIO4 IIO7 IIO9 1 1 12 III4 III7 III9

.05

1122 1125 1127 1130 1132 1135 1138 1140 1143 IT46

.06

1148 1151 1153 1156 1159 1161 1164 1167 1169 1172

.07

1175 1178 1180 1183 1186 1189 1191 1194 1197 1199

.08

I2O2 I2O5 I2O8 I2II 1213 I2l6 1219 1222 1225 1227

.09

1230 1233 1236 1239 1242 1245 1247 1250 1253 1256

.10

1259 1262 1265 1268 1271 1274 1276 1279 1282 1285

.11

1288 1291 1294 1297 1300 1303 1306 1309 1312 1315

.12

1318 1321 1324 1327 1330 1334 1337 1340 1343 1346

•13

1349 1352 1355 1358 1361 1365 1368 1371 1374 1377

.14

1380 1384 1387 1390 1393 1396 1400 1403 1406 1409

.15

1413 I4l6 1419 1422 1426 1429 1432 1435 1439 1442

.16

1445 *449 US2 M55 J459 1462 1466 1469 1472 1476

•17 .18

1479 J483 J486 1489 1493 M96 1500 1503 1507 1510 i5H 15*7 *521 *524 i528 I531 ^535 J538 *542 1545

2 2

.19

1549 1552 1556 1560 1563 1567 1570 1574 1578 1581

.20

J585 *589 T592 T59^ 1600 1603 1607 1611 1614 1618

.21

1622 1626 1629 1633 1637 1641 1644 1648 1652 1656

.22

1660 1663 1667 1671 1675 J679 1683 1687 1690 1694

•23

1698 1702 1706 1710 1714 1718 1722 1726 1730 1734

.24

1738 1742 1746 1750 1754 1758 1762 1766 1770 1774

.25

1778 1782 1786 1791 1795 J799 l8°3 l8°7 l8ir l8l6

.26

1820 1824 1828 1832 1837 1841 1845 1849 1854 1858

.27

1862 1866 1871 1875 l879 l884 l888 l892 l897 I9°I

.28

1905 1910 1914 1919 1923 1928 1932 1936 1941 1945

.29

J95° 1954 T959 J9^3 I9^8 *972 J977 I982 J98^ J99*

.30

•3i

1995 2000 2004 2009 2014 2018 2023 2028 2032 2037 2042 2046 2051 2056 2061 2065 2070 2075 2°8° 2084

I I

2 2

•32

2089 2094 2099 2104 2109 2113 2118 2123 2128 2133

•33

2138 2143 2148 2153 2158 2163 2168 2173 2178 2183

•34

2188 2193 2198 2203 2208 2213 2218 2223 2228 2234

.35

2239 2244 2249 2254 2259 2265 2270 2275 2280 2286

•36

2291 2296 2301 2307 2312 2317 2323 2328 2333 2339

2344 2350 2355 2360 2366 2371 2377 2382 2388 2393

.38

2399 2404 2410 2415 2421 2427 2432 2438 2443 2449

•39

2455 2460 2466 2472 2477 2483 2489 2495 25°° 25°6

.40

2512 2518 2523 2529 2535 2541 2547 2553 2559 2564

.41

2570 2576 2582 2588 2594 2600 2606 2612 2618 2624

.42

2630 2636 2642 2649 2655 266r 2667 2673 2^79 2685

•43

2692 2698 2704 2710 2716 2723 2729 2735 2742 2748

.44

2754 2761 2767 2773 2780 2786 2793 2799 2805 2§I2

.45

2818 2825 2831 2838 2844 285r 2858 28^4 287r 2877

.46

2884 2891 2897 2904 2911 2917 2924 2931 2938 2944

•47

2951 2958 2965 2972 2979 2985 2992 2999 3006 3013

.48

3020 3027 3034 3041 3048 3055 3062 3069 3076 3083

•49

3090 3097 3105 3112 3119 3126 3133 3141 3H8 3'55

SMITHSONIAN TABLES.

TABLE 9 (continued).

ANTILOGARITHMS.

0 123 456 789

]

P.I

.50

3162 3170 3177 3184 3192 3199 3206 3214 3221 3228

•Si

3236 3243 3251 3258 3266 3273 3281 3289 3296 3304

•52

3311 33*9 3327 3334 3342 3350 3357 3365 3373 3381

•53

3388 3396 3404 3412 3420 3428 3436 3443 3451 3459

•54

3467 3475 3483 3491 3499 35°8 3516 3524 3532 3540

.55

3548 3556 3565 3573 358i 35§9 3597 3606 3614 3622

.56

363r 3639 3648 3656 3664 3673 3681 3690 3698 3707

•57

37i5 3724 3733 374i 3750 3758 3767 3776 3784 3793

.58

3802 3811 3819 3828 3837 3846 3855 3864 3873 3882

•59

3890 3899 3908 3917 3926 3936 3945 3954 39^3 3972

.60

3981 3990 3999 4009 4018 4027 4036 4046 4055 4064

.61

4074 4083 4093 4102 4111 4121 4130 4140 4150 4159

.62

4169 4178 4188 4198 4207 4217 4227 4236 4246 4256

•63 .64

4266 4276 4285 4295 4305 4315 4325 4335 4345 4355 4365 4375 4385 4395 44°6 44^ 4426 4436 4446 4457

2 2

3 3

4 4

5 5

.65

.66

4467 4477 4487 4498 4508 4519 4529 4539 4550 4560 4571 4581 4592 4603 4613 4624 4634 4645 4656 4667

2 2

3 3

4 4

5 5

.67

4677 4688 4699 4710 4721 4732 4742 4753 4764 4775

.68

4786 4797 4808 4819 4831 4842 4853 4864 4875 4887

•69

4898 4909 4920 4932 4943 4955 4966 4977 4989 5000

.70

5012 5023 5035 5047 5058 5070 5082 5093 5105 5117

•71

5129 5140 5152 5164 5176 5188 5200 5212 5224 5236

.72

5248 5260 5272 5284 5297 5309 5321 5333 5346 5338

•73

5370 5383 5395 5408 5420 5433 5445 5458 5470 5483

•74

5495 55°8 552i 5534 5546 5559 5572 5585 5598 5610

.75

5623 5636 5649 5662 5675 5689 5702 5715 5728 5741

.76

5754 5768 5781 5794 5808 5821 5834 5848 5861 5875

5888 5902 5916 5929 5943 5957 5970 5984 5998 6012 6026 6039 6053 6067 6081 6095 6109 6124 6138 6152

3 3

4 4

7 7

•79

6166 6180 6194 6209 6223 6237 6252 6266 6281 6295

.80

6310 6324 6339 6353 6368 6383 6397 6412 6427 6442

.81

6457 6471 6486 6501 6516 6531 6546 6561 6577 6592

.82

6607 6622 6637 6653 6668 6683 6699 6714 6730 6745

•83

6761 6776 6792 6808 6823 6839 6855 6871 6887 6902

.84

6918 6934 6950 6966 6982 6998 7015 7031 7047 7063

.85

7079 7096 7112 7129 7145 7161 7178 7194 7211 7228

.86

7244 7261 7278 7295 7311 7328 7345 7362 7379 7396

.87 .88

7413 7430 7447 7464 7482 7499 75J6 7534 755' 7568 7586 7603 7621 7638 7656 7674 7691 7709 7727 7745

2 2

3 4

5 5

9 9

.89

7762 7780 7798 7816 7834 7852 7870 7889 7907 7925

.90

7943 7962 7980 7998 8017 8035 8054 8072 8091 8110

.91

8128 8147 8166 8185 8204 8222 8241 8260 8279 8299

.92

83l8 8337 8356 8375 8395 8414 8433 8453 8472 8492

•93

8511 8531 8551 8570 8590 8610 8630 8650 8670 8690

•94

8710 8730 8750 8770 8790 8810 8831 8851 8872 8892

.95

.96

8913 8933 8954 8974 8995 9016 9036 9057 9078 9099 9120 9141 9162 9183 9204 9226 9247 9268 9290 9311

2 2

4 4

6 6

'9l

9333 9354 9376 9397 9419 9441 9462 9484 9506 9528

.98 •99

955° 9572 9594 9616 9638 9661 9683 9705 9727 9750 9772 9795 9817 9840 9863 9886 9908 9931 9954 9977

2 2

9 9

II II

SMITHSONIAN TABLES.

TABLE 1O. ANTILOGARITHMS.

.900

7943

7945

7947

7949

7951

7952

7954

7956

7958

7960

7962

.901

7962

7963

7965

7967

7969

7971

7973

7974

7976

7978

7980

.902

7980

7982

7984

7985

7987

7989

7993

7995

7997

7998

•9°3

7998

8000

8002

8004

8006

8008

Sou

8013

8015

8017

.904

8017

8019

8020

8022

8024

8026

8030

8032

8033

8035

.905

8035

8037

8039

8041

8043

8045

8046

8048

8050

8052

8054

.906

8054

8056

8057

8059

8061

8063

8065

8067

8069

8070

8072

.907

8072

8074

8076

8078

8080

8082

8084

8085

8087

8089

8091

.908

8091

8093

8095

8097

8098

8100

8102

8104

8106

8108

8110

.909

8110

8111

8113

8115

8117

8119

8121

8123

8125

8126

8128

.910

8128

8130

8132

8i34

8136

8138

8140

8141

8143

8145

8147

.911

8147

8149

8151

8i53

8i55

8156

8158

8168

8162

8164

8166

.912

8166

8168

8170

8171

8i73

8i75

8i77

8179

8181

8183

8185

•9*3

8185

8187

8188

8190

8192

8194

8196

8198

8200

8202

8204

.914

8204

8205

8207

8209

8211

8213

8215

8217

8219

8221

8222

.915

8222

8224

8226

8228

8230

8232

8234

8236

8238

8239

8241

.916

8241

8243

8245

8247

8249

8251

8253

8255

8257

8258

8260

.917

8260

8262

8264

8266

8268

8270

8272

8274

8276

8278

8279

.918

8279

8281

8283

8285

8287

8289

8291

8293

8295

8297

8299

.919

8299

8300

8302

8304

8306

8308

8310

8312

8314

8316

8318

.920

.921

8318 8337

8320 8339

8321 8341

8323 8343

8325 8344

8327 8346

8329

8348

833i 8350

8333 8352

8335 8354

8337 8356

.922

8356

8358

8360

8362

8364

8366

8368

8370

8371

8373

8375

•923

8375

8377

8379

8381

8383

8385

8387

8389

8391

8393

8395

.924

8395

8397

8398

8400

8402

8404

8406

8408

8410

8412

8414

.925

8414

8416

8418

8420

8422

8424

8426

8428

8429

8431

8433

.926 .927

8433 8453

8435 8455

8437 8457

8439 8459

8441 8461

8443 8463

8445 8464

8447 8466

8449 8468

8451 8470

8453 8472

.928

8472

8474

8476

8478

8480

8482

8484

8486

8488

8490

8492

.929

8492

8494

8496

8498

8500

8502

8504

8506

8507

8509

8511

.930

•931

8511 8531

8513 8533

8515 8535

851?

8537

8519 8539

8521 8541

8523

8543

8525 8545

8527 8547

8529

8549

853i

8551

•932

8551

8553

8555

8557

8559

8561

8562

8564

8566

8568

8570

•933

8570

8572

8574

8576

8578

8580

8582

8584

8586

8588

8590

•934

8590

8592

8594

8596

8598

8600

8602

8604

8606

8608

8610

.935

8610

8612

8614

8616

8618

8620

8622

8624

8626

8628

8630

•936

8630

8632

8634

8636

8638

8640

8642

8644

8646

8648

8650

•937

8650

8652

8654

8656

8658

8660

8662

8664

8666

8668

8670

•938

8670

8672

8674

8676

8678

8680

8682

8684

8686

8688

8690

•939

8690

8692

8694

8696

8698

8700

8702

8704

8706

8708

8710

.940

8710

8712

8714

8716

8718

8720

8722

8724

8726'

8728

8730

.941

8730

8732

8734

8736

8738

8740

8742

8744

8746

8748

8750

.942

875°

8752

8754

8756

8758

8760

8762

8764

8766

8768

8770

•943

8770

8772

8774

8776

8778

8780

8782

8784

8786

8788

8790

.944

8790

8792

8794

8796

8798

8800

8802

8804

8806

8808

8810

.945

8810

8813

8815

8817

8819

8821

8823

8825

8827

8829

8831

.946

8831

8833

8835

8837

8839

8841

8843

8845

8847

8849

8851

•947

8851

8853

8855

8857

8839

8861

8863

8865

8867

8870

8872

.948

8872

8874

8876

8878

8880

8882

8884

8886

8888

8890

8892

•949

8892

8894

8896

8898

8900

8902

8904

8906

8908

8910

8913

SMITHSONIAN TABLES.

TABLE 1 O (continued). ANTILOGARITHMS,

.950

•95 i

8913

8933

8915 8935

8917

8937

8919

8939

8921 8941

8923 8943

8925 8945

8927 8947

8929 8950

8931 8952

8933 8954

•952

8954

8956

8958

8960

8962

8964

8966

8968

8970

8972

8974

•953

8974

8976

8978

8980

8983

8985

8987

8989

8991

8993

8995

•954

8995

8997

8999

9001

9003

9005

9007

9009

9012

9014

9016

.955

9016

9018

9020

9022

9024

9026

9028

9030

9032

9034

9036

•956

9036

9°39

9041

9043

9045

9047

9049

9051

9053

9055

9057

•957

9057

9°59

9061

9064

9066

9068

9070

9072

9074

9076

9078

•958

9078

9080

9082

9084

9087

9089

9091

9°93

9095

9097

9099

•959

9099

9101

9103

9J°5

9108

9110

9112

9114

9116

9118

9120

.960

9120

9122

9124

9126

9129

9i3i

9133

9135

9U7

9J39

9141

.961

9141

9H3

9M5

9H7

9r5°

9152

9!54

9*56

9158

9160

9162

.962

9162

9164

9166

9169

9171

9*73

917S

9177

9179

9181

9183

•963

9183

9185

9188

9190

9192

9194

9196

9198

9200

9202

9204

.964

9204

9207

9209

9211

9213

9215

9217

9219

9221

9224

9226

.965

9226

9228

9230

9232

9234

9236

9238

9241

9243

9245

9247

.966

9247

9249

9251

9253

9256

9258

9260

9262

9264

9266

9268

•967

9268

9270

9273

9275

9277

9279

9281

9283

9285

9288

9290

.968

9290

9292

9294

9296

9298

9300

9303

9305

93°7

93°9

9311

.969

9311

9313

9315

93i8

9320

9322

9324

9326

9328

9330

9333

.970

9333

9335

9337

9339

934i

9343

9345

9348

935°

9352

9354

.971

9354

9356

9358

9361

9363

9365

9367

9369

937i

9373

9376

.972

9376

9378

9380

9382

9384

9386

9389

939i

9393

9395

9397

•973

9397

9399

9402

9404

9406

9408

9410

9412

94i5

94i7

9419

•974

9419

9421

9423

9425

9428

943°

9432

9434

9436

9438

9441

.975

9441

9443

9445

9447

9449

9451

9454

9456

9458

9460

9462

.976

9462

9465

9467

9469

947i

9473

9475

9478

9480

9482

9484

•977

9484

9486

9489

949 1

9493

9495

9497

9499

9502

9504

9506

.978

9506

9508

9510

95*3

9515

951?

95J9

952i

9524

9526

9528

•979

9528

953°

9532

9535

9537

9539

954i

9543

9546

9548

9550

980

955°

9552

9554

9557

9559

956i

95g3

9565

9568

9570

9572

.981

9572

9574

9576

9579

95»i

9583

9585

9587

959°

9592

9594

.982

9594

9598

9601

9603

9605

9607

9609

9612

9614

9616

•983

9616

9618

9621

9623

9625

9627

9629

9632

9634

9636

9638

.984

9638

9641

9643

9645

9647

9649

9652

9654

9656

9658

9661

.985

9661

9663

9665

9667

9669

9672

9674

9676

9678

9681

9683

.986

9683

9685

9687

9689

9692

9694

9696

9698

9701

9703

9705

! -987

9705

9707

9710

9712

97 T4

9716

97 19

9721

9723

9725

9727

.988 .989

9727 9750

9730 9752

9732 9754

9734 9757

9736 9759

9739 9761

974i 9763

9743 9766

9745 9768

9748 9770

975° 9772

.990

9772

9775

9777

9779

9781

9784

9786

9788

9790

9793

9795

•99 i

9795

9797

9799

9802

9804

9806

9808

9811

9813

9815

9817

.992

9817

9820

9822

9824

9827

9829

9831

9833

9838

9840

•993

9840

9842

9845

9847

9849

9851

9854

9856

9858

9861

9863

•994

9863

9865

9867

9870

9872

9874

9876

9879

9881

9883

9886

.995

9886

9888

9890

9892

9895

9897

9899

9901

9904

9906

9908

.996

9908

9911

9913

9915

9917

9920

9922

9924

9927

9929

993i

•997

993 i

9933

9936

9938

9940

9943

9945

9947

9949

9952

9954

•998

9954

99 56

9959

9961

9963

9966

996g

9970

9972

9975

9977

•999

9977

9979

9982

9984

9986

9988

999 i

9993

9995

9998

oooo

SMITHSONIAN TABLES.

TABLE 11. CIRCULAR (TRIGONOMETRIC) FUNCTIONS.

(Taken from B. O. Peirce's " Short Table of Integrals," Ginn & Co.)

1 .

i ™

SINES.

COSINES.

TANGENTS.

COTANGENTS.

§£

«W

Nat. Log.

o.oooo

0°00'

.OOOO 00

I. OOOO O.OOOO

.OOOO 00

00 00

9o°oo'

1.5708

0.0029

.0029 7.4637

I. OOOO .OOOO

.0029 7.4637

343-77 2.5363

5°

0.0058

.0058 .7648

I. OOOO .OOOO

.0058 .7648

171.89 .2352

1.565°

0.0087

3°

.0087 .9408

I. OOOO .OOOO

.0087 .9409

114.59 -0591

3°

1.5621

0.0116

.0116 8.0658

.9999 .0000

.0116 8.0658

85.940 1.9342

J«5592

0.0145

.0145 .1627

.9999 .0000

.0145 .1627

68.750 .8373

^5563

0.0175

I°00'

.0175 8.2419

.9998 9.9999

.0175 8.2419

57.290 1.7581

S9°oo'

1-5533

0.0204

.0204 .3088

.9998 .9999

.0204 .3089

49.104 .6911

5°

1-5504

0.0233

.0233 .3668

•9997 -9999

•0233 .3669

42.964 .6331

!-5475

0.0262

.0262 .4179

•9997 -9999

.0262 .4181

38.188 .5819

3°

1.5446

0.0291

.0291 .4637

.9996 .9998

.0291 .4638

34.368 .5362

1 0.0320

.0320 .5050

•9995 -9998

•0320 .5053

31.242 .4947

1^5388

0.0349

2°00'

.0349 8.5428

•9994 9-9997

•0349 8.5431

28.636 1.4569

88°oo'

J-5359

0.0378

-0378 .5776

•9993 -9997

•0378 .5779

26.432 .4221

5°

I-533°

0.0407

.0407 .6097

.9992 .9996

.0407 .6101

24-542 .3899

0.0436

3°

.0436 .6397

.9990 .9996

.0437 .6401

22.904 .3599

3°

1.5272

0.0465

.0465 .6677

•9989 -9995

.0466 .6682

21.470 .3318

1-5243

0.0495

.0494 .6940

.9988 .9995

.0495 .6945

20.206 .3055

1-5213

0.0524

3°oo'

.0523 8.7188

.9986 9.9994

.0524 8.7194

19.081 1.2806

87°oo'

1.5184

0-0553

.0552 .7423

•9985 -9993

•0553 -7429

18.075 -2571

5°

I-5I55

0.0582

.0581 .7645

.9983 .9993

.0582 .7652

17.169 .2348

1.5126

0.06 u

3°

.0610 .7857

.9981 .9992

.0612 .7865

16.350 .2135

3°

I-5097

0.0640

.0640 .8059

-998o .9991

.0641 .8067

15.605 .1933

1.5068

0.0669

5°

.0669 .8251

.9978 .9990

.0670 .8261

14.924 .1739

T-5°39

0.0698

4°oo/

.0698 8.8436

.9976 9.9989

.0699 8.8446

14.301 1.1554

86°oo'

1.5010

0.0727

.0727 .8613

•9974 -9989

.0729 .8624

J3-727 .1376

5°

1.4981

0.0756

.0756 .8783

.9971 .9988

•0758 .8795

13.197 .1205

1.4952

0.0785

.0785 .8946

•9969 -9987

.0787 .8960

12.706 .1040

1.4923

0.0814

.0814 .9104

•9967 -9986

.0816 .9118

12.251 .0882

1.4893

0.0844

.0843 .9256

.9964 -9985

.0846 .9272

11.826 .0728

1.4864

0.0873

5°oo'

•0872 8.9403

.9962 9.9983

.0875 8.9420

11.430 1.0580

85°oo'

1.4835

0.0902

•0901 .9545

•9959 -9982

.0904 .9563

11.059 .0437

i .4806

0.0931

•0929 .9682

•9957 -9981

•0934 .9701

10.712 .0299

1-4777

0.0960 0.0989

3°

.0958 .9816 •0987 -9945

-9954 .9980 •995 i -9979

.0963 .9836

.0992 .9966

10.385 .0164 10.078 .0034

1.4748 1.4719

0.1018

.1016 9.0070

.9948 .9977

.1022 9.0093

9.7882 0.9907

1.4690

0.1047 0.1076

6°oo

.1045 9.0192 .1074 .0311

•9945 9-9976 •9942 .9975

.IO5I 9.0216 .I080 .0336

9.5144 0.9784 9.2553 .9664

84°oo' 50

1.4661 1.4632

0.1105

.1103 .0426

•9939 -9973

.1110 .0453

9.0098 .9547

1.4603

0.1134

."32 .0539

.9936 .9972

.1139 .0567

8.7769 .9433

3°

1-4574

0.1164

.1161 .0648

•9932 -997I

.1169 .0678

8-5555 -9322

1-4544

0.1193

.1190 -.0755

.9929 .9969

.1198 .0786

8.3450 .9214

I-45I5

0.1222

7°oo'

.1219 9.0859

.9925 9.9968

.1228 9.0891

8.1443 0.9109

83°oo'

1.4486

O.I25I O.I28O

.1248 .0961 .1276 .1060

.9922 .9966 .9918 .9964

•I257 .0995 .1287 .1096

7.9530 -9005 7.7704 .8904

50 40

1-4457 1.4428

0.1309

.13% .1157

.9914 .9963

.1317 .1194

7.5958 .8806

3°

1-4399

0.1338

.1334 .1252

.9911 .9961

.1346 .1291

7.4287 .8709

1.4370

0.1367

•1363 -1345

•9907 -9959

•1376 .1385

7.2687 .8615

I.434I

0.1396

8°oo'

•1392 9-1436

•9903 9-9958

.1405 9.1478

7.1154 0.8522

82°00'

1.4312

0.1425

.1421 .1525

.9899 .9956

-1435 ^569

6.9682 .8431

1.4283

0.1454

.1449 .1612

•9894 -9954

.1465 .1658

6.8269 .8342

1.4254

0.1484

3°

.1478 .1697 .1507 .1781

.9890 .9952 .9886 .9950

• 1495 -T745 .1524 .1831

6.6912 .8255 6.5606 .8169

1.4224

0.1542

.1536 .1863

.9881 .9948

.1554 .1915

6.4348 .8085

1.4166

0.1571

9°oo'

.1564 9.1943

.9877 9.9946

.1584 9.1997

6.3138 0.8003

8i°oo'

I.4I37

Nat. Log.

c/5

COSINES.

SINES.

COTAN- GENTS.

TANGENTS.

Qti

•j?

SMITHSONIAN TABLES.

TABLE 1 1 (continued). CIRCULAR (TRIGONOMETRIC) FUNCTIONS.

C/3

SINES.

COSINES.

TANGENTS.

COTANGENTS.

Nat. Log.

0.1571

9°oo/

.1564 9.1943

.9877 9.9946

.1584 9.1997

6.3138 0.8003

8i°oo/

1.4137

0.1600

.1593 .2022

.9872 .9944

.1614 .2078

6.1970 .7922

5°

1.4108

0.1629

.1622 .2100

.9868 .9942

.1644 .2158

6.0844 .7842

1.4079

0.1658 0.1687

30 40

.1650 .2176 .1679 -2251

.9863 .9940 •985» .9938

.1673 -2236 .1703 .2313

5.9758 .7764 5.8708 .7687

1.4050 1.4021

0.1716

.1708 .2324

•9853 ^936

.1733 -2389

5.7694 .7611

1.3992

0.1745

I0°00'

.1736 9-2397

.9848 9.9934

.1763 9.2463

5.6713 0.7537

8o°oo'

J-3963

0.1774

.1765 .2468

•9843 -993 i

.1793 -2536

5-5764 .7464

J-3934

0.1804

.1794 .2538

.9838 .9929

.1823 .2609

5.4845 .7391

1.3904

0.1833

.1822 .2606

•9833 -9927

.1853 .2680

5-3955 -7320

I-3875

0.1862

.1851 .2674

.9827 .9924

.1883 .2750

5-3093 -7250

1.3846

0.1891

5°

.1880 .2740

.9822 .9922

.1914 .2819

5.2257 .7181

1.3817

0.1920

1 I°00'

.1908 9.2806

.9816 9.9919

.1944 9.2887

5.1446 0.7113

79°oo'

1.3788

0.1949

.1937 .2870

.9811 .9917

•1974 .2953

5.0658 .7047

1-3759

0.1978

.1965 .2934

.9805 .9914

.2004 .3020

4.9894 .6980

1-3730

0.2007

3°.

.1994 .2997

•9799 -99 J 2

•2035 -3085

4.9152 .6915

3°

1.3701

0.2036

.2022 .3058

•9793 -9909

.2065 .3149

4.8430 .68 c i

1.3672

0.2065

.2051 .3119

.9787 .9907

.2095 -3212

4.7729 .6788

1-3643

0.2094

I2°00/

.2079 9.3179

.9781 9.9904

.2126 9.3275

4.7046 0.6725

78°oo'

1.3614

0.2123

.2108 .3238

•9775 -9901

.2156 .3336

4.6382 .6664

5°

1-3584

0-2153

.2136 .3296

.9769 .9899

.2186 .3397

4.5736 .6603

J-3555

0.2182

3°

•2164 -3353

.9763 .9896

•2217 .3458

4.5107 .6542

1-3526

0.221 1

.2193 .3410

•9757 .9893

-2247 .35J7

4.4494 .6483

O.224O

.2221 .3466

.9750 .9890

.2278 .3576

4.3897 .6424

1^3468

O.2269

13000'

.2250 9.3521

.9744 9.9887

.2309 9.3634

4.3315 0.6366

77°oo'

J-3439

0.2298

•2278. .3575

•9737 -9884

•2339 .3691

4.2747 .6309

5°

1.3410

0.2327

.2306 .3629

.9730 .9881

.2370 .3748

4.2193 .6252

0.2356

.2334 .3682

.9724 .9878

.2401 .3804

4.1653 .6196

3°

J-3352

0.2385

-2363 -3734

•9717 -9875

.2432 .3859

4.1126 .6141

J-3323

0.2414

5°

•2391 -3786

.9710 .9872

.2462 .3914

4.0611 .6086

i-3294

0.2443

i4°oo'

.2419 9.3837

•97°3 9-9869

.2493 9.3968

4.0108 0.6032

76°oo'

1-3265

0.2473

.2447 .3887

.9696 .9866

.2524 .4021

3-96i7 -5979

1.3235

O.25O2

-2476 -3937

.9689 .9863

.2555 .4074

3.9136 .5926

1.3206

0.2531

.2504 .3986

.9681 .9859

.2586 .4127

3.8667 .5873

3°

1.3177

0.2560 0.2589

40 50

.2532 .4035 .2560 .4083

.9674 .9856 .9667 .9853

.2617 .4178 .2648 .4230

3.8208 .5822 3.7760 .5770

1.3148 1.3119

0.26l8

i5°oo>

.2588 9.4130

.9659 9-9849

.2679 9.4281

3.7321 0.5719

75°oo'

1.3090

0.2647

.2616 .4177

.9652 .9846

.2711 .4331

3.6891 .5669

5°

1.3061

0.2676

.2644 4223

.9644 .9843

.2742 .4381

3.6470 .5619

1.3032

0.2705

.2672 .4269

.9636 .9839

•2773 -443°

3-6059 -5570

1-3003

0.2734

.2700 .4314

.9628 .9836

.2805 .4479

3.5656 .5521

1.2974

0.2763

.2728 .4359

.9621 .9832

.2836 .4527

3.5261 .5473

1.2945

0.2793

i6°oo'

.2756 9.4403

.9613 9.9828

.2867 94575

3.4874 0.5425

74°oo'

1.2915

O.2822

.2784 .4447

.9605 .9825

.2899 .4622

3-4495 -5378

1.2886

0.2851

.2812 .4491

.9596 .9821

.2931 .4669

3.4124 .5331

1.2857

0.2880

.2840 .4533

.9588 .9817

.2962 .4716

3-3759 -5284

1.2828

0.2909

.2868 .4576

.9580 .9814

.2994 .4762

3.3402 .5238

1.2799

0.2938

5°

.2896 .4618

.9572 .9810

.3026 .4808

3.3052 .5192

1.2770

0.2967 0.2996

i7°oo'

.2924 9.4659 .2952 .4700

•9563 9-98o6 •9555 -9802

.3057 9-4853 .3089 .4898

3.2709 0.5147 3.2371 .5102

73°oo/ 50

1.2741 1.2712

0.3025

.2979 .4741

•9546 .9798

.3121 .4943

3.2041 .5057

1.2683

0.3054

3°

.3007 .4781

•9537 .9794

•3 '53 -4987

3.1716 .5013

1.2654

0.3083

.3035 .4821

.9528 .9790

•3185 .5031

3.1397 .4969

1.2625

0.3H3

.3062 .4861

.9520 .9786

•3217 .5075

3.1084 .4925

1-2595

0.3142

i8°oo'

.3090 9.4900

.9511 9.9782

.3249 9.5118

3.0777 0.4882

72°00'

1.2566

Nat. Log.

QC/5

COSINES.

SINES.

COTAN- GENTS.

TANGENTS

SMITHSONIAN TABLES.

TABLE 1 1 (continued). CIRCULAR (TRIGONOMETRIC) FUNCTIONS..

SINES.

COSINES.

TANGENTS.

COTANGENTS.

Get o

Nat. Log.

0.3142

i8°oo'

.3090 9.4900

.9511 9.9782

.3249 9.5118

3.0777 0.4882

72°00'

.2566

0.3171

.3118 .4939

.9502 .9778

.3281 .5161

3.0475 .4839

5°

•2537

0.3200

.3145 .4977

•9492 -9774

•3314 .5203

3.0178 .4797

.2508

0.3229

3°

•3J73 -5OI5

.9483 .9770

•3346 .5245

2.9887 .4755

.2479

0.3258

.3201 .5052

•9474 -9765

•3378 .5287

2.9600 .4713

•2450

0.3287

5°

.3228 .5090

•9465 .9761

•3411 .5329

2.9319 .4671

.2421

0.3316

i9°oo'

.3256 9.5126

•9455 9-9757

•3443 9-5370

2.9042 0.4630

7i°oo/

.2392

0-3345

•3283 -5163

.9446 .9752

•3476 .5411

2.8770 .4589

5°

.2363

0-3374

•33" -5199

.9436 .9748

.3508 .5451

2.8502 .4549

•2334

0.3403

•3338 .5235

.9426 .9743

•3541 -5491

2.8239 .4509

•2305

0-3432

.3365 -5270

•9417 -9739

•3574 -5531

2.7980 .4469

.2275

0.3462

•3393 -5306

.9407 .9734

•3607 -557I

2.7725 .4429

.2246

0.3491

20°00'

.3420 9.5341

•9397 9-9730

.3640 9.5611

2.7475 04389

7o°oo'

.2217

0.3520

•3448 .5375

•9387 -9725

•3673 -5650

2.7228 .4350

.2188

Q-3549

-3475 -5409

•9377 -9721

.3706 .5689

2.6985 .4311

.2159

0.3578

.3502 .5443

.9367 .9716

•3739 -5727

2.6746 .4273

.2130

0.3607

.3529 -5477

.9356 .9711

.3772 .5766

2.6511 .4234

.2IOI

0.3636

•3557 .55*0

•934$ -9706

.3805 .5804

2.6279 .4196

.2072

0.3665

2I°00'

•3584 9-5543

.9336 9.9702

•3839 9-5842

2.6051 0.4158

69°oo'

.2043

0.3694

•3611 -5576

•9325 -9697

.3872 .5879

2.5826 .4121

5°

.2014

0.3723

.3638 .5609

.9315 .9692

.3906 .5917

2.5605 .4083

.1985

0.3752

.3665 .5641

.9304 .9687

•3939 -5954

2.5386 .4046

.1956

0.3782

.3692 .5673

.9293 .9682

•3973 .5991

2.5172 .4009

.1926

0.3811

.3719 .5704

.9283 .9677

.4006 .6028

2.4960 .3972

.1897

0.3840

22°OO'

.3746 9-5736

.9272 9.9672

.4040 9.6064

2.4751 0.3936

68°oo'

.1868

0.3869

•3773 -5767

.9261 .9667

.4074 .6100

2-4545 -3900

5°

.1839

0.3898

.3800 .5798

.9250 .9661

.4108 .6136

2.4342 .3864

.1810

0.3927

3°

.3827 .5828

.9239 -9656

.4142 .6172

2.4142 .3828

.1781

0.3956

.3854 -5859

.9228 .9651

.4176 .6208

2-3945 -3792

•1752

0.39^5

.3881 .5889

.9216 .9646

.4210 .6243

2.3750 -3757

•1723

0.4014

23°00'

.3907 9-59!9

.9205 9.9640

.4245 9.6279

2-3559 0.3721

67°oo'

.1694

0.4043

•3934 .5948

.9194 .9635

.4279 .6314

2.3369 -3686

.1665

0.4072 0.4102

20 3°

.3961 .5978 .3987 .6007

.9182 .9629 .9171 .9624

.4314 .6348 .4348 .6383

2.3183 .3652 2.2998 .3617

40 30

.1636 .1606

0.4131 0.4160

40 50

.4014 .6036 .4041 .6065

.9159 .9618 .9147 .9613

.4383 .6417 .4417 .6452

2.2817 .3583 2-2637 .3548

20 IO

.1577 .1548

0.4189

24°00'

.4067^ 9.6093

•9135 9-96o7

.4452 9.6486

2.2460 0.3514

66°oo'

-1S19

0.4218

.4094 .6121

.9124 .9602

.4487 .6520

2.2286 .3480

5°

.1490

0.4247

.4120 .6149

.9112 .9596

.4522 .6553

2.2113 .3447

.1461

0.4276

3°

.4147 .6177

.9100 .9590

.4557 .6587

2.1943 .3413

.1432

0-4305 0.4334

40 50

.4173 .6205 .4200 .6232

.9088 .9584 •9075 -9579

.4592 .6620 .4628 .6654

2-1775 -3380 2.1609 .3346

20 10

.1403 •1374

o-4363

25°00'

.4226 9.6259

•9063 9-9573

.4663 9.6687

2.1445 o-33T3

65°oo'

•1345

0.4392

.4253 .6286

.9051 .9567

.4699 .6720

2.1283 .3280

•'316

0.4422

.4279 .6313

.9038 .9561

•4734 -6752

2.1123 .3248

.1286

0.4451

•43°5 -6340

.9026 .9555

.4770 .6785

2.0965 .3215

3°

•1257

0.4480

.4331 .6366

.9013 .9549

.4806 .6817

2.0809 -3l83

.1228

0.4509

.4358 .6392

.9001 .9543

.4841 .6850

2.0655 -3150

.1199

0.4538

26°00'

.4384 9.6418

•8988 9-9537

.4877 9.6882

2.0503 0.3118

64°oo'

.1170

0.4567

.4410 .6444

•8975 -953°

.4913 .6914

2-0353 -3086

.1141

0.4596

.4436 .6470

.8962 .9524

.4950 .6946

2.0204 -3°54

.1112

0.4625

.4462 .6495

.8949 .9518

.4986 .6977

2.0057 .3023

3°

.1083

0.4654

.4488 .6521

.8936 .9512

.5022 .7009

1.9912 .2991

.1054

0.4683

.4514 .6546

•8923 -95°5

•5059 .7040

1.9768 .2960

.'1025

0.4712

27°00'

.4540 9.6570

.8910 9.9499

.5095 9.7072

1.9626 0.2928

63°oo'

1.0996

Nat. Log.

' w

Q^"

COSINES.

SINES.

COTAN- GENTS.

TANGENTS.

SMITHSONIAN TABLES.

TABLE 1 1 (continued). CIRCULAR (TRIGONOMETRIC) FUNCTIONS.

SINES.

COSINES.

TANGENTS.

COTANGENTS.

Qp< O

Nat. Log.

0.4712

27°00'

.4540 9.6570

.8910 9.9499

.5095 9.7072

1.9626 0.2928

63°00'

1.0996

0.4741

.4566 .6595

.8897 .9492

.5132 .7103

1.9486 .2897

1.0966

0.4771

.4592 .6620

.8884 .9486

.5169 .7134

1.9347 .2866

I-°937

0.4800

.4617 .6644

.8870 .9479

.5206 .7165

1.9210 .2835

3°

0.4829 0.4858

40 50

.4643 .6668 .4669 .6692

•885? -9473 .8843 .9466

.5243 .7196 .5280 .7226

1.9074 .2804 1.8940 .2774

2O 10

1.0879 1.0850

0.4887

28°00'

.4695 9.6716

•8829 9.9459

•5317 9.7257

1.8807 °-2743

62°00'

1.0821

0.4916

.4720 .6740

.8816. .9453

-5354 -7287

1.8676 .2713

5°

1.0792

0.4945

.4746 .6763

.8802 .9446

•5j92 -73 i 7

1.8546 .2683

1.0763

0.4974

.4772 .6787

.8788 .9439

.5430 .7348

1.8418 .2652

3°

1-0734

0.5003

.4797 .6810

.8774 .9432

-5467 .7378

1.8291 .2622

1.0705

0.5032

.4823 .6833

.8760 .9425

.5505 .7408

1.8165 .2592

1.0676

0.5061

29°00'

.4848 9.6856

.8746 9.9418

§3 9-7438

1.8040 0.2562

6i°oo'

1.0647

0.5091

.4874 .6878

.8732 .9411

i .7467

I-79I7 .2533,

5°

1.0617

0.5120

.4899 .6901

.8718 .9404

9 -7497

1.7796 .2503

1.0588

0.5149

3°

.492^ .6923

•8704 -9397

.5658 .7526

1.7675 .2474

I-°559

0.5178

.4950 .6946

.8689 .9390

-5696 .7556

1.7556 .2444

1.0530

0.5207

•4975 -6968

•8675 -9383

-5735 7585

1.7437 , .2415

1.0501

0.5236

3o°oo'

.5000 9.6990

.8660 9.9373

•5774 9-76i4

I.7J&I 0.2386

6o°oo'

1.0472

0.5265

.5025 .7012

.8646 .9368

.5812 .7644

1.7205 .2356

5°

1.0443

0.5294

•5050 -7033

•8631 .9361

•5851 -7673

1.7090 .2327

1.0414

0-5323

•5°75 -7055

•8616 .9353

.5890 .7701

1.6977. -2299

1.0385

0-5352

.5100 .7076

.8601 .9346

•5930 -7730

1.6864 .2270

1.0356

0.5381

.5125 .7097

•8587 .9338

•5969 -7759

1.6753 -2241

1.0327

0.5411

3I°00'

.5150 9.7118

•8572 9-9331

.6009 9.7788

1.6643 O.22I2

59°oo'

1.0297

0.5440

•5'75 -7139

•8557 .9323

.6048 .7816

1.6534 .2184

5°

1.0268

0.5469

.5200 .7160

•8542 .93'5

.6088 .7845

1.6426 .2155

1.0239

0.5498

3°

.5225 .7181

.8526 .9308

.6128 .7873

1.6319 .2127

1. 02 10

0-5527

.5250 .7201

.8511 .9300

.6168 .7902

I.62I2 .2098

I.OlSl

0.5556

.5275 .7222

.8496 .9292

.6208 .7930

I.6lO7 .2O7O

I.OI52

0-5585

32°00'

.5299 9.7242

.8480 9.9284

•6249 9-7958

1.6003 O.2O42

58°oo'

I.OI23

0.5614

.5324 .7262

.8465 .9276

.6289 .7986

1.5900 .2014

5°

1.0094

0.5643

.5348 .7282

.8450 .9268

.6330 .8014

1.5798 .1986

1.0065

0.5672

3°

•5373 -7302

.8434 .9260

.637 i .8042

1.5697 .1958

3°

1.0036

0.5701

.5398 .7322

.8418 .9252

.6412 .8070

'•5597 -193°

I.OOO7

0.5730

5°

.5422 .7342

.8403 .9244

.6453 .8097

1.5497 .1903

0.9977

0.5760

33000'

•5446 9-736i

-8387 9-9236

.6494 9.8125

1-5399 0.1875

57°oo'

0.9948

0.5789

•5471 -738°

.8371 .9228

•6536 .8153

1.5301 .1847

0.9919

0.5818

•5495 -7400

.8355 .9219

.657^ .8180

1.5204 .1820

0.9890

0.5847

3°

•55*9 -7419

.8339 .9211

.6619 .8208

1.5108 .1792

3°

0.9861

0.5876 0-5905

40 50

•5544 -743s .5568 .7457

-8323 -9203 .8307 .9194

.6661 .8235 .6703 .8263

1.5013 .1765 1.4919 .1737

0.9832 0.9803

0-5934

34°oo'

•5592 9-7476

.8290 9.9186

.6745 9.8290

1.4826 0.1710

56°oo'

0.9774

0-5963

.5616 .7494

.8274 .9177

.6787 .8317

1.4733 -l683

5°

0-9745

0.5992

.5640 .7513

.8258 .9169

.6830 .8344

1.4641 .1656

0.9716

0.6021

3°

.5664 .7531

.8241 .9160

.6873 .8371

1.4550 .1629

0.9687

0.6050

.5688 .7550

.8225 .9151

.6916 .8398

1.4460 .1602

0.9657

0.6080

5°

.5712 .7568

.8208 .9142

.6959 .8425

1.4370 .1575

0.9628

0.6109

35<>oo'

.5736 9.7586

.8192 9.9134

.7d02 9.8452

1.4281 0.1548

55°oo'

0.9599

0.6138 0.6167

10- 20

.5783 .7622

•8i7S -9125 .8158 .9116

.7046 .8479 .7089 .8506

1.4193 .1521 1.4106 .1494

50 40

0.9570 0.9541

0.6196

3°

.5807 .7640

.8141 .9107

•7133 -8533

1.4019 .1467

0.9512

0.6225

.5831 .7657

.8124 .9098

•7i77 -8559

1.3934 .1441

0.9483

0.6254

.5854 .7675

.8107 .9089

.7221 .8586

1.3848 .1414

0.9454

0.6283

36°oo'

.5878 9.7692

.8090 9.9080

.7265 9.8613

1.3764 0.1387

54°oo'

0.9425

Nat. Log.

5</5

COSINES.

SINES.

COTAN- GENTS.

TANGENTS.

a W

SMITHSONIAN TABLES.

TABLE 1 1 (continued). CIRCULAR (TRIGONOMETRIC) FUNCTIONS,

§*

c/5 \M

SINES.

COSINES.

TANGENTS.

COTANGENTS

«<

Qp4

Nat. Log.

0.6283

36°oo'

.5878 9.7692

.8090 9.9080

.7265 9.8613

1.3764 0.1387

54°00'

0.9425

0.6312

.5901 .7710

.8073 .9070

.7310 .8639

1.3680 .1361

0.9396

0.6341

.5925 .7727

.8056 .9061

.7355 -8666

J-3597 -1334

0.9367

0.6370

3°

.5948 .7744

•8039 .9052

.7400 .8692

1.3514 .1308

3°

0-9338

0.6400

.5972 .7761

.8021 .9042

•7445 -8718

1.3432 .1282

0.9308

0.6429

5°

•5995 -7778

.8004 .9033

.7490 .8745

1.3351 .1255

0.9279

0.6458

37°oo'

•6018 9.7795

.7986 9.9023

•7536 9-877I

1.3270 0.1229

53°oo'

0.9250

0.6487

.6041 .7811

.7969 .9014

.7581 ,8797

1.3190 .1203

5°

0.9221

0.6516

.6065 .7828

•795 l -9004

.7627 .8824

1.3111 .1176

0.9192

0.6545

.6088 .7844

•7934 -8995

.7673 .8850

1.3032 .1150

0.9163

0.6574 0.6603

.6m .7861 .6134 -7877

.7916 .8985 .7898 .8975

.7720 .8876 .7766 .8902

1.2954 .1124 1.2876 .1098

20 IO

0.9134 0.9105

0.6632

38°oo'

•6157 97893

.7880 9.8965

.7813 9.8928

1.2799 0.1072

52°00'

0.9076

0.6661

.6180 .7910

.7862 .8955

.7860 .8954

1.2723 .1046

5°

0.9047

0.6690

.6202 .7926

.7844 .8945

.7907 .8980

1.2647 •102<^>

0.9018

0.6720

3°

.6225 .7941

•7826 .8935

•7954 •9°°6

1.2572 ^994

3°

0.8988

0.6749

.6248 .7957

.7808 .8925

.8002 .9032

1.2497 70968

0.8959

0.6778

.6271 .7973

.7790 .8915

.8050 .9058

1.2423 .0942

0.8930

0.6807

39°oo'

.6293 9.7969

.7771 9.8905

.8098 9.9084

1.2349 0.0916

5i°oo'

0.8901

0.6836

.6316 .8004

7753 -8895

.8146 .9110

1.2276 .0890

0.8872

0.6865

.6338 .8020

-7735 -8884

•8i95 -9!35

1.2203 .0865

0.8843

0.6894

3°

.6361 .8035

.7716 .8874

.8243 .9161

1.2131 .0839

3°

0.8814

0.6923 0.6952

40 50

.6383 .8050 .6406 .8066

.7698 .8864 .7679 .8853

.8292 .9187 .8342 .9212

1.2059 .0813 1.1988 .0788

0.8785 0.8756

0.6981

40°oo'

.6428 9.8081

.7660 9.8843

•8391 9-9238

1.1918 0.0762

5o°oo'

0.8727

0.7010

.6450 .8096

.7642 .8832

.8441 .9264

1.1847 -0736

5°

0.8698

0.7039

.6472 .8m

.7623 .8821

.8491 .9289

1.1778 .0711

0.8668

.6494 .8125

.7604 .8810

•8541 .93r5

1.1708 .0685

0.8639

0.7127

40 50

.6517 .8140 •6539 -8155

.7585 .8800 .7566 .8789

•8591 -934I .8642 .9366

1.1640 .0659 1.1571 .0634

20 10

0.8610 0.8581

0.7156

4i°oo'

.6561 9.8169

•7547 9-8778

•8693 9-9392

1.1504*0.0608

49°oo'

0.8552

0.7185

.6583 .8184

.7528 .8767

.8744 .9417

1.1436 .0583

0.8523

0.7214

.6604 .8198

•7509 -8756

.8796 .9443

1.1369 .0557

0.8494

0.7243

3°

.6626 .8213

.7490 .8745

.8847 .9468

1-1303 -0532

0.8465

0.7272

.6648 .8227

.7470 .8733

.8899 .9494

1.1237 .0506

0.8436

0.7301

.6670 .8241

.7451 .8722

.8952 .9519

1.1171 .0481

0.8407

0-733°

42°00'

.6691 9.8255

.7431 9.8711

.9004 9.9544

1.1106 0.0456

48°oo'

0.8378

0-7359

.6713 .8269

.7412 .8699

•9°57 -9570

1.1041 .0430

5°

0.8348

0.7389

.6734 .8283

.7392 .8688

.9110 .9595

1.0977 .0405

0.8319

0.7418

.6756 .8297

•7373 -8676

.9163 .9621

1.0913 .0379

0.8290

0.7447

.6777 .8311

•7353 -8665

.9217 .9646

1.0850 .0354

0.8261

0.7476

5°

.6799 -8324

•7333 -8653

.9271 .9671

1.0786 .0329

0.8232

0-7505

43°oo'

.6820 9.8338

.7314 9.8641

•9325 9-9697

1.0724 0.0303

47°oo'

0.8203

0-7534

.6841 .8351

.7294 .8629

.9380 .9722

1.0661 .0278

0.8174

0.7563

.6862 .8365

.7274 .8618

•9435 -9747

1.0599 .0253

0.8145

0.7592

.6884 .8378

.7254 .8606

-949° -9772

1.0538 .0228

0.8116 |

0.7621

.6905 .8391

.7234 .8594

•9545 -9798

1.0477 .0202

0.8087

0.7650

.6926 .8405

.7214 .8582

.9601 .9823

1.0416 .0177

0.8058

0.7679

44°oo'

.6947 9.8418

.7193 9.8569

.9657 9-9848

1.0355 0.0152

46°oo'

0.8029

0.7709

.6967 .8431

.7173 .8557

.9713 .9874

1.0295 .0126

0.7999

0.7738

.6988 .8444

•7153 -8545

•9770 .9899

1.0235 .0101

0.7970

0.7767 0.7796

30 40

.7009 .8457 .7030 .8469

•7133 -8532 .7112 .8520

.9827 .9924 .9884 .9949

1.0176 .0076 1.0117 .0051

3°

0.7941 0.7912

0.7825

.7050 .8482

.7092 .8507

•9942 -9975

1.0058 .0025

0.7883

0.7854

45°oo'

.7071 9.8495

I.OOOO O.OOOO

45°oo'

0.7854

Nat. Log.

Nat Log.

Nat. Log.

' W

COSINES.

SINES.

COTAN- GENTS.

TANGENTS.

W y

a* o

(4**

SMITHSONIAN TABLES.

TABLE 1 2. CIRCULAR (TRIGONOMETRIC) FUNCTIONS.*

RADIANS.

SINES.

COSINES.

TANGENTS.

COTANGENTS.

DEGREES.

Nat Log.

Nat. Log.

0.00

.01 .02

.03 .04

O.OOOOO — 00

.01000 7-99999 .02000 8.30100 .03000 .47706 .03999 -60194

1. 00000 0.00000

0-99995 9.99998 .99980 .99991 •99955 -99980 .99920 .99965

— oo — oo o.oiooo 8.OOOOI .02000 .30109

.03001 .47725

.04002 .60229

.00 00

99-997 1-99999 49.993 .69891

33-323 .52275 24.987 .39771

00°00'

0034

oi 09 oi 43 02 18

°s

.07 .08 .09

0.04998 8.69879

•05996 .77789 .06994 .84474 .07991 .90263 .08988 .95366

0-99875 9-99946 .99820 .99922

•99755 -99894 .99680 .99861

•99595 -99824

0.05004 8.69933

.06007 -77867 .07011 .84581 .08017 .90402

.09024 .95542

J9-983 1.30067 16.647 .22133 14.262 .15419 12473 -09598 11.081 .04458

02°52'

03 26 04 oi

0435 0509

O.IO

.11

.12

•13 .14

0.09983 8.99928 .10978 9.04052 .11971 .07814 .12963 .11*72 .13954 .14471

0.99500 9.99782 .99396 .99737 .99281 .99687 .99156 .99632 •99022 .99573

0.10033 9.00145

.11045 -043I5 .12058 .08127 .13074 .11640 .14092 .14898

9.9666 0.99855 9.0542 .95685 8.2933 -91873 7.6489 .88360 7.0961 .85102

Sl°4f

06 18

0653 07 27 0801

o-'S

.16

•17 .18 .19

0.14944 9- * 7446 .15932 .20227 .16918 .22836

!i8886 .27614

0.98877 9.99510 .98723 .99442 .98558 .99369 .98384 .99293 .98200 .99211

0.15114 9.17937 .16138 .20785 .17166 .23466 .18197 .26000 .19232 .28402

6.6166 0.82063 6.1966 .79211; 5.8256 .76534 54954 .74000 S-I997 -71598

o8°36' 09 10

0944 10 19

1053

0.20 .21

.22

•23 .24

0.19867 9.29813 .20846 -31902 .21823 .33891 •22798 -35789 •23770 .37603

0.98007 9.99126 .97803 .99035 .97590 .98940 •97367 .98841 .97134 .98737

0.20271 9.30688 .21314 .32867 .22362 .34951 .23414 .36948 .24472 .38866

4-9332 0.69312 4.6917 .67133 4.4719 .65049 4.2709 .63052 4.0864 .61134

II°28' 12 02 12 36 I3 II 1345

°:ll :%

.29

0.24740 9.39341 .25708 .41007 .26673 .42607 .27636 .44147 .28595 .45629

0.96891 9.98628

•96639 -98515 .96377 .98397 .96106 .98275 .95824 .98148

0-25534 9-40712 .26602 42491 .27676 .44210 .28755 -45872 .29841 47482

3.9163 0.59288 3-7592 .57509 3-6i33 .55790 3.4776 .54128 3.3511 .52518

I4°i9'

H 54 15 28 16 03 1637

0.30 •31 •32 •33 •34

0-29552 9.47059 .30506 .48438

•3'457 -49771 .32404 .51060

•33349 -52308

0-95534 9-98016 •95233 -97879 .94924 .97737 .94604 .97591 .94275 .97440

0.30934 9.49043 •32033 '50559 .33139 .52034 .34252 .53469 -35374 .54868

3.2327 0.50957 3.1218 49441 3.0176 47966 2.9195 46531 2.8270 45132

17°! i' 17 46 18 20

1854 19 29

o-35 •36

•39

0.34290 9.53516 •35227 .54688 .36162 .55825 .37092 .56928 .38019 .58000

0-93937 9-97284 •9359° -97123 •93233 .96957 .92866 .96786 .92491 .96610

0-36503 9-56233 .37640 .57565 .38786 .58868 .39941 .60142 41105 .61390

2-7395 0.43767 2.6567 42435 2.5782 41132

2-5037 .39858 2.4328 .38610

20°03' 20 38 21 12 21 46 22 21

0.40 .41 .42 •43 •44

0.38942 9.59042 .39861 .60055 .40776 .61041 41687 .62000 .42594 .62935

0.92106 9.96429 .91712 .96243 .91309 .96051 •90897 -95855 •90475 -95653

0.42279 9.62-613 43463 .63812 44657 .64989 .45862 .66145 47078 .67282

2.3652 0.37387 2.3008 .36188

2.2393 -350II 2.1804 .33853 2.1241 .32718

22°55' 23 29 24 04 2438 25 13

o.4S .46

•47 .48

•49

0.43497 9-63845 .44395 .64733 45289 .65599 .46178 .66443 47063 .67268

0.90045 9.95446 .89605 .95233 .89157 -95015 .88699 -94792 -88233 .94563

0.48306 9.68400 •49545 .69500 •50797 -70583 .52061 .71651

•53339 -72704

2.0702 0.31600 2.0184 .30500 1.9686 .29417 1.9208 .28349 1.8748 .27296

25°47'

26 21 2656 27 30 28 04

o 50

0-47943 9-68072

0.87758 9-94329

0.54630 973743

1.8305 0.26257

28°39'

SMITHSONIAN TABLES.

* Arranged and computed by C. E. Van Orstraud.

TABLE i 2 (continued).

CIRCULAR (TRIGONOMETRIC) FUNCTIONS.

RADIANS.

SINES.

COSINES.

TANGENTS.

COTANGENTS.

DEGREES.

Nat. Log.

0.50 •51 •52 •53 •54

0-47943 9-68072 .48818 .68858 .49688 .69625

•50553 70375 .51414 71108

0.87758 9.94329 .87274 .94089 .86782 .93843 .86281 .93591 •85771 -93334

0.54630 973743 •55936 74769 •57256 75782 .58592 .76784 •59943 77774

1.8305 0.26257 7878 .25231 .7465 .24218 .7067 .23216 .6683 .22226

29 U 2948

3O 22 3056

-59

0.52269 9.71824 .53119 72525 •53963 73210 .54802 73880 .55636 74536

0.85252 9.93071 .84726 .92801 .84190 .92526 .83646 .92245 .83094 .91957

0.61311 978754 .62695 79723 .64097 .80684 •65517 -81635 .66956 .82579

1.6310 0.21246

.5950 .20277 .5601 .19316 .5263 .18365 .4935 -I742I

3205 32 40

33 14 3348

0.60 .61 .62

•63 .64

0.56464 975T77 .57287 75805 .58104 76420 •589H 77022 .59720 .77612

0.82534 9.91663 .81965 .91363 .81388 .91056 .80803 -90743 .80210 .90423

0.68414 9.83514 .69892 .84443

•7I391 -85364 .72911 .86280 74454 .87189

1.4617 0.16486 •43°8 .15557 .4007 .14636 '3715 -13720 .3431 .12811

34°23'

34 57

3606 3640

0.60519 9.78189 .61312 78754 .62099 79308 .62879 79851 .63654 .80382

0.79608 9.90096 .78999 .89762 .78382 .89422

•77757 -89074 .77125 .88719

076020 9.88093 77610 .88992 79225 .89886 .80866 .90777 .82534 .91663

1.3154 0.11907 .2885 .11008 .2622 .10114 .2366 .09223 .2Il6 .08337

3749 3823 3858 3932

0.70

72 73 74

0.64422 9.80903 .65183 .81414 .65938 .81914 .66687 -82404 .67429 .82885

076484 9.88357 .75836 .87988 75181 .87611 74517 .87226 .73847 .86833

0.84229 9.92546 •85953 -93426 •87707 -94303 .89492 .95178 .91309 .96051

1.1872 0.07454 .1634 .06574 .1402 .05697 .1174 .04822 .0952 .03949

40°o6' 40 41 41 15

41 5° 42 24

075 76

0.68164 9.83355 .68892 .83817 .69614 .84269 .70328 .84713 .71035 .85147

073169 9.86433 .72484 .86024 .71791 .85607 .71091 .85182 -70385 .84748

0.93160 9.96923

•95045 -97793 .96967 .98662 .98926 9.9953 1 i .0092 0.00400

1.0734 0.03077 .0521 .02207 .0313 .01338 1.0109 .00469 0.99084 9.99600

42°58'

4333 44 07 44 41 45 l6

0.80 .81 .82

0.71736 9.85573 .72429 .85991 .73115 .86400 73793 -86802 .74464 .87195

0.69671 9.84305 .68950 .83853 .68222 .83393 .67488 .82922 .66746 .82443

1.0296 0.01268 .0505 .02138 .0717 .03008 .0934 .03879 .1156 .04752

0.97121 9.98732

•95T97 -97862 •93309 -96992 .91455 .96121

•89635 -95248

45°5o' 4628 46 59 47 33 48 08

.'87 .88 .89

075128 9.87580 75784 .87958 .76433 .88328 77074 .88691 .77707 .89046

0.65998 9.81953 .65244 .81454 .64483 .80944 .63715 .80424 .62941 .79894

1.1383 0.05627 .1616 .06504 -1853 .07384 .2097 .08266 .2346 .09153

0.87848 9.94373 .86091 -93496 .84365 .92616 .82668 .91734 .80998 .90847

49 16 49 51 5° 25 51 oo

0.90 .91 .92 •93 •94

078333 9-89394 7895° -89735 .79560 .90070 .80162 .90397 .80756 .90717

0.62161 979352

•61375 78799 .60582 78234

•59783 77658 .58979 77070

1.2602 0.10043 .2864 .10937 •3*33 -"835 .3409 .12739 .3692 .13648

079355 9-89957 7773s .89063 .76146 .88165 .74578 .87261 73034 .86352

5'034' 52 08

52 43 53 17 53 5i

0-95 .96

•97 •98 •99

0.81342 9.91031 .81919 .91339 .82489 .91639 .83050 .91934 .83603 .92222

0.58168 9.76469

•57352 75855 .56530 .75228 .55702 .74587 •54869 73933

1.3984 0.14563 .4284 .15484 .4592 .16412 .4910 .17347 .5237 .18289

0.71511 9.85437 .70010 -84516 .68531 .83588 .67071 .82653 .65631 .81711

54°26'

5535 5609

5643

1. 00

0.84147 9.92504

0.54030 973264

J-5574 0.19240

0.64209 9.80760

57°i8'

SMITHSONIAN TABLES.

TABLE 12 (continued), CIRCULAR (TRIGONOMETRIC) FUNCTIONS.

RADIANS. II

SINES.

COSINES.

TANGENTS.

COTANGENTS.

DEGREES.

Nat. Log-

Nat. Log.

Nat. Log,

1. 00 .Ol

.02

•03 .04

0.84147 9.92504 .84683 .92780 .85211 .93049

•8573° -933 i 3 .86240 .93571

0.54030 9.73264 .53186 .72580 •52337 -71881 .51482 .71165 .50622 .70434

1.5574 0.19240

.5922 .20200 .6281 ,2Il69 .6652 .22148 .7036 .23137

0.64209 9.80760 .62806 .79800 .61420 .78831 .60051 .77852 .58699 .76863

57°i8' 57 52 5827 59oi 5935

•a a

.09

0.86742 9.93823 .87236 .94069 .87720 -94310 .88196 .94545 .88663 -94774

0-49757 9-69686 .48887 .68920 .48012 -68135

•47133 -67332 .46249 .66510

1.7433 0.24138 .7844 .25150 .8270 -26175 .8712 .27212 .9171 .28264

0.57362 9-75862 .56040 .74850

•54734 .73825 .53441 .72788 .52162 .71736

6o°io' 6044 61 18

61 53 62 27

I.IO

.11

.12

•J3

.14

0.89121 9.94998 .89570 .95216 .90010 -95429 .90441 .95637 .90863 .95839

0.45360 9.65667 .44466 .64803 .43568 -63917 .42666 .63008 .41759 .62075

1.9648 0,29331 2.0143 .30413 .0660 .31512 .1197 .32628 ..1759 .33763

0.50897 ' 9.70669 .49644 .69587 .48404 .68488 .47175 -67372 •45959 -66237

63°o2' 6336 64 10 64 45 65 19

.16

•17 .18 .19

0.91276 9-96036 .91680 .96228 .92075 .96414 .92461 -96596 .92837 .96772

0.40849 9.61118 -39934 -60134 •39015 -59123 .38092 .58084 .37166 .57015

2.234C 0.34918 .2958 .36093 .3600 .37291 .4273 .38512

•4979 -39757

0-44753 9-65082 43558 -63907 .42373 .62709 .41199 .61488 ,40034 .60243

65°53' 6628 67 02

67 37 68 ii

i. 20

.21 .22

•23 .24

0.93204 9-96943 •93562 .97110 .93910 .97271 .94249 .97428 •94578 -97579

0.36236 9.55914

•353°2 .54780 •34365 -53611 .33424 .52406 .32480 .51161

2.5722 0.41030 .6503 .42330 .7328 .43660 .8198 .45022 .9119 .46418

0.38878 9.58970 •37731 -57670 .36593 -56340 •35463 .54978 •34341 -53582

68°45' 69 20 6954 70 28 7i 03

•a

.27 .28 .29

0.94898 9.97726 .95209 .97868 .95510 .98005 .95802 .98137 .96084 .98265

o-3 r 532 949875 .30582 .48546 .29628 .47170 .28672 .45745 .27712 .44267

3.0096 0.47850 •"33 -49322 .2236 .50835

•3413 -52392 .4672 .53998

o.33227 9-52I5° .32121 .50678 .31021 .49165 .29928 .47608 .28842 .46002

7i°37' 72 12

'72 46 73 20 73 55

1.30 •31 •32

•33 •34

0.96356 9-98388 .96618 .98506 .96872 .98620 .97115 .98729 •97348 -98833

0.26750 9.42732 .25785 .41137 .24818 .39476 .23848 .37744 •22875 .35937

3.6021 0.55656 •7471 -57369 .9033 «59*44 4.0723 .60984 .2556 .62896

0.27762 944344 .26687 -42631 .25619 .40856 .24556 .39016 •23498 .37104

74°29' 7503 7538

76 12 7647

'3 9

•39

0-97572 9-98933 .97786 .99028 .97991 .99119 .98185 .99205 .98370 .99286

0.21901 9.34046 .20924 .32064 .19945 .29983 .18964 .27793 .17981 .25482

4.4552 0.64887 .6734 .66964 .9131 .69135 5.1774 .71411 .4707 .73804

0.22446 9-35113 .21398 .33036 .20354 .30865 .19315 .28589 .18279 .26196

77°2i' 77 55 7830

79 °4 7938

1.40 .41 .42 •43 •44

0-98545 9-99363 .98710 .99436 .98865 .99504 .99010 .99568 .99146 .99627

0.16997 9.23036 .16010 .20440 .15023 .17674 .14033 .14716 .13042 .11536

5-7979 0.76327 6.1654 .78996 6.5811 .81830 7.0555 .84853 7.6018 .88092

0.17248 9.23673 .16220 .21004 .15195 .18170 .14173 .15147 .13155 .11908

8o°i3' 8047

8l 22

81 56 82 30

.46

48 .49

0.99271 9.99682

•99387 -99733 .99492 .99779 .99588 .99821 .99674 .99858

0.12050 9.08100 .11057 .04364 .10063 .00271 .09067 8.95747 .0807 1 .90692

8.2381 0.91583 8.9886 .95369 9.8874 .99508 10.983 1.04074 12.350 .09166

0.12139 9.08417 .11125 .04631 .10114 .00492 .09105 8.95926 .08097 .90834

83°o5' 8339 84 13 8448

85 22

1.50

0-99749 9-9989I

0.07074 8.84965

14.101 1.14926

0.07091 8.85074

85°57'

SMITHSONIAN TABLES.

TABLES 1 2 (continued) AND 1 2A. CIRCULAR FUNCTIONS AND FACTORIALS.

TABLE 12 (continued). — Circular (Trigonometric) Functions.

RADIANS.

SINES.

COSINES.

TANGENTS.

COTANGENTS.

DEGREES. 1

Nat. Log.

1.50

•52 •53 •54

0.99749 9.99891 .99815 .99920 .99871 .99944 .99917 .99964

•99953 -99979

0.07074 8.84965 .06076 .78361 .05077 .70565 .04079 .61050 .03079 .48843

14.101 1.14926 16.428 .21559 19.670 .29379 24.498 .38914 32.461 .51136

0.07091 8.85074 .06087 .78441 .05084 .70621 .04082 .61086 .03081 .48864

S5°57' 8631 87 05 87 40 88 14

•56

•59

0.99978 9.99991 0.99994 9-99997

I .OOOOO O.OOOOO

0.99996 9.99998 0.99982 9-99992

0.02079 8.31796 .01080 8.03327 .00080 6.90109 -.00920 7.96396n -.01920 8.2833611

48.078 1.68195 92.621 1.96671 1255.8 3.09891 108.65 2.03603 52.067 1.71656

0.02o8o 8.31805 .01080 8.03330 .00080 6.90109 -.00920 7. 96397 n -.01921 8.28344n

88°49' 8923 8957 9032 91 06

1.60

0-99957 9-9998i

-0.02920 8.46538n

34-233 1-53444

-0.02921 8-46556n

9i°4o'

90°= i. 570 7963 radians.

TABLE 12a. -Factorials.

Logarithms of the products 1.2.3 w» n ^rom i to 100.

See Table 30 for log. T (n + 1 ), values of n between i and 2.

!*(->

£W>

log.(«0

•*w

o.oooooo

26.605619

66.190645

111.275425

0.301029

28.036982

67.906648

113.161916

0.778151

29.484140

69.630924

115.054010

1.380211

30.946538

71.363318

116.951637

2.079181

32.423660

73.103680

118.854727

2.857332

33-9I502I

74.851868

120.763212

3.702430

35.420171

76.607743

122.677026

4.605520

36.938685

78.371171

124.596104

5-559763 6.559763

34 35

38.470164 40.014232

59 60

80.142023 81.920174

126.520383 128.449802

7.601155 8.680336

4L570535 43-T38736

83.705504 85.497896

1 30-38430 1 132.323820

9.794280

10.940408

38 39

44.718520 46.309585

63 64

87.297236 89.103416

88 89

134.268303 136.217693

12.116499

47.911645

90.916330

I38.I7I935

13.320619

49.524428

92.735874

140.130977

14.551068

51.147678

94.561948

142.094765

15.806341

52.781146

96-394457

144.063247

17.085094

54.424599

98.233306

146.036375

18.386124

56.07781 1

100.078405

148.014099

19.708343

57.740569

101.929663

149.996370

21.050766

59.412667

103.786995

151.983142

22.412494

61.093908

105.650318

'53-974368

23-792705

62.784104

I°7-5I955°

155.970003

25.190645

5°

64.483074

109.394611

IOO

157.970003

SMITHSONIAN TABLES.

TABLE 13. HYPERBOLIC FUNCTIONS.*

Hyperbolic sines. Values of

•8

0.0

o.oooo

O.OIOO

O.02OO

0.0300

0.0400

0.0500

0.0600

0.0701

0.0801

0.0901

.1002

.IIO2

.I2O3

.1304

.1405

.1506

.1607

.1708

.1810

.1911

0.2

.2013

.2115

.2218

.2320

.2423

.2526

.2629

•2733

•2837

.2941

o-3

•3°45

•3150

•3255

•336o

•3466

•3572

.3678

•3785

.3892

.4000

0.4

.4108

.4216

•4325

•4434

•4543

•4653

•4764

•4875

.4986

.5098

0.5

0.6

0.7

0.5211 .6367

°'P«4 .6485

.7712

0.5438 .6605 .7838

0-5552 •6725 .7966

0.5666 .6846 .8094

°:$*

.8223

0.5897 .7090 •8353

0.6014 .7213 .8484

0.6131

•7336 .8615

0.6248 .7461 .8748

0.8

.8881

.9015

.9150

.9286

•9423

.9561

.9700

.9840

.9981

.0122

0.9

1.0265

1.0409

1-0554

1.0700

1.0847

1.0995

1.1144

1.1294

1.1446

1.1598

1.0

1.1752

I.I907

1.2063

I.222O

1-2379

1-2539

1.2700

1.2862

1.3025

1.3190

i.i

•3356

•3524

-3693

•3863

•4035

.4208

.4382

•4558

•4735

.4914

1.2

•5°95

.5276

.5460

•5645

•5831

.6019

.6209

.6400

•6593

.6788

i-3

.6984

.7182

.7381

•7583

•7786

.7991

.8198

.8406

.8617

.8829

1.4

•9043

•9259

•9477

.9697

.9919

2.0143

2.0369

2.0597

2.0827

2.1059

1.5

2.1293

2.1529

2.1768

2.2008

2.2251

2.2496

2.2743

2.2993

2-3245

2-3499

1.6

i-7

•3756 .6456

.4015 .6740

.4276 .7027

•4540 •7317

.4806 .7609

•5075 •7904

•5346 .8202

.5620 •8503

Jsol

•6175 .9112

1.8

.9422

•9734

3.0049

3-0367

3.0689

3-IOI3

3-J340

3.1671

3.2005

3-234I

1.9

3.2682

3-3025

•3372

.3722

•4075

•4432

•4792

•5156

•5523

.5894

2.0

3.6269

3.6647

3.7028

3-74I4

3-7803

3.8196

3-8593

3-8993

3-9398

3.9806

2.1

4.0219

4-0635

4.1056

4.1480

4.1909

4-2342

4.2779

4.3221

4.3666

4.4117

2.2

4-4571

4-5030

4-5494

4.5962

4.6434

4.6912

4-7394

4.7880

4.8372

4.8868

2-3 2.4

4-9370 5.4662

4.9876 5.5221

5-0387 5-5785

5-0903 5-6354

5-I425 5.6929

5-I951 5-75Jo

5-2483 5.8097

5.3020 5.8689

5-3562 5.9288

5.4109 5-9892

2.5

2.6

6.0502 6.6947

6.1118 6.7628

6.1741 6-8315

6.2369 6.9009

6.3004 6.9709

6-3645 7.0417

6.4293 7.1132

6.4946 7-1854

6.5607 7-2583

6.6274 7-33I9

2.7

7.4063

7.4814

7-5572

7.6338

7.7112

7.7894

7-8683

7.9480

8.0285

8.1098

2.8

2.9

8.1919 9.0596

8.2749 9.1512

8.3586 9-2437

8.4432 9-3371

8.5287 9-43I5

8.61 50 9.5268

8.7021 9.6231

8.7902 9.7203

8.8791 9.8185

8.9689 9-9I77

3.0

10.018

10.119

IO.22I

10.324

11.429

JI-534

11.640

11.748

11.856

11.966

3-i

11.076

11.188

11.301

II.4I5

n-530

12.647

12.764

12.883

12.003

12.124

3-2

12.246

12.369

12.494

12.620

12.747

12.876

13.006

I3-I37

13.269

1 3-403

3-3

I3-538

I3-674

I3.8I2

J3-95I

14.092

14.234

14-377

14.522

14.668

14.816

14.965

15.116

15.268

15.422

15-577

15-734

I5-893

16.053

16.214

16.378

3.5

16.543

16.709

16.877

17.047

17.219

17-392

17-567

17-744

17-923

18.103

3-6

18.285

18.470

18-655

18.843

1 9-°33

19.224

19.418

19.613

19.811

2O.OIO

2O.2II

20.415

2O.62O

20.828

21.037

21.249

21.463

21.679

21.897

22.117

3-8

22.339

22.564

22.791

23.020

23.252

23.486

23.722

23.961

24.202

24.445

3-9

24.691

24.939

25.190

25.444

25.700

25.958

26.219

26.483

26.749

27.018

4.0

27.290

27.564

27.842

28.122

28.404

28.690

28.979

29.270

29-564

29.862

4.1

30.162

30-465

30.772

31.081

31-393

31.709

32.028

32.350

32-675

33-004

4.2

33-336

33-67I

34.009

34-351

34-697

35-046

35-398

35-754

36-113

36.476

4-3

36.843

37-214

37.588

37.966

38.347

38.733

39.122

39-5J5

39-9»3

40.314

4.4

40.719

41.129

41.542

41.960

42-382

42.808

43-238

43-673

44.112

44-555

4.5

45-003

45-455

45-912

46.374

46.840

47-311

47-787

48.267

48.752

49.242

4.6

49-737

50-237

50.742

51.252

5^767

52.288

52-813

53-344

53.880

54.422

4-7

54.969

55.522

56.080

56-643

57-213

57788

58.369

58-955

59-548

60.147

4.8

60.751

61.362

61.979

62.601

63-231

63.866

64.508

65-I57

65.812

66.473

4.9

67.141

67.816

68.498

69.186

69.882

70.584

71.293

72.010

72-734

73-465

* Tables 38-41 are quoted from " Des Ingenieurs Taschenbuch," herausgegeben vom Akademischen Verein (Hiitte). SMITHSONIAN TABLES.

TABLE 14. HYPERBOLIC FUNCTIONS.

Common logarithms -f 10 of the hyperbolic sines.

0.0

8.0000

3011

4772

6022

6992

7784

8455

9036

9548

O.I

9.0007

0423

0802

1152

1475

1777

2060

2325

2576

2814

O.2

3°39

3254

3459

3656

3844

4025

4199

4366

4528

4685

0.4

9.6136

4983 6249

5^25 6359

5264 6468

5398 6574

5656 6780

578i 6880

5902 6978

6020 7074

0.5

9.7169

7262

7354

7444

7533

7620

7707

7791

7875

7958

0.6

8039

8119

8199

8277

8334

8431

8506

8581

8655

8728

0.7

8800

8872

8942

9012

9082

9150

9218

9286

9353

9419

0.8

9485

9550

9614

9678

9742

9805

9868

9930

9992

°°53

0.9

10.0114

0174

0234

0294

0353

0412

0470

0529

0586

0644

1.0

10.0701

0758

0815

0871

0927

0982

1038

1093

1148

1203

i.i

1257

1311

1365

1419

1472

1525

1578

1631

1684

1736

1.2

1788

1840

1892

1944

I995

2046

2098

2148

2199

2250

1.3

2300

2351

2401

2451

2501

2551

2600

2650

2699

2748

1.4

2797

2846

2895

2944

2993

304i

3090

3138

3186

3234

1.5

10.3282

3330

3378

3426

3474

3521

3569

3616

3663

37H

1.6

3758

3805

3852

3899

3946

3992

4039

4086

4J32

i-7

4225

4272

4364

4411

4457

45°3

4549

4595

4641.

1.8

4687

4733

4778

4824

4870

49*5

4961

5007

5052

5098

1.9

5H3

5188

5234

5279

5324

5370

5415

5460

55°5

5550

2.0

2.1

10.5595 6044

5640 6089

5685 6134

5730 6178

5775 6223

5820 6268

5865 6312

59™ 6357

5955 6401

5999 6446

2.2

6491

6535

6580

6624

6668

6713

6757

6802

6846

6890

2-3

6935

6979

7023

7067

7112

7156

7200

7244

7289

7333

2.4

7377

7421

7465

7509

7553

7597

7642

7686

773°

7774

2.5

10.7818

7862

7906

795°

7994

8038

8082

8126

8169

8213

2.6

8257

8301

8345

8389

8433

8477

8521

8564

8608

8652

2.7

8696

8740

8784

8827

8871

8915

8959

9003

9046

9090

2.8

9134

9178

9221

9265

9309

9353

9396

9440

9484

9527

2.9

957i

9615

9658

9702

9746

9789

9833

9877

9920

9964

3.0

11.0008

0051

0095

0139

0182

0226

0270

0313

°357

0400

3-1

0444

0488

0531

°575

0618

0662

0706

0749

0793

0836

3-2

0880

0923

0967

ion

1054

1098

1141

1185

1228

1272

3-3

1316

1359

1403

1446

1490

1533

1577

1620

1664

1707

3-4

I751

1794

1838

1881

1925

1968

2OI2

2056

2099

2143

3.5

11.2186

2230

2273

2317

2360

2404

2447

2491

2534

2578

3-6

2621

2665

2708

2752

2795

2839

2882

2925

2969

3012

3-7

3056

3°99

3H3

3186

3230

3273

3317

3360

3404

3447

3-8

3534

3578

3621

3665

3708

3752

3795

3838

3882

3-9

3925

3969

4012

4056

4099

4H3

4186

4230

4273

4317

4.0

11.4360

4403

4447

4490

4534

4577

4621

4664

4708

4751

4.1

4795

4838

4881

4925

4968

5012

5055

5°99

5H2

5186

4.2

5229

5273

5316

5359

5403

5446

5490

5533

5577

5620

4-3

4.4

5707 6141

575° 6185

5794

5837 6272

5881 6315

5924 6359

5968 6402

6011 6446

6055 6489

4.5

11.6532

6576

6619

6663

6706

6750

6793

6836

6880

6923'

4.6

6967

7010

7054

7097

7141

7184

7227

7271

73J4

7358

4-7

7401

7445

7488

7531

7575

7618

7662

7705

7749

7792

4.8

7836

7879

7922

7966

8009

8053

8096

8140

8183

8226

4.9

8270

8313

8357

8400

8444

8487

8530

8574

8617

8661

SMITHSONIAN TABLES.

TABLE 1 5. HYPERBOLIC FUNCTIONS.

Hyperbolic cosines. Values of

•

0.0

1. 0000

1. 000 1

1.0002

1.0005

1.0008

1.0013

1.0018

1.0025

1.0032

1.0041

O.I

.0050

.0061

.OO72

.0085

.0098

.0113

.0128

.0145

.0162

.0181

O.2

.0201

.0221

.0243

.0266

.0289

.0314

.0340

.0367

•0395

.0423

o-3

•0453

.0484

.0516

.0549

.0584

.0619

•0655

.0692

.0731

.0770

0.4

.0811

.0852

•0895

.0939

.0984

.1030

.1077

.1125

.1174

.1225

0.5

1.1276

1.1329

LI383

1.1438

1.1494

I.I55I

1.1609

1.1669

1.1730

1.1792

0.6

•1855

.1919

.1984

.2051

.2119

.2188

.2258

•2330

.2402

.2476

0.7

•2552

.2628

.2706

.2785

.2865

•2947

•3030

•3114

•3J99

.3286

0.8

•3374

•3464

-3555

•3647

-3740

•3835

•3932

.4029

.4128

.4229

0.9

433 i

4434

4539

.4645

4753

.4862

4973

.5085

•5199

•5314

1.0

i-543i

1-5549

1.5669

1-5790

J-59I3

1.6038

.6164

1.6292

1.6421

1.6552

.6685 .8107

.6820 .8258

!S4?2

.7093 .8568

•7233

.8725

•m

.7517 •9045

.7662 .9208

.7808 •9373

•7956 -9540

•3

.9709

.9880

2.0053

2.0228

2.0404

2.0583

2.0764

2.0947

2.1132

2.1320

2.1509

.1700

.1894

T2090

.2288

.2488

.2691

.2896

•3!03

•3312

1.5

2.3524

2.3738

2.3955

2.4174

2-4395

2.4619

2.4845

2.5073

2-5305

2.5538

•5775

•6013

•6255

•6499

.6746

•6995

•7247

.7502

.7760

.8020

•7

.8283

.8549

.8818

.9090

-9364

.9642

.9922

3.0206

3.0492

3.0782

1.8

3-^075

3-i37i

3-1669

3.1972

3.2277

3-2585

3.2897

.3212

•353°

•3852

1.9

4177

.4506

4838

•5'73

•5512

•5855

.6201

•6551

.6904

.7261

2.0

3.7622

37987

3-8355

3.8727

3-9 i 03

3-9483

3.9867

4-0255

4.0647

4.1043

2.1

4-1443

4.1847

4.2256

4.2668

4-3085

4-3507

4-3932

4.4362

4-4797

4-5236

2.2

4- 5679

4.6127

4.6580

4-7037

4-7499

4.7966

4.8437

4.8914

4-9395

4.9881

2-3 2.4

5-0372 5-5569

5.0868 5.6119

5-1370 5.6674

5-1876 5-7235

5.2388 5-78oi

5-2905 5-8373

5-3427 5-895I

5-3954 5-9535

5-4487 6.0125

5-5026 6.0721

2.5

2.6

6.1323 6.7690

6.1931 6.8363

6.2545 6.9043

6.3166 6.9729

6-3793 7.0423

6.4426 7.1123

6.5066 7.1831

6.5712 7-2546

6.6365 7.3268

6.7024 7-3998

74735 8.2527

7-5479 8-3351

7.6231

8.4182

7.6990 8.5022

77758 8.5871

7-8533 8.6728

7.9316 8.7594

8.0106 8.8469

8.0905 8.9352

8.1712 9.0244

2.9

9.1146

9.2056

9.2976

9-3905

9.4844

9-5791

9.6749

9.7716

9.8693

9.9680

3.0

10.068

10.168

10.270

10.373

10.476

10.581

10.687

10.794

10.902

1 1. Oil

3-i

II. 121

12.233

"•345

"459

11.574

11.689

1 1. 806

11.925

12.044

12.165

3-2

12.287

12.410

12.534

12.660

12.786

12.915

13.044

I3-I75

I3-307

13.440

3-3

!3-575

i3-7ii

13.848

13-987

14.127

14.269

14.412

14-556

14.702

14.850

3-4

14.999

I5-I49

I5-30I

15455

15.610

15-766

15.924

16.084

16.245

16.408

3.5

16.573

16.739

16.907

17.077

17.248

17.421

I7-596

17.772

I7-951

18.131

3-6

18.313

18.497

18.682

18.870

19.059

19.250

19.444

19.639

19.836

20.035

3-7

20.236

20.439

20.644

20.852

21.061

21.272

21.486

21.702

21.919

22.139

3-8

22.362

22.586

22.813

23.042

23-273

23-507

23-743

23.982

24.222

24.466

3-9

24.711

24.959

25.210

25.463

25-719

25-977

26.238

26.502

26.768

27.037

4.0

27.308

27-582

27.860

28.139

28.422

28.707

28.996

29.287

29.581

29.878

4.1

30.178

30.482

30.788

3I-°97

31.409

31-725

32-044

32.365

32.691

33019

4.2 4-3 44

33-351 36-857 40.732

33-686 37.227 41.141

34.024 37.601 4L554

34.366

37-979 41.972

34-7" 38.360 42.393

35.060 38.746 42.819

35412 39-135 43-25°

35768 39-528 43.684

36.127

39-925 44.123

36.490 40.326 44.566

4.5

45.014

45.466

45-923

46.385

46.851

47-321

47-797

48.277

48.762

49.252

4.6

49-747

50.247

50-752

51.262

5T-777

52.297

52.823

53-354

53-890

54431

4-7 4.8

54.978 60.759

55-531 61.370

56.089 61.987

56.652 62.609

57-221

63.239

57-796 63.874

64.516

58.964 65.164

59-556 65.819

60.155 66.481

4-9

67.149

67.823

68.505

69.193

69.889

70.591

71.300

72.017

72.741

73472

SMITHSONIAN TABLES.

TABLE 1 6. HYPERBOLIC FUNCTIONS.

Common logarithms of the hyperbolic cosines.

0.0

o.oooo

oooo

OOOI

OOO2

0003

0005

0008

OOII

0014

0018

O.I

0022

0026

0031

0037

0042

0049

0055

0062

0070

0078

0.2

0086

0095

0104

OII4

0124

0134

0145

0156

0168

0180

°-3

0193

0205

0219

0232

0246

0261

0276

0291

0306

0322

0.4

0339

0355

0372

0390

0407

0426

0444

0463

0482

0502

0.5

0.0522

0542

0562

0583

0605

0626

0648

0670

0693

0716

0.6

0739

0762

0786

0810

0835

0859

0884

0910

0935

0961

0.7

0987

1013

1040

1067

1094

1122

"49

"77

1206

1234

0.8

1263

1292

1321

'35°

1380

I4IO

1440

1470

1501

I532

0.9

1563

1594

1625

1657

1689

1721

1753

1785

1818

1851

1.0

0.1884

1917

1950

1984

2018

2O5I

2086

2I2O

2154

2189

i.i

2223

2258

2293

2328

2364

2399

2435

2470

2506

2542

1.2

2578

2615

2651

2688

2724

276!

2798

2835

2872

2909

1.3

2947

2984

3022

3°59

3°97

3135

3173

321!

3249

3288

1.4

3326

3365

3403

3442

348i

3559

3598

3637

3676

1.5

0.3715

3754

3794

3833

3873

3913

3952

3992

4032

4072

1.6

4112

4152

4192

4232

4273

43J3

4353

4394

4434

4475

1-7

4515

4556

4597

4637

4678

4719

4760

4801

4842

4883

1.8

4924

4965

5006

5048

5089

5130

5172

5213

5254

5296

1.9

5337

5379

5421

5462

55°4

5545

5587

5629

5671

5713

2.0

0.5754

5796

5838

5880

5922

5964

6006

6048

6090

6132

2.1

2.2

6175 6597

6217 6640

6259 6682

6301 6724

6343 6767

6386 6809

6428 6852

6470 6894

6512 6937

6555 6979

2-3

7022

7064

7107

715°

7192

7235

7-278

7320

7363

7406

2.4

7448

749i

7534

7577

7619

7662

7705

7748

7791

7833

2.5

0.7876

7919

7962

8005

8048

8091

8i34

8176

8219

8262

2.6

8305

8348

8391

8434

8477

8520

8563

8606

8649

8692

2-7

8735

8778

8821

8864

8907

8951

8994

9°37

9080

9123

2.8

9166

9209

9252

9295

9338

9382

9425

9468

95"

9554

2.9

9597

9641

9684

9727

9770

9813

9856

9900

9943

9986

3.0

1.0029

0073

0116

0159

0202

0245

0289

0332

0375

0418

3.1

0462

°5°5

0548

0591

0635

0678

0721

0764

0808

08 ci

3-2 3-3

0894 1327

0938 J37i

0981 1414

1024

1457

1067 I5OI

mi 1544

"54

1587

"97 1631

1241 1674

1284 1717

3-4

1761

1804

1847

1891

J934

1977

202 1

2064

2107

2151

3.5

1.2194

2237

2281

2324

2367

2411

2454

2497

2541

2584

3-6

2628

2671

2714

2758

2801

2844

2888

2931

2974

3018

3061

3io5

3*48

3235

3278

3322

3365

3408

3452

3-8

3495

3538

3582

3625

3712

3755

3799

3842

3886

3-9

3929

3972

4016

4059

4103

4146

4189

4233

4278

4320

4.0

14363

4406

4450

4493

4537

4580

4623

4667

4710

4754

4.1

4797

4840

4884

4927

497i

5014

5057

5101

5M4

5188

4.2

5231

5274

5318

5361

5405

5448

5492

5535

5578

5622

4-3 4.4

5665 6099

5709 6143

5795 6230

5839 6273

5882 6316

5926 6360

5969 6403

6012 6447

6056 6490

4.5

I-6533

6577

6620

6664

6707

6751

6794

6837

6881

6924

4.6

6968

7011

7055

7098

7141

7185

7228

7272

7315

7358

4-7

7402

7445

7489

7532

7576

7619

7662

7706

7749

7793

4.8

7836

7880

7923

7966

8010

8053

8097

8140

8184

8227

4.9

8270

8314

8357

8401

8444

8487

8574

8618

8661

SMITHSONIAN TABLES.

TABLE 1 7. 43

EXPONENTIAL FUNCTIONS.

Values of e* and e~* intermediate to those here given may be found by adding or subtracting the values of the hyperbolic cosine and sine given in Tables 15 and 13.

logio(ea:)

logio(e')

tr*

0.0

0.00000

1. 0000

I.OOOOOO

5.0

2.17147

148.41

0.006738

•04343

.1052

0.904837

.21490

164.02

.006097

.08686

.2214

.818731

.25833

181.27

•005517

•3

• 13^29

•3499

.740818

•3

.30176

200.34

.004992

•4

•17372

.4918

.670320

221.41

.004517

0.5

0.2^715

1.6487

0.606531

5.5

2.38862

244.69

0.004087

.29058

.8221

.548812

$3205

270.43

.003698

.30401

2.0138

.496585

•7

•47548

298.87

.003346

•9

•34744 .39087

•2255 •4596

•449329 •406570

.8 •9

•$1891 •56234

330.30 365-04

.003028 .002739

1.0

0.43429

2.7183

0.367879

6.0

2.60577

403-43

0.002479

.47772

3.0042

•332871

.64920

445-86

.002243

.52115

.3201

.301194

•69263

492.75

.002029

•3 •4

:$p

.6693 4-0552

•272532 .246597

•3 •4

.73606 .77948

544-57 601.85

.001836 .001662

1.5

0.65144

4-48i7

0.223130

6.5

2.82291

665.14

0.001503

.69487

•953°

.201897

.86634

735-10

.001360

•7

.73830

5-4739

.182684

•7

.90977

812.41

.001231

.8 •9

•78173 .82516

6.0496 6.6859

.165299 .149569

.8 •9

•95320 .99663

897.85 992.27

.001114 .001008

2.0

0.86859

7-3891

0-135335

7.0

3.04006

1096.6

0.000912

.91202

8.1662

.122456

.08349

J2I2.O

.000825

•95545

9.0250

.110803

.12692

13394

.000747

•3

.99888

9.9742

.100259

•3

•17035

1480.3

.000676

•4

1.04231

11.023

.09071.8

•4

.21378

1636.0

.000611

2.5

1.08574

12.182

0.082085

7.5

3.25721

1 808.0

0.000553

.12917

13.464

•074274

.30064

I998.2

.000500

•7

.17260

14.880

.067206

•7

•34407

2208.3

•000453

.21602

16.445

.060810

•38750

2440.6

.000410

•9

•25945

18.174

•055023

•9

•43°93

2697.3

.000371

1.30288

20.086

0.049787

3-47436

2981.0

0.000335

•34631

22.198

.045049

.51779

3294-5

.000304

•3

•38974 •433 i 7

24-533 27.113

.040762

.036883

• 2

•3

.56121 .60464

3641.0 4023.9

.000275 .000249

•4

.47660

29.964

•033373

.64807

4447-1

.000225

3.5

1.52003

33.115

0.030197

8.5

3.69150

4914.8

0.000203

•56346

36.598

.027324

•73493

543 '-7

.000184

.60689

40.447

.024724

•7

•77836

6002.9

.000167

.65032

44.701

.022371

.82179

6634.2

.000151

•9

•69375

49.402

.020242

•9

.86522

7332.0

.000136

4.0

I.737I8 .78061

54.598 60.340

0.018316

•016573

9.0

3-90865 .95208

8103.1

0.000123

.000112

.82404

66.686

.014996

•995 5 *

9897.1

.000101

•3

.86747

73-700

.013569

•3

4.03894

10938.

.000091

•4

.91090

81.451

.012277

•4

.08237

12088.

.000083

4.5

1 -95433

90.017

0.011109

9.5

4.12580

13360.

0.000075

•99775

99.484

.010052

.16923

14765-

.000068

•7

2.04118

109.95

.009095

•7

.21266

16318.

.000061

.08461

121.51

.008230

.25609

18034.

.000055

•9

.12804

134.29

.007447

•9

.29952

19930.

.OOOO5O

5.0

2.17147

148.41

0.006738

10.0

4-34294

22026.

0.000045

Taken from Glaisher's ' Tables of the Exponential Function,' Trans. Cambridge Phil. Soc. vol. xiii. 1883. This volume also contains a ' Table of the Descending Exponential to Twelve or Fourteen Places of Decimals,' by F. W. Newman.

SMITHSONIAN TABLES.

44 TABLE 18.

EXPONENTIAL FUNCTIONS, LOG e*.

logIoM

>0g,o(,)

logrfO

1 0.0

4.34294

15-0

6.51442

20.0

8.68589

25.0

10.85736

.38637

.55785

.72932

.90079

.42980

.60128

.77275

.94422

•3

•47323

•3

.64471

•3

.8l6l8

•3

.98765

•4

.51666

•4

.68814

•4

•85961

•4

11.03108

loi

4.56009 .60352

I5i

6.73*56 -77499

20.5

8.90304 .94647

25-5

11.07451 .11794

•7

•64695

•7

.81842

•7

.98990

•7

.16137

.8 •9

.69038 •7338i

.8 -9

.86185 .90528

.8 •9

9-03333 .07675

.8 •9

.20480

.24823

II.O

4.77724

1 6.0

6.94871

2I.O

9.I2OI8

26.0

11.29166

.82067

.99214

.l636l

•33509

.86410

7.03557

•2

.20704

•37852

•3

•90753

•3

.07900

•3

•25047

•3

.42194

.95096

•4

.12243

•4

•29390

' -4

.46537

"•5

4-99439

16.5

7.16586

21.5

9-33733

26.5

11.50880

5.03782

.20929

.38076

.55223

•7

.08125

•7

.25272

•7

.42419

•7

•59566

.12467

•29615

.46762

.63909

•9

.16810

•9

.33958

•9

.51105

•9

.68252

I2.O

5.21153

17.0

7-38301

22.0

9-55448

27.0

11.72595

.25496

.42644

•59791

.76938

.29839

.46987

.64134

.81281

•3

.34182 •38525

•3

•4

•51329 •55672

•3

•4

•68477 .72820

•3 •4

.85624 .89967

12.5

5.42868

J7-5

7.60015

22.5

9.77163

27-5

11.94310

.47211

•64358

.8 1 506

•98653

•7

•51554

•7

.68701

•7

.85848

•7

12.02996

•55897

•73044

.90191

•07339

•9

.60240

•9

•77387

•9

•94534

•9

.11682

13.0

5.64583

18.0

7.81730

23.0

9.98877

28.0

12.16025

.68926

.86073

10.03220

.20367

.73269

.90416

•07563

.24710

•3

.77612

•3

•94759

•3

.11906

•3

.29053

•4

•8i955

•4

.99102

•4

.16249

•4

.33396

13.5

5.86298

18.5

8-03445

23-5

10.20592

28.5

12.37739

.90640

.07788

•24935

.42082

.94983 5-99326

.12131

.16474

.29278 • -33621

.46425 .50768

•9

6.03669

•9

.20817

•9

•37964

•9

•55111

14.0

6.08012

19.0

8.25160

24.0

10.42307

29.0

12.59454

•J2355

.29502,

.46650

.63797

.16698

•33845

•5°993

.68140

•3

.21041

•3

.38188

•3

•55336

•3

.72483

•4

•25384

•4

•42531

-4

•59679

•4

.76826

14-5

6.29727 .34070

<9;5

8.46874 .51217

24:I

10.64021 .68364

29:l

12.81169 .85512

•38413

•5556o

•72707

•7

.89855

•42756

•59903

•77050

.94198

•9

•47099

•9

.64246

•9

.98541

15.0

6.51442

2O.O

8.68589

25.0

10.85736

30.0

13.02883

SMITHSONIAN TABLES.

TABLE 19. EXPONENTIAL FUNCTIONS.

Value of e*a and e-«3 and their logarithms.

The equation to the probability curve is y =. , negative, between zero and infinity.

*a, where x may have any value, positive or

log ex*

log e-J?

0.1

I.OIOI

0.00434

0.99005

1.99566

1.0408

01737

96079

98263

.0904

03909

9*393

96091

•1735

06949

85214

93051

.2840

10857

77880

89*43

0.6

-4333

0.15635

0.69768

1.84365

21280

61263

78720

.8965

27795

52729

72205

2.2479

35178

44486

64822

I.O

2-7183

43429 .

36788

56571

1.1

3-3535

0.52550

0.29820

1.47450

4.2207

62538

2:3693

37462

5-4I95

73396

18452

26604

7.0993

85122

14086

14878

9.4877

97716

10540

02284

1.6

1.2936 X io

1. 11179

0.77306 X io-1

2.88821

1-7993

255*1

55576 "

74489

2-5534 "

40711

39*64 "

59289

3.6996 "

56780

27052

43220

2.0

54598 "

18316 "

26282

2.1

8.2269 "

1.91524

0.12155 "

2.08476

1.2647 X io2

2.10199

79070 X io-2

3^89801

1.9834 «

29742

50418

70258

5OI54

3*5**

49846

5.1802 "

7*434

19304

28566

2.6

8.6264 " i. 4656 X io3

2.93583 3.16601

0.11592 "

68233 X 10-3

3.06417 4.83400

2.5402 "

40487

39367 "

595*3

4.4918 "

65242

22263 "

34758

3-°

8.1031 «

90865

I234I

09*35

3.1

1.4913 X io4

4-17357

0.67055 X io~4

5.82643

2.8001 " 5-3638 "

447*8 72947

357*3 18644

55283 27053

1.0482 X io5

5-02044

95402 X io~5

6.97956

2.0898 "

32011

47851 •«

67989

3.6

4-2507 "

5.62846

0.23526 "

6-37*54

8.8205 " 1.8673 X io«

94549 6.27121

1*337 " 53554 X io-«

0545* 7.72879

4.0329 "

60562

24796

39438

4.0

8.8861 "

94871

11254

05129

4.1

1.9976 X io7

7.30049

0.50062 X io~7

5.69951

4.5809 "

66095

21829 "

33905

1.0718 X io8

8.03011

93303 X IO-8

9.96989

2.5583

40796

39088 «

59204

6.2297

79447

16052 "

20553

4.6

1.5476 X io9

9.18967

0.64614 X io-*

10.81033

3.9228 " 1.0143 X io10

59357 10.00615

25494 98595 X 10-10

40643 II-99385

2.6755 "

42741

37376 "

57259

S-o

7.2005 «

85736

v i 3888 "

14264

SMITHSONIAN TABLES.

TABLE 20. EXPONENTIAL FUNCTIONS.

w vf

Values ol 0** and 6 * and their logarithms.

rr 0**

log 8**

e~^*

log*"**

2-1933

0.34109

0.45594

1.65891

4.8105

.68219

.20788

.31781

1.0551 X 10

1.02328

.94780 X io-1

2.97672

2.3141

-36438

.43214

.63562

5-0754

•70547

.19703

•29453

1.1132 X io2

2.04656

0.89833 X 10-2

3-95344

2.4415 "

.38766

.40958 «

.61234

5-3549 "

-72875

.18674 "

.27125

1.1745 X io3

3.06985

.85144 X IO-3

4-930I5

2.5760 «

.41094

.38820 "

.58906

5.6498 « 1.2392 X io*

3-75204 4-093!3

0.17700 " .80699 X io~4

4.24796 5.90687

2.7168 "

.43422

.36794 "

.56578

14 15

5.9610 " 1.3074 X io5

•77532 5.11641

.16776 •'

.76487 X io~5

.22468 6.88359

2.8675 "

5-45751

0.34873 "

6.54249

6.2893 «

.79860

.15900 '•

.20140

1-3794 X io6

6.13969

.72495 X I0~6

7.86031

!9 20

3-0254 6.6356 «

.48079 .82189

•33053 .15070

.51921 .17812

TABLE 21 . EXPONENTIAL FUNCTIONS.

Values of 0 <• * and

and their logarithms.

e~r"

,g^

'•5576

0.19244

0.64203

1.80756

2.4260

.38488

.41221

.61512

3.7786

•57733

.26465

.42267

5-8853

•76977

.16992

.23023

9.1666

.96221

.10909

•03779

14.277

1.15465

0.070041

2.84535

22.238

•34709

.044968

.65291

34-636

•53953

.028871

.46047

53-948

•73198

.018536

.26802

84.027

.92442

.011901

•07558

130.87

2.11686

0.0076408

3.88314

203.85

.30930 .50174

.0049057 .0031496

.69070 .49826

494.52

.69418

.OO2O222

.30582

770.24

.88663

.0012983

•IJ337

1199.7

3.07907

0.00083355

4.92093

1868.5

.27151

.00053517

.72849

2910.4

•46395

.00034360

•53605

4533-1

.00022060

7060.5

.00014163

.15117

SMITHSONIAN TABLES.

TABLES 22 AND 23. EXPONENTIAL FUNCTIONS AND LEAST SQUARES. 47

TABLE 22. —Exponential Functions. Value of e* and e~* and their logarithms.

log**

log<?*

.,-

i/64

1.0157

0.00679

0.98450

i/3

I-3956

0.14476

0-71653

1/32

.0317

.01357

.96923

1/2

.6487

.21715

.60653

i/i6

.0645

.02714

•93941

3/4

2.1170

•32572

.47237

I/IO

.1052

.04343

.90484

•7183

.43429

.36788

J/9

."75

.04825

.89484

5/4

3-4903

•54287

.28650

1/8

1.1331

0.05429

0.88250

3/2

4.4817

0.65144

0.22313

1/7

.1536

.06204

.86688

7/4

5-7546

.76002

•I7377

1/6

.1814

.07238

.84648

7.3891

.86859

1/5

.2214

.08686

.81873

9/4

9.4877

.97716

.10540

1/4

.2840

.10857

.77880

5/2

12.1825

1.08574

.08208

TABLE 23. —Least Squares. Values of P = -

This table gives the value of P, the probability of an observational error having a value posi- tive or negative equal to or less than x when h is the measure of precision, P = — T f-<hx)

\ir*J O

d(hx}, For values of the inverse function see the table on Diffusion.

0.0

.01128

.02256

•03384

.04511

•05637

.06762

.07886

.09008

.10128

.11246

.12362

•13476

•14587

•^695

.16800

.17901

.18999

.20094

.21184

.22270

•23352

.24430

.25502

•26570

•27633

.28690

.29742

.30788

.31828

•32863

•3

•33891

•349 i 3

•35928

•36936

•37938

•38933

.39921

.40901

.41874

•42839

•4

•43797

•44747

•45689

.46623

•47548

.48466

•49375

•50275

.51167

•52050

0.5

.52924 .61168

•5379° .61941

.54646 •62705

•55494 •63459

•56332 .64203

.57162 .64938

.57982 .65663

•58792 •66378

•59594 .67084

.60386 .67780

.68467

.69143

.69810

.70468

.71116

•71754

•72382

•73001

.73610

.74210

.74800

.75952

.76514

.77067

.77610

.78144

.78669

.79184

.79691

•9

.80188

.80677

.81156

.81627

.82089

.82542

.82987

•83423

.83851

.84270

1.0

.84681

.85084

.85478

.85865

.86244

.86614

.86977

•87333

.87680

.88021

•88353

.88679

.88997

.89308

.89612

.89910

.90200

.90484

.90761

.91031

.91296

•9I553

.91805

•92051

.92290

•92524

•9275!

•92973

.93190

.93401

•3

.93606

.93807

.94002

.94191

•94376

•94556

•947 3 i

.94902

•95067

•95229

•4

•95385

•95538

.95686

•95830

•95970

.96105

•96237

•96365

.96490

.96611

1.5

.96728

.96841

•96952

•97059

.97162

.97263

•97360

•97455

•97546

•97635

.97721

.97804

.97884

.97962

.98038

.98110

.98181

.98249

•98315

•98379

•7

.98441

.98500

•98558

•98613

.98667

.98719

.98769

.98817

.98864

.98909

.8 •9

•98952 .99309

•98994 •9933s

•99035 •99366

•99074 •99392

.99111 .99418

.99147 •99443

•99182 .99466

.99216 .99489

.99248 •995 "

.99279 •99532

2.0

•99552

•99572

•99591

.99609

.99626

.99642

.99658

•99673

.90688

.99702

•99715

.99728

.99741

•99753

.99764

•99775

•99785

•99795

•99805

.99814

.99822

.99831

.99839

.99846

.99854

.99861

.99867

.99874

.99880

.99886

•3

.99891

•99897

.99902

.99906

.9991 1

.99920

•99924

.99928

•9993 i

•4

•99935

•99938

.99941

•99944

•99947

.99950

•99952

•99955

•99957

•99959

2.5

.99961

.99963

.99965

•99967

•99969

.99971

•99972

•99974

•99975

.99976

•99978

•99979

.99980

.99981

.99982

.99983

.99984

•99985

.99986

•99987

•7

•99987

.99988

•99989

•99990

.99991

.99992

•99992

•99993

•99994

•99995

•99996

•9

•99996

.99996

•99997

.99997

•99998

3.0

-99999

•99999

I.OOOOO

Taken from a paper by Dr. James Burgess ' on the Definite Integral JL f* er-& dty with Ex-

•y 7JY/ O

tended Tables of Values.' Trans. Roy. Soc. of Edinburgh, vol. xxxix, 1900, p. 257. SMITHSONIAN TABLES.

48 TABLE 24.

LEAST SQUARES.

This table gives the values of the probability P, as defined in last table, corresponding to different values of x I r where r is the " probable error." The probable error r is equal to 0.476947 Jt.

an r

0.0

.00000

.00538

.01076

.01614

.02152

.02690

.03228

.03766

•04303

.04840

O.I

•05378

.05914

.06451

.06987

•07523

.08059

•08594

.09129

.09663

.10197

O.2

.10731

.11264

.11796

.12328

.12860

I339I

.13921

•I4451

.14980

• 15508

o-3

.16035

.16562

.17088

.17614

.18138

.18662

.19185

.19707

.20229

.20749

0.4

.21268

.21787

.22304

.22821

.23336

.23851

.24364

.24876

•25388

.25898

0.5

.26407

.26915

.27421

.27927

.28431

.28934

.29436

•29936

•30435

•30933

0.6

•3H30

•31925

.32419

.32911

•33402

•33892

.34380

.34866

•35352

•35835

0.7

•363 i 7

.36798

•37277

•37755

•38231

.38705

•39*78

•39649

.40118

.40586

0.8

.41052

•4i5r7

.41979

.42440

.42899

•43357

•43813

.44267

.44719

.45169

0.9

.45618

.46064

.46509

.46952

•47393

.47832

.48270

48605

•49139

•49570

1.0

i.i

.50000 .54188

.50428 •54595

•50853 .55001

•5I277 .55404

.51699 .55806

.52119 .56205

•52537 .56602

•52952 • 56998

•53366 •57391

•53778 •57782

1.2

.58171

•58558

•58942

•59325

•59705

.60083

.60460

•60833

.61205

•6i575

i-3

.61942

.62308

.62671

.63032

.63391

•63747

.64102

•64554

.64804

•65152

1.4

.65498

.65841

.66182

.66521

.66858

•67193

.67526

•67856

.68184

.68510

1.5

.68833

•691 55

.69474

.69791

.70106

.70419

.70729

.71038

.71344

.71648

1.6

.71949

.72249

•72546

.7284!

•73134

•73425

.73714

.74000

.74285

•74567

i-7

.74847

•75I24

.75400

•75674

•75945

.76214

.76481

.76746

.77009

.77270

1.8

.77528

•77785

.78039

.78291

•78542

.78790

.79036

.79280

•79522

.79761

1.9

•79999

•80235

.80469

.80700

.80930

.81158

•81383

.81607

.81828

.82048

2.0

.82266

.82481

.82695

.82907

.83117

•83324

•83530

•83734

•83936

•84137

2.1

2.2

•84335 .86216

•84531 .86394

.84726 .86570

.84919 .86745

.85109 .86917

.85298 .87088

.85486 .87258

.85671 •87425

.85854 •87591

.86036 •87755

2-3

.87918

.88078

.88237

•88395

•88550

.88705

.88857

.89008

.89157

.89304

2.4

.89450

•89595

.89738

.89879

.90019

•90157

•90293

.90428

.90562

.90694

2.5

.90825

.90954

.91082

.91208

•9J332

.91456

•9*578

.91698

.91817

•91935

2.6

.92051

.92166

.92280

.92392

•92503

.92617

.92721

.92828

•92934

•93038

2.7

•93I4I

•93243

•93344

•93443

•93541

•93638

•93734

.93828

.93922

.94014

2.8

.94105

•94195

.94284

•94371

•94458

•94543

.94627

.94711

•94793

.94874

2.9

•94954

•95033

.95111

•95187

•95263

•95338

.95412

•95484

•95557

.95628

.95698

.96346

.96910

•97397

.97817

.98176

.98482

•98743

.98962

.99147

.99302

•99431

•99539

99627

.99700

.99760

.99808

.99848

•99879

.99905

.99926

•99943

.99956

.99966

•99974

.99980

•99985

.99988

.99991

•99993

TABLE 25. LEAST SQUARES.

Values of the factor o.6745\/-^r .

\»»— 1

This factor occurs in the equation * =r o.6j4S\ — for tne probable error of a single observation, and other

j| n — i similar equations.

n =

0-6745

0.4769

0.3894

0.3372

0.3016

0.2754

0.2549

0.2385

0.2248 •1547

0.2133 .1508

.2029 .1472

.1947 •1438

.1871 .1406

.1803 •1377

.1742 •1349

.1686 •!323

.1636 .1298

.1590 •1275

3°

.1252

.1231

.1211

.1192

.1174

."57

.1140

.1124

.1109

.1094

.1080

.1066

•1053

.1041

.1029

.1017

.1005

.0994

.0984

.0974

0.0964

0.0954

0.0944

0.0935

0.0926

0.0918

0.0909

0.0901

0.0893

0.0886

.0878

.0871

.0864

.0857

.0850

.0843

.0837

.0830

.0824

.0818

.0812

.0806

.0800

•0795

.0789

.0784

•0778

•0773

.0768

.0763

80 90

•0759 •0715

•0754 .0711

.0749 .0707

•0745 .0703

.0740 .0699

.0736 .0696

•0731 .0692

.0727 .0688

.0723 .0685

,0719 .0681

SMITHSONIAN TABLES.

TABLE 26. LEAST SQUARES

Values of the factor 0.6745

_--.

This factor occurs in the equation e = o.6j4$\ ^ for the probable error of the arithmetic mean.

\ n(n — i)

» =

0.0711

0.0643

0.4769 .0587

0.2754 .0540

0.1947 .0500

0.1508 .0465

0.1231 •0435

0.1041 .0409

0.0901 .0386

0.0795 •0365

.0346

.0329

.0314

.0300

.0287

.0275

.0265

.0255

.0245

.0237

0.0229

0.0221

0.0214

0.0208

0.0201

0.0196

0.0190

0.0185

0.0180

0.0175

.0171

.0167

.0163

.0159

•0155

.0152

.0148

.0145

.0142

.0139

.0136

.0134

.0131

.0128

.OI26

.0124

.0122

.0119

.0117

.0115

TABLE 27. LEAST SQUARES.

Values of the factor 0.8453-v/ *

This factor occurs in the equation et = 0.8453

for the probable error of a single observation.

n -

0.0891

0.0806

0.5978 .0736

o.345i .0677

0.2440 .0627

0.1890 •°583

o.i543 .0546

0.1304 •0513

0.1130 .0483

0.0996

•0457

•0434

.0412

•0393

.0376

.0360

•0345

•0332

.0319

.0307

.0297

0.0287

0.0277

0.0268

0.0260

0.0252

0.0245

0.0238

0.0232

0.0225

0.0220

.0214

.0209

.0204

.0199

.0194

.0190

.0186

.0182

.0178

.0174

.0171

.0167

.0164

.0161

.0158

.0155

.0152

.0150

.0147

.0145

TABLE 28. LEAST SQUARES,

Values of 0.8453^

This table gives the average error of the arithmetic mean when the probable error is one.

n =

0.4227

0.1993

0.1220

0.0845

0.0630

0.0493

0.0399

0.0332

0.0282 .0097

0.0243 .0090

.0212 .0084

.0188 .0078

.0167 .0073

.0151 .0069

.0136 .0065

.0124 .0061

.0114 .0058

.0105 •0055

0.0052

0.0050

0.0047

0.0045

0.0043

0.0041

0.0040

0.0038

0.0037

0.0035

.0034

•0033

.0031

.0030

.OO29

.0028

.0027

.0026

.0025

.0024

.0023

.0022

.OO22

.0021

.0020

.0019

SMITHSONIAN TABLES.

50 TABLE 29.

DIFFUSION.

2 /-« C& da. Inverse* values of v fc = i — ^f~J0

log x = log (2?) + log\//£A t expressed in seconds. = log 8 + \og\/ki. t expressed in days. = log 7 -f- log \/kt. " " years.

j, k = coefficient of diffusion.! *• = initial concentration. v = concentration at distance x, time t.

V/C

log 2?

log 3

logy

0.00

+ 00

.01

0.56143

3.6428

3.02970

1070.78

4.31098

20463.

.02

.51719

3.2900

2.98545

967.04

.26674

18481.

•03

.48699

3.0690

.95525

902.90

•23654

17240.

.04

.46306

2.9044

.93132

853-73

.2I26l

16316.

0.05

0.44276

2.7718

2.91102

814.74

4.19231

I557I-

.06

.07

.42486

.40865

2.6598

2.5624

.89311 .87691

781.83

753-20

.17440 .15820

14942. H395-

.08

.39372

2.4758

.86198

72775

•14327

13908.

.09

•37979

2.3977

.84804

704.76

•12933

13469.

0.10

.11

0.36664 .35414

2.3262

2.2602

2-83490 .82240

683.75 664.36

4.11619 .10369

13067. 12697.

.12

.34218

2.1988

.81044

646.31

.09173

12352.

•13

•33067

2.1413

•79893

629.40

.08022

12029.

.14

•31954

2.0871

.78780

613-47

.06909

11724.

0.15

0.30874

2.0358

2.77699

598.40

4.05828

11436.

.16

.29821

1.9871

•76647

584.08

.04776

11162.

.17

.28793

1.9406

•75619

570.41

.03748

10901.

.18

.27786

1.8961

.74612

557-34

.02741

10652.

.19

.26798

1.8534

.73624

544.80

•01753

10412.

0.20

.21

0.25825 .24866

1.8124

1.7728

2.72651 .71692

532.73 521.10

4.00780 3.99821

10181. 9958.9

.22

.23919

1-7346

.70745

509.86

.98874

9744.1

•23

.22983

1.6976

.69808

498.98

•97937

9536.2

.24

.22055

1.6617

.68880

488.43

.97010

9334-6

0.25

0.21134

1.6268

2.67960

478.19

3.96089

9138.9

.26

.20220

I-593°

.67046

468.23

•95T75

8948.5

.27

.19312

1.5600

•66137

458-53

.94266

8763.2

.28

.18407

1.5278

.65232

449.08

.93361

8582.5

.29

•17505

1.4964

.64331

439-85

.92460

8406.2

0.30

0.16606

1-4657

2.63431

430.84

3.91560

8233.9

•32

.15708 .14810

1-4357 1.4064

•62533 .61636

422.02 4I3-39

[89765

8065.4 7900.4

•33

.13912

1.3776

.60738

404-93

.88867

7738.8

•34

.13014

1-3494

.59840

396.64

.87969

7580.3

0.35

0.12114

1.3217

2.58939

388.50

3.87068

7424.8

•36

.11211

1.2945

.58037

380-51

.86166

7272.0

•10305

1.2678

•57I3I

372.66

.85260

7122.0

.38

.09396

1.2415

.56222

364-93

.84351

6974.4

•39

.08482

1.2157

•55308

357-34

.83437

6829.2

0.40

0.07563

1.1902

2.54389

349-86

3.82518

6686.2

.41

.06639

1.1652

.53464

342.49

•8i593

6545-4

.42

.05708

1.1405

•52533

335-22

.80662

6406.6

•43

.04770

1.1161

.5T595

328.06

•79724

6269.7

•44

.03824

1.0920

.50650

320.99

.78779

6134.6

0.45

0.02870

1.0683

2.49696

314.02

3-77825

6001.3

.46

.01907

1.0449

48733

307-13

.76862

5869.7

•47

.00934

1.0217

.47760

.75889

5739-7

.48

9-9995 1'

0.99886

•46776

293.60

•749°5

5611.2

.98956

0.97624

.45782

286.96

•73911

5484.1

0.50

9-97949

0.95387

2-44775

280.38

3.72904

5358.4

* Kelvin, Mathematical and Physical Papers, vol. III. p. 428 ; Becker, Am. Jour, of Sci. vol. III. 1897, p. 280. t For direct values see table 23.

Taken from unpublished manuscript of C. E. Van Orstrand. SMITHSONIAN TABLES,

TABLE 29 (continued). DIFFUSION.

v/c

log zq

,OgJ

logy

0.50

9-97949

0.95387

2.44775

280.38

3.72904

5358.4

.51

.96929

.93J74

•43755

273-87

.71884

5234.1

.52 •53

.95896 .94848

.90983 .88813

.42722 .41674

267.43 261.06

.70851 .69803

5111.0 4989.1

•54

.93784

.86665

.40610

25474

•68739

4868.4

0.55

•56

9.92704 .91607

0.84536 .82426

2-3953° •38432

248.48 242.28

3-67659 .66561

4748.9 4630.3

•57

.90490

•80335

236.13

•65445

4512.8

•58

.89354

.78260

.36180

230.04

.64309

4396.3

•59

.88197

.76203

•35023

223.99

.63152

4280.7

0.60

9.87018

0.74161

2.33843

217.99

3-6I973

4166.1

.61

.85815

•72135

.32640

212.03

.60770

4052.2

.62

.84587

.70124

.31412

206.12

•59541

3939-2

•63

•83332

.68126

•3OI57

200.25

.58286

3827.0

.64

.82048

.66143

.28874

194.42

•57003

3715.6

0.65

.66 •67

9.80734 .79388 .78008

0.64172 .62213 .60266

2.27560 .26214 •24833

188.63 182.87 177.15

3.55689

•54343 .52962

3604-9 3494-9 3385.4

.68

•76590

•58331

.23416

171.46

3276.8

.69

•75133

.56407

.21959

165.80

.50088

3168.7

0.70

.72

973634

.72089

•70495

2.20459 .18915 .17321

160.17 154.58 149.01

3.48588 .47044 4545°

3061.1 2954.2 2847.7

•73

.68849

.48808

.15675

143-47

•43804

2741.8

•74

.67146

.46931

.13972

'37-95

.42101

2636.4

0.75

9.65381

0.45062

2.12207

132.46

3-40336

253*4

•76

•63550

.43202

.10376

126.99

•38505

2426.9

•77

.61646

.41348

.08471

121.54

.36600

2322.7

•78

.59662

.39502

.06487

n6.ii

.34616

2219.0

•79

•57590

.37662

.04416

110.70

.32545

2115.7

0.80

9.55423

0.35829

2.02249

105-31

3-30378

2012.7

,8 1

•5315°

.34001

1.99975

99-943

.28104

1910.0

.82

.50758

.32180

•97584

94-589

•25713

1807.7

•83 .84

•48235 •45564

.30363

.28552

.95061 •92389

89.250 83.926

.23190 .20518

1705-7 1603.9

0.85

9.42725

0.26745

I-8955I

78.615

3.17680

1502.4

.86 .87

•39695 •36445

.24943 .23145

.86521 .83271

73-3!7 68.032

.14650 .11400

1401.2 1300.2

.88

•32940

.21350

.79766

62.757

.07895

1199.4

.89

•29135

.19559

.7596i

57492

3-04090

1098.7

0.90

.91

9.24972

•20374

0.17771 .15986

1.71797 .67200

52.236 46.989

2.99926 •95329

99|3i

.92

•15239

.14203

.62065

4I-750

.90194

797.89

•93

.09423

.12423

.56249

36-516

•84378

697-88

•94

9.02714

.10645

•49539

31.289

.77668

597-98

0.95

8.94783

0.08868

1.41609

26.067

2.69738

498.17

.96

.85082

.07093

.31907

20.848

.60036

398.44

•97

.72580

.05319

.19406

15.633

•47535

298.78

•98

•54965

.03545

.01791

10.421

.29920

199.16

•99

.24859

.01773

9.71684

5.21007

1.99813

99-571

1.00

— 00

o.ooooo

— oo

o.ooooo

— 00

0.000

SMITHSONIAN TABLES.

TABLE 30. GAMMA FUNCTION.*

Value of log I e—af^dx + 10.

Values of the logarithms + 10 of the " Second Eulerian Integral " (Gamma function) | e-*x*-*dx or log T(n )4-ro

for values of n between i and 2. When n has values not lying between i and 2 the value of the function can be readily calculated from the equation r(»+i) = nT(n) =. «(»— i) . . . («— r)T(n— r).

1.00

9>99

97497

95ooi

92512

90030

87555

85087

82627

80173

77727

I.OI 1.02

75287 51279

4891!

70430 46561

68011 44212

65600

41870

63196 39535

60799 37207

58408 34886

56025

32572

53648 30265

1.03 1.04

27964 05334

25671 03108

23384

21104 98677

18831 96471

16564 94273

1 2052 89895

09806 87716

07567 85544

1.05

9-9883379

81220

79068

76922

74783

72651

70525

68406

66294

64188

i. 06

62089

59996

579*0

55830

53757

51690

49630

47577

45530

43489

1.07

41469

39428

37407

35392

33384

31382

29387

27398

25415

23449

i. 08 1.09

21469 02123

19506 00223

17549 98329

15599 96442

13655 9456i

11717 92685

07860 89856

05941 87100

04025 3-5250

1.10

9.9783407

81570

79738

779*4

76095

74283

72476

70676

68882

67095

i. ii

65313

63538

61768

60005

58248

56497

54753

53014

51281

49555

1. 12

47834

46120

44411

42709

41013

39323

37638

34288

32622

I.I3

30962

29308

27659

26017

24381

22751

21126

19508

17896

16289

I.I4

14689

13094

11505

09922

08345

06774

05209

03650

02096

00549

1.15

9.9699007

97471

95941

94417

92898

91386

89879

88378

86883

85393

1.16

83910

82432

80960

79493

78033

76578

75I29

73686

72248

70816

!:!$

69390 55440

67969 54076

66554 52718

65H5 51366

63742 50019

48$

60952 47341

59566 46011

58185 44687

56810 43368

1.19

42054

40746

39444

36856

35570

34290

33OI6

3*747

30483

1.20

9.9629225

27973

26725

25484

24248

23017

21792

20573

19358

18150

1. 21

16946

15748

^369

12188

IIOII

09841

08675

06361

1.22

05212

04068

02930

01796

00669

99546

98430

973*8

96212

95* **

1.23

594015

92925

91840

90760

89685

88616

87553

86494

8544*

84393

1.24

83350

82313

81280

80253

79232

78215

77204

76198

75*97

74201

1.25

1.26

9-95732" 63592

72226 62658

71246 61730

70271 60806

69301

59888

68337 58975

67377 58067

66423

57*65

6|474 56267

6453°

55374

1.27

54487

53604

52727

51855

50988

50126

49268

48416

47570

46728

1.28

45891

45059

44232

434io

42593

41782

40975

40173

39376

38585

1.29

37798

37016

36239

35467

34700

33938

32439

31682

30940

1.30

9.9530203

29470

28743

28021

27303

26590

25883

25180

24482

23789

1.31

23100

22417

21739

21065

20396

19732

19073

18419

17770

17125

1.32

16485 10353

15850 09766

15220 09184

*4595 08606

13975 08034

'3359 07466

12748 06903

12142 06344

11540 0579*

10944 05242

i-34

04698

04158

03624

03094

02568

02048

01532

OIO2I

00514

00012

1.35

9-94995I5

99023

98535

98052

97573

97100

96630

96166

95706

95251

1.36

94800

94355

939*3

93477

92617

92194

91776

91362

90953

*-37

9°549

90149

89754

89363

88977

88595

88218

87846

87478

87II5

1.38

86756

86402

86052

85707

85366

85030

84698

84371

84049

83731

83417

83108

82803

82503

82208

81916

81630

81348

81070

80797

1.40

1.41

9.9480528 78084

80263 77864

80003 77648

79748 77437

79497 7723°

79250 77027

79008 76829

78770 76636

7?S3£ 76446

78308 76261

1.42

76081

75905

75733

75565

75402

75243

75089

74939

74793

74652

1-43 1.44

74515 73382

74382 73292

74254 73207

74130 73^5

74010 73°49

73894 72976

73783 72908

73676 72844

73574 72784

73746 72728

* Quoted from Carr's " Synopsis of Mathematics," and is there quoted from Legendre's " Exercises de Calcul Integral," tome ii.

SMITHSONIAN TABLES. s

TABLE 30 (continued}.

GAMMA FUNCTION.

1.45

9.9472677

72630

72587

72549

725H

72484

72459

72437

72419

72406

1.46

72397

72393

72392

72396

72404

72416

72432

72452

72477

72506

1.47

72539

72576

72617

72662

72712

72766

72824

72886

72952

73022

1.48

73097

73*75

73258

73345

73436

73531

73630

73734

73841

73953

1.49

74068

74188

743 i 2

74440

74572

74708

74848

74992

75Hi

75293

1.50

9-9475449

75610

75774

75943

76116

76292

76473

76658

76847

77040

i-5«

1.52

77237 79426

77438 79667

77642 79912

77851 80161

78064 80414

78281 80671

78502 80932

78727 81196

78956 81465

79189 81738

i-54

82015 84998

82295

82580 85642

82868 85970

83161 86302

83457 86638

83758 86977

84062 87321

84370 87668

84682 88019

1.55

9.9488374

88733

89096

89463

89834

90208

90587

90969

9!355

9I745

1.56

92139

92537

92938

93344

93753

94166

94583

95004

95429

95.857

1-57

96289

96725

97165

97609

98056

98508

98963

99422

99885

00351

1.58

500822

01296

01774

02235

02741

03230

03723

04220

04720

05225

i-59

05733

06245

06760

07280

07803

08330

08860

09395

09933

10475

1.60

9.9511020

11569

I2I22

12679

13240

13804

H372

H943

I55I9

16098

1.61

16680

17267

17857

18451

19048

19650

20254

20862

2H75

22O9I

1.62

22710

23333

23960

24591

25225

25863

26504

27149

27798

28451

1.63 1.64

29107 35867

29767 36563

30430 37263

31097 37966

31767 38673

32442 39383

33120 40097

33foi 40815

34486 41536

35175 4226O

1.65

9.9542989

43721

44456

45195

45938

46684

47434

48187

48944

49704

1.66

50468

51236

52007

52782

5356o

54342

55127

55916

56708

57504

1.67

58303

59106

59913

60723

61536

62353

63174

63998

64826

65656

1.68

66491

67329

68170

69015

69864

70716

7I57I

7243°

73293

74159

1.69

75028

75901

76777

77657

78540

79427

80317

81211

82108

83008

1.70

9.9583912

84820

85731

86645

87536

88484

89409

90337

21268

92203

1.71

93 HI

94083

95028

95977

96929

97884

98843

99805

00771

01740

1.72

602712

03688

04667

06636

07625

08618

09614

I06I3

Il6l6

12622

13632

H645

1 5661

16681

17704

18730

19760

2O793

21830

1.74

22869

23912

24959

26009

27062

28118

29178

30241

3*308

32377

1.75

9-963345I

34527

35607

36690

37776

38866

39959

41055

42155

43258

1.76

44364

§473

46586

47702

48821

49944

51070

52200

53331

54467

1.77

556o6

749

57894

59043

60195

61350

62509

63671

64836

66004

1.78

67176

35i

69529

70710

71895

73082

74274

75468

76665

77866

1.79

79070

80277

81488

82701

83198

85138

86361

87588

88818

90051

1.80

9.9691287

92526

93768

950H

96263

97515

98770

00029

01291

0255;

1.81

703823

05095

06369

07646

08927

IO2II

11498

12788

14082

1.82

16678

17981

19287

20596

21908

23224

24542

25864

27189

28517

1 1-83

29848

31182

32520

33860

35204

36551

37900

39254

40610

41969

1.84

43331

44697

46065

47437

48812

50190

S'571

52955

54342

55733

1.85

1.86

9.9757126 71230

58522 72657

59922 74087

61325

75521

62730 76957

64140 78397

65551 79839

66966 81285

68384 82734

69805 84186

1.87 1.88 1.89

85640 800356

87098 01844 16893

88559 03335 I84H

90023 04830 !9939

91490 06327 21466

92960 07827 22996

94433 09331 2453°

95910 10837 26066

97389 12346 27606

98871

13859 29148

1.90

9.9830693

32242

33793

35348

36905

38465

40028

41595

43l64

44736

1.91 1.92

46311 62226

47890 63834

4947 i 65445

67058

^8675

54232 70294

55825 71917

5742i 73542

59020 75170

60622 76802

1.93

78436

80073

81713

83356

85002

86651

88302

89957

93275

1.94

9493s

96605

98274

99946

01621

03299

04980

06663

0835°

10039

1.95

9.9911732

13427

15125

16826

18530

20237

21947

23659

25375

27093

1.96

28815

3°539

32266

33995

35728

37464

39202

40943

42688

1.97

46185

47937

49693

53213

54977

56744

58513

60286

62062

1.98 1.99

63840 81779

65621 83588

67405 85401

69192 87216

70982 89034

72774 90854

74570 92678

76368 94504

78169 96333

79972 98165

SMITHSONIAN TABLES.

54 TABLE 31 .

ZONAL HARMONICS.*

The values of the first seven zonal harmonics are here given for every degree between 6 = o° and 0 = 90°.

0°

I.OOOO

1°

0.9998

0.9995

0.9991

0.9985

0.9977

0.9967

0-9955

•9994

.9982

•9963

•9939

.9909

.9872

.9829

.9986 .9976

•9959

.9918 .9854

.9863 •9758

•9795 .9638

•9713

•9495

.9617 •9329

.9962

.9886

•9773

.9623

•9437

.9216

.8961

6°

•9945

.9836

.9674

•9459

.9194

.8881

.8522

•9925

•9777

•9557

.9267

.8911

.8476

.7986

•9903

.9709

•9423

.9048

.8589

•8053

.7448

.9877 .9848

•9633 .9548

•9273 .9106

.8803 •8532

.8232 .7840

•7571 •7045

.6831 .6164

11°

.9816

•9454

.8923

.8238

•7417

.6483

.5461

.9781

•9352

.8724

.7920

.6966

•5892

•4732

•9744

.9241

.8511

•7582

.6489

•5273

•3940

•9703

.9122

.8283

.7224

•5990

•4635

.3219

.9659

•8995

.8042

.6847

•5471

•3982

•2454

16°

.9613 •9563

.8860 .8718

.7787

.6046

•4937 •4391

•3322 .2660

.1699 .0961

•9511

.8568

.7240

.5624

•3836

.2002

.0289

•9455

.8410

.6950

.5192

.3276

•1347

—•0443

•9397

.8245

.6649

•475°

•2715

.0719

— .1072

21°

•9336

.8074

•6338

.4300

.2156

.0107

—.1662

.9272

•7895

.6019

•3845

.1602

— .0481

— .2201

.9205

.7710

.5692

•3386

•1057

—.1038

—.2681

•9135

•7518

•5357

.2926

•0525

— .1559

—•3095

.9063

.7321

.5016

.2465

.0009

—•2053

—•3463

26°

.8988

.7117

.4670

.2007

—.0489

—.2478

—•3717

.8910

.6908

•4319

•1553

—.0964

-.2869

—.3921

.8829 .8746

.6694 .6474

.3964 .3607

.1105 .0665

—.1415 -.1839

—.3211 —•35°3

—.4052 —.4114

.8660

.6250

.3248

.0234

—•2233

—•3740

— .4101

31°

•8572

.6021

.2887

—.0185

—•2595

—•3924

— .4022

.8480

•5788

.2527

—.0591

—•2923

—.4052

-•38/6

•8387

•5551

.2167

—.0982

—.3216

— .4126

—.3670

.8290

•5310

.1809

—•1357

—•3473

—.4148

—•3409

.8192

•5065

•1454

—.1714

—.3691

—•4115

—.3096

36°

.8090

.4818

.1102

— .2052

—•3871

—.4031

—2738

.7986

•4567

•0755

—.2370

— .4011

—3898

—•2343

.7880

•43H

.0413

—.2666

—.4112

—•3719

— .1918

.7771

•4059

.0077

—.2940

—.4174

—•3497

—.1469

.7660

.3802

—.0252

—.3190

—.4197

—•3234

—•1003

41°

•7547

•3544

—.0574

—.3416

—.4181

—2938

— -°534

•7431

.3284

—.0887

—.3616

—.4128

— .2611

— .0065

•7314

•3023

— .1191

— -3791

—•4038

—•225?

•°395

•7193

.2762

-.I485

—•3940

— -39J4

—.1878

.0846

.7071

.2500

—.1768

— .4062

—•3757

—.1485

.1270-

* Calculated by Prof. Perry (Phil. Mag. Dec. 1891). See also A. Gray, "Absolute Measurements in Electricity and Magnetism," vol. ii., part 3. —

SMITHSONIAN TABLES.

TABLE 31 (continued). ZONAL HARMONICS.

46°

0.6947

0.2238

— .2040

—.4158

-.3568

—.1079

0.1666

.6820

.1977

— .2300

—4252

—•3350

—.0645

.2054

48 49

.6691 .6561

.1716 .1456

-.2547 — .2781

—.4270 —.4286

—•3105 —.2836

-.0251 .0161

•2349 .2627

.6428

.1198

— .3002

—4275

—•2545

•0563

.2854

51°

.6293

.0941

—.3209

—4239

—•2235

-0954

•3°3r

.6157 .6018

.0686 •0433

—.3401 —3578

-.4178

— .1910 —•I57I

.1326 .1677

•3153 .3221

54 55

.5878 •5736

.0182 — .0065

-•$6

-3852

—.1223 —.0868

.2002 -.2297

•3234 •3*91

56°

•5592

— .0310

40l6

—3698

—.0510

•2559

•3095

•5446

—•0551

.4131

—•3524

— .0150

.2787

.2949

•5299

—.0788

—4229

— -3331

.0206

.2976

•2752

•5I5°

— .1021

—4310

— -3«9

.0557

•3125

.2511

.5000

— .I25O

—4375

—.2891

.0898

.3232

.2231

61°

.4848

—.1474

—4423

—.2647

.1229

.3298

.1916

.4695

— .1694

—4455

—.2390

.1545

•3321

•1571

•4540

— .1908

—.4471

—.2121

.1844

•3302

.1203

4384

—.2117

—.4470

—.1841

.2123

,3240

.0818

.4226

—.2321

—4452

—•1552

.2381

•3138

.0422

66°

.4067

—.2518

—.4419

— .1256

.2615

.2996

.0021

67 68

•3907 •3746

— .2710 —.2896

—4370 —4305

—•0955 — .0650

.2824 •3005

.2819 .2605

— -°375 —.0763

•3584

— -3°74

—4225

—•0344

•3158

.2361

.3420

—•3245

—.4130

—.0038

.3281

.2089

—•$5

71°

•3256

—.3410

— .4021

.0267

•3373

.1786

—.1811

72 73

.3090 .2924

-.3568

-.3898 —•376i

^64

•3434 •3463

.1472 .1144

—.2099 —•2347

.2756

—.'3860

— .3611

•"53

.3461

•0795

—•2559

.2588

—•3995

—•3449

•1434

•3427

.0431

—.2730

76°

.2419

— .4112

—•3275

•1705

•3362

.0076

—.2848

.2250

—.4241

—.3090

.1964

.3267

— .0284

—.2919

.2079

—4352

—.2894

.2211

•3143

—.0644

—•2943

.1908

—4454

—.2688

•2443

.2990

—.0989

—.2913

•1736

—4548

—•2474

.2659

.2810

—.1321

-•2835

81°

.1564

—4633

—.2251

.2859

.2606

—.1635

—.2709

.1392

—.4709

— .2020

.3040

.2378

— .1926

—•2536

.1219

—4777

—1783

•3203

.2129

—.2193

—•2321

.1045

— 4836

— -'539

.1861

—.2431

— .2067

.0872

—.4886

— .1291

.3468

•1577

-.2638

—.1779

86°

.0698

—4927

—.1038

.3569

.1278

—.2811

— .1460

•0523

—4959

—.0781

.3648

.0969

—.2947

—.1117

•0349

—.4982

— .0522

•3704

.0651

—•3045

—0735

.0175

—4995

— .0262

•3739

.0327

—•3105

—.0381

.0000

— .5000

— .0000

•3750

.0000

—•3125

— .0000

SMITHSONIAN TABLES.

TABLE 32.

MUTUAL INDUCTANCE.* M

Table of values of log — 17= for facilitating the calculation of the mutual inductance M of two coaxial circles of

4*-V«*' f(a_a/)2_l_£2> J

radii a, a', at distance apart b. The table is calculated for intervals of 6/ in the value of cos-1 \ (g_a/\a _L ^2 j from 60° to 90°.

12'

18'

24'

30'

36'

42'

48'

54'

60°

1.4994783

5022651

5050505

5078345

5106173

5133989

5161791

5189582

5217361

5245128

5272883

5300628

5328361

5356084

5383796

5411498

5439*90

5466872

5494545

5522209

5549864

55775io

5605147

5632776

5660398

5688011

5715618

5743217

5770809

5798394

5825973

5853546

5881113

5908675

5936231

5963782

5991322

6018871

6046408

6073942

6101472

6128998

6156522

6184042

6211560

6239076

6266589

6294101

6321612

6349121

65°

1.6376629

6404137

6431645

6459153

6486660

6514169

6541678

6569189

6596701

6624215

6651732

6679250

6706772

6734296

6761824

6789356

6816891

6844431

6871976

6899526

6927081

6954642

6982209

7009782

7037362

7064949

7092544

7120146

7H7756

7175375

7203003

7230640

7258286

7285942

7313609

7341287

7368975

7396675

7424387

7452111

7479848

7507597

753536i

7563138

7590929

7618735

7646556

7674392

7702245

7730114

70°

1.7758000

7785903

7813823

7841762

7869720

7897696

7925692

7953709

7981745

8009803

8037882

8065983

8094107

8122253

8150423

8178617

8206836

8235080

8263349

8291645

8319967

8348316

8376693

8405099

8433534

8461998

8490493

8519018

8547575

8576164

8604785

8633440

8662129

8690852

8719611

8748406

8777237

8806106

8835013

8863958

8892943

8921969

8951036

8980144

9009295

9038489

9067728

9097012

9126341

9I557I7

75°

7.9185141

9214613

9244i35

9273707

9303330

9333005

9362733

93925 l 5

9422352

9452246

9482196

9512205

9542272

9572400

9602590

9632841

9663157

9693537

9723983

9754497

9785079

98I5731

9846454

9877249

9908118

9939062

9970082

0001181

0032359

0063618

0.0094959

0126385

0157896

0189494

0221181

0252959

0284830

0316794

0348855

0381014

04i3273

0445633

0478098

0510668

0543347

0576136

0609037

0642054

0675187

0708441

80°

0.0741816

07753J6

0808944

0842702

0876592

0910619

0944784

0979091

1013542

1048142

1082893

1117799

1152863

1188089

1223481

1259043

1294778

1330691

1366786

1403067

J439539

1476207

1513075

I550H9

1587434

1624935

1662658

1700609

1738794

1777219

1815890

1854815

1894001

1933455

1973184

2013197

2053502

2094108

2135026

2176259

2217823

2259728

2301983

2344600

2387591

2430970

2474748

2518940

2563561

2608626

85°

0.2654152

2700156

2746655

2793670

2841221

2889329

2938018

2987312

3037238

3087823

3139097

3191092

3243843

3297387

3351762

3407012

3463184

3520327

3578495

3637749

3698153

3759777

3822700

3887006

3952792

4020162

4089234

4160138

4233022

4308053

4385420

4465341

4548064

4633880

4723127

4816206

4913595

5015870

5123738

5238079

5360007

5490969

5632886

5788406

5961320

6i5737o

6385907

6663883

7027765

7586941

* Quoted from Gray's SMITHSONIAN TABLES.

'Absolute Measurements in Electricity and Magnetism," vol. ii., p. 852.

TABLE 33. ELLIPTIC INTEGRALS.

Values ol I 3(1- sin2* sin2 $)**<*£.

This table gives the values of the integrals between o and ir / 2 of the function (i — sin2 0 sin2 $) d$ for different val- ues of the modulus corresponding to each degree of 6 between o and 90.

rs d*

Cl #

f *(i sin^sin »</

^/O (i— sin20sin2<J>)*

J0 (,-»»*«**)»

Number.

Log.

Number.

Log.

Number.

Log.

Number.

Log.

0°

1.5708

0.196120

1.5708

0.196120

45°

1.8541

0.268127

I-3506

0.130541

5709

I96I53

5707

196087

8691

271644

3418

127690

5713

196252

5703

195988

8848

275267

3329

124788

5719

196418

195822

9OII

279001

3238

121836

5727

196649

5^9

I9559I

9180

282848

3*47

118836

5°

1. 5738

0.196947

1.5678

0.195293

50°

L9356

0.2868II

1-3055

0.115790

5751

197312

5665

194930

9539

290895

2963

112698

5767 5785

197743 198241

5649 5632

194500 194004

9729 9927

295IOI 299435

2870 2776

109563 106386

5805

198806

5611

193442

2.0133

303501

2681

103169

10°

1.5828

0.199438

L5589

0.192815

55°

2-0347

0.308504

1.2587

0.099915

5854

200137

5564

192121

0571

313247

2492

096626

5882

200904

5537

1913(32

0804

318138

2397

093303

3 4

5913 5946

201740 202643

5507 5476

190537 189646

1047 1300

323182 328384

2301 2206

089950 086569

15°

I.598I

0.203615

1.5442

0.188690

60°

2.1565

0-333753

I.2III

0.083164

6O2O

204657

5405

187668

1842

339295

2OI5

079738

6061

205768

5367

I8658I

2132

345020

1920

076293

6105

206948

5326

185428

2435

350936

1826

072834

6151

208200 .

5283

184210

2754

357053

1732

069364

20°

1.6200

0.209522

1.5238

0.182928

65°

2.3088

0.363384

1.1638

0.065889

6252

210916

181580

3439

369940

*545

062412

6307

212382

5141

180168

3809

376736

1453

058937

6365

213921

5090

178691

4198

383787

1362

055472

6426

215533

5037

I77I50

4610

39III2

1272

052020

25°

1.6490

0.217219

1.4981

0.175545

70°

2.5046

0.398730

1.1184

0.048589

6557

218981

4924

173876

55°7

406665

1096

045l83

6627

2208l8

4864

172144

5998

4M943

ion

041812

6701

222732

4803

170348

6521

423596

0927

038481

6777

224723

4740

168489

7081

432660

0844

035200

30°

1.6858 6941

0.226793 228943

1.4673 4608

0.166567 164583

75°

2.7681 8327

0.442176 452196

1.0764 0686

0.031976 028819

7028

23H73

4539

162537

9026

462782

0611

025740

7119

233485

4469

160429

9786

474008

0538

022749

7214

235880

4397

158261

3-0617

485967

0468

019858

35°

I.73I2

0.238359

1.4323

0.156031

80°

3.1534

0.498777

1.0401

0.017081

7415

240923

4248

153742

2553

5I259I

0338

014432

7522

243575

4171

I5I393

3699

527613

0278

011927

7633

246315

4092

148985

5004

544120

0223

009584

7748

249146

4013

146519

6519

562514

0172

007422

40°

1.7868 7992

0.252068

255085

I-393I

0.143995 141414

85°

3-8317 4.0528

0.583396 607751

1.0127 0086

0.005465 003740

8122

258197

3765

138778

3387

637355

0053

002278

8256

261406

3680

136086

7427

676027

0026

OOII2I

8396

264716

3594

1 33 340

5-4349

735192

0008

OOO326

45°

1.8541

0.268127

1.3506

0.130541

90°

1. 0000

SMITHSONIAN TABLES.

$8 TABLE 34.

MOMENTS OF INERTIA, RADII OF GYRATION, AND WEIGHTS.

In each case the axis is supposed to traverse the centre of gravity of the body. The axis is one of symmetry. The mass of a unit of volume is w.

Body.

Axis.

Weight.

Moment of Inertia Io.

Square of Ra- dius of Gyra- tion p2.

Rir"?>r&

Sphere of radius r

Diameter

47Tw//

£r

Spheroid of revolution, po- lar axis 20, equatorial di-

Polar axis

4irwar2

Sinvar*

2^2

ameter 2r

Ellipsoid, axes 20, 2b, 20

Axis 20,

qirwabc 3

4.irwabc(bz+c* )

^2+^2

Spherical shell, external ra- dius r, internal r'

Diameter

47rw(r3 — r's)

Sirzvir6 — r'5)

2(r5 — ^/5)

S(r8—r/s)

Ditto, insensibly thin, ra- dius r, thickness dr

Diameter

<»*&.

&irwr*dr

2r*

Circular cylinder, length 2a, radius r

Longitudinal axis 20,

«*H

irwar*

Elliptic cylinder, length 20, transverse axes 2b, 2c

Longitudinal . axis 20,

2-trwabc

•Kwabc(P-\-c*)

jy-^j

Hollow circular cylinder, length 2a, external ra- dius r, internal r1

Longitudinal axis 20,

~-w

~+*-^

Ditto, insensibly thin, thick- ness dr

Longitudinal axis 20

qmuardr

Vrw^dr

Circular cylinder, length 2a, radius r

Transverse diameter

2-tnuar2

invar1 ( yz-\- 40?)

r* a*

4+3

Elliptic cylinder, length 2a,

Transverse

irwabc(y'1-{-4[a'2')

c<i i ^

transverse axes 2a, 2b

axis 2b

Hollow circular cylinder,

Transverse

mva 1 ^-r'*) )

r2+r'2 «2

dius r> internal r1

diameter

r )

6 } +4^2(^2-^) ;

4 "*" 3

Ditto, insensibly thin, thick- ness dr

Transverse diameter

^irwardr

¥+3

Rectangular prism, dimen- sions 2a, 2b, 2c

Axis 20,

Swabc

8wabc(l>2+^)

Rhombic prism, length 20, diagonals 2b, 2c

Axis 2a

qwabc

zwabctft+c1)

^2-fr2

Ditto

Diagonal 2b

tpvabc

2wabc(c 2+2a2)

^ai

(Taken from Rankine.)

SMITHSONIAN TABLES.

TABLES 35-36. BRITISH GAUGE NUMBERS AND SIZES OF WIRES.

For Brown & Sharp American Gauge and Electrical Constants see Tables 40 and 41. TABLE 35. —British Standard Wire Gauge. TABLE 36. — Birmingham Wire Gauge.

Diameter in I Inches.

Section in Sq. Inches.

Diameter in Centi- metres.

Section in Sq. Cms.

7-0

0.500

0.1963

1.2700

1.267

6-0

.464

.1691

.1786

.091

5-o

0.432

0.1466

1.0973

0.9456

4-0

.400

•1257

.0160

.8107

3-o

•372

.I087

0.9449

.7012

2-0

.348

.0951

.8839

.6136

.324

.0825

.8230

•53*9

0.300

0.07069

0.7620

0.4560

.276

•05983

.7010

.3858

.252

.04988

.6401

.3218

.232

.04227

•5893

.2727

.212

•03530

•5385

.2277

0.192

0.02895

0.4877

0.18679

.176

•02433

.4470

.15696

.100

.O2OIO

.4064

•12973

.144

.01629

.3658

.10507

.128

.01287

•3251

.08302

0.116

0.010568

0.2946

0.06818

.104

.00849^

.2642

.05480

.092

.006648

•2337

.04289

.080

.005027

.2032

.03243

.072

.00407 1

.1829

.02627

0.064

0.003217

0.16256

0.020755

.056

.002463

.14224

.015890

.048

.OOlSlO

.12192

•011675

.040

.001257

.I0l6o

.008107

.036

.OOIOlS

.09144

.006567

0.032

0.0008042

0.08128

0.005189

.028

.00061 58

.07112

.003973

.024

.0004524

.06096

.002922

.022

.0003801

.05588

.002452

.020

.0003142

.05080

.002027

0.0180

0.0002545

0.04572

0.0016417

.0164

.0002112

.04166

.0013628

.0148

.OOOI728

•03759

.0011099

.0136

.0001453

•03454

.0009363

.0124

.0001208

.03150

.0007791

0.0116

0.00010568

0.02946

0.0006818

.0108

.OOOO9l6l

.02743

.0005910

.0100

.00007854

.02540

.0005067

.0092

.00006648

.02337

.0004289

.0084

.00005542

.02134

.0003575

0.0076

0.00004536

0.01930

0.0002927

.0068 .0060

.00003632 .OOOO2827

.01727 .01524

.0002343 .0001824

39 40

.0052 .0048

.00002124 .OOOOlSlO

.01321 .01219

.0001370 .0001167

0.0044

O.OOOOI52I

O.OIIlS

0.0000982

.0040

.OOOOI257

.OIOl6

.0000811

.0036

.OOOOIOlS

.00914

.0000656

.0032

.OOOOO8O4

.00813

.0000519

.0028

.0000o6l6

.00711

.0000397

0.0024

0.00000452

0.00610

0.0000292

4£

.0020

.OOOOO3I4

.00508

.0000203

.0016

.OOOOO2OI

.00406

.0000129

.0012

.OOOOOII3

.00305

.0000073

.0010

.OOOOOO79

.00254

.0000051

4> tU

P *&

Diameter in I Inches.

Sections in Sq. Inches.

Diameter in Centi- metres.

Section in Sq. Cms.

0000

0-454

0.16188

I-I532

1.0444

000

425

.14186

•0795

.9152

.380

.11341

0.9652

•7317

•340

.09079

.8636

•5858

0.300

0.07069

0.7620

0.4560

.284

.06335

.7214

.4087

•259

.05269

.6579

•3399

.238

.04449

.6045

.2870

.220

.03801

•5588

.2452

0.203

0.03237

0.5156

0.20881

.180 .165

•0254<> .02138

•4572 .4191

.16417 •J3795

.148

.01720

•3759

.11099

•134

.01410

•3404

.09098

0.120

0.011310

0.3048

0.07297

.I09

.009371

.2769

.06160

•095

.007088

.2413

•04573

.083

.005411

.2108

.03491

.072

.004072

.1829

.02627

0.065

0.0033183

0.16510

0.021409

.058

.0026421

•14732

.017046

18 J9

.049 .042

.0018857 .0013854

.12446 .10668

.012166 .008938

•035

.0009621

.08890

.006207

0.032

0.0008042

0.08128

0.005189

.028

.0006158

.07112

•003973

.025

.0004909

•06350

.003167

.022

.0003801

.05588

.002452

.020

.0003142

.05080

.002027

0.018

0.0002545

0.04572

0.0016417

.Ol6

.0002011

.04064

.0012972

.014

.0001539

•03556

.0009932

.013

.0001327

.03302

.0008563

.012

.OOOIlSl

.03048

.0007297

0.010 .009

0.00007854 .00006362

0.02540 .02286

0.0005067 .0004104

33 34 35

.008 .007 .005

.00005027 .00003848 .00001963

.02032 .01778 .01270

.0003243 .0002483 .0001267

0.004

0.00001257

0.01016

0.0000811

SMITHSONIAN TABLES.

TABLE 37. BRITISH UNITS.

Cross sections and weights of wires.

This table gives the cross section and weights in British units of copper, iron, and brass wires of the diameters iven in the first column. For one tenth the diameter divide section and weights by 100. For ten times the iameter multiply by 100, and so on.

gi di

Area of cross section in Sq. Mils.

Copper — Density 8.90.

Iron — Density 7.80.

Brass— Density 8.56.

Pounds per Foot.

Log.

Feet per Pound.

Pounds per Foot.

Log.

Feet per Pound.

Pounds per Foot.

Log.

Feet per Pound.

78.54

.000303

4.48150

33°0-

.0002656

4.42420

3765.

.000291 5

4.46458

343 1-

95-03

0367

•56429

2727.

03214

•50697

3II2.

03527

54735

2836.

113.10

0436

.63986

2291.

0382;

-58257

2615.

04197

62295

2383-

132.73

0512

•70939

1953.

04488

.65208

2228.

04926

69246

2030.

153-94

0594

•77376

1683.

05206

.71646

1921.

05713

75684

1750-

176.71 2OI.O6

.000682 0776

4.83368 .88974

1467. 1289.

.0005976 06799

4.77637 .83244

1674. 1471.

.0006558 07461

4.81675

.87282

1525- 1340.

226.98

0876

.94240

1142.

07675

.88510

I3°3-

08423

.92548

1187.

25447

0982

•99205

1018.

08605

•93475

1162.

09443

•975*3

1059.

283.53

1094

3-03902

914.

09588

.98171

1043.

.0010522

3.02209

950-

314.16

.OOI2I2

3-08357

825.1

.001062

3.02626

941.4

.OOIl66

3.06664

857-7

346.36 380.13

J336 1467

.12594 .16634

748.3 681.8

II7I 1286

.06864 .10904

777-8

1285 I4II

.10902 .14942

778.0 708.9

415.48

1603

.20496

623.8

1405

.14766

711.7

1542

.18804

648.6

452-39

1746

.24192

572.9

1S3°

.18463

653-7

1679

.22500

595-7

490.87

.001894

3.27738

528.0

.001660

3.22008

602.4

.OOl822

3.26046

549-o

530-93

2046

.31146

488.1

1795

•25415

557-o

1970

•29453

507-5

572.56

2209

•34423

452.6

1936

.28693

5*6-5

2125

•3273*

470.6

6I5-75

2376

•37583

420.9

2082

•31852

480.3

2285

•35890

437-6

660.52

2549

.40630

3924

2234

.34900

447-7

245 !

•38938

408.0

706.86

.002727

3-43575

366.7

.002390

3-37845

418.4

.002623

3.41882

381.2

754-77 804.25

2912 3I03

.46424 .49181

343-4 322.2

2552 2720

.40693 •4345°

391.8

2801 2985

-44731 .47488

357-0 335-1

33 34

855-30 907.92

35°3

•51854 •54446

303-0 285.4

2892 3070

.46123 .48716

345-8

3369

.50161

•52754

3I5-1 296.8

962.11

.003712

3-56964

269.4

•003253

3-5I233

307-4

.003570

3-5527I

280.1

1017.88

3927

.59412

254.6

3442

•53691

290.5

3777

•57719

264.7

1075.21

4149

.61791

241.0

3636

.56061

275-o

3990

.60098

250.6

38 39

1134.11 1194.59

4376 4609

.64108 .66364

228.5 216.9

3844 4040

.58476 •60633

260.2 247.6

4218 4433

.62514 .64671

237-1 225.6

1256.64

.004849

3-68563

206.2

.004249

3-62833

235-3

.004664

3.66871

214.4

1320.25

5°94

.70708

196.3

4465

•64977

224.0

4900

.69015

204.1

I385-44

5346

.72801

187.1

4685

.67070

213-5

5141

.71108

1^4.5

1452.20

5603

.74845

178.5

4911

.69114

203.6

5389

.73152

185.6

1520.53

5867

.76842

170.4

5J42

.71111

'94-5

5643

•75*49

177.2

159043 1661.90

.006137 6412

3-78793 •80703

162.9 J55-9

.005378 5620

3-73063

.74972

185.9 177.9

.005902

3.77101 .79010

169.4 162.1

1734-94

6694

.82569

149.4

5867

.76840

170.5

6438

.80878

J55-3

1809.56

6982

•84399

143.2

6119

.78669

163.4

6715

.82706

148.9

1885.74

7276

.86189

137-4

6377

.80459

156.8

6o9S

•84497

142.9

1963.50 2042.82

•007576 7882

3-87945 .89664

132.0 126.9

.006640 6908

3.82214 •83934

150.6

144.8

.007287 7581

3.86252 .87972

137-2

52 53 54

2123.72 2206.18 2290.22

8194 8512 8837

•91352 •93005 .94630

I22.O

"7-5

113.2

7181 746o 7744

.85621

•87275 .88899

139.2 134.0 129.1

8187 8499

.89659 •92937

126.9

I22.I 1177

2375.83

.009167

3.96223

109.1

.008034

3-90493

124.5

.008817

3-94531

"3-4

SMITHSONIAN TABLES.

TABLE 37 (continued).

BRITISH UNITS.

Cross sections and weights of wires.

2 ^ Q

Area of cross section

Sq. Mils.

Copper — Density 8.90.

Iron — Density 7.80.

Brass — Density 8.56.

Pounds per Foot.

Log.

Feet per Pound.

Pounds per Foot.

Log.

Feet per Pound.

Pounds per Foot.

Log.

Feet per Pound.

2375-83 2463.01 255I-76 2642.08

.009167 09504 09846 10195

3.96223 .97789

.99325 2.00837

109.1 105.2

101.6 98.1

.008034 08329 08629 08934

3-90493 .92058

-93595 .95106

124.5 1 20. 1

"5-9 111.9

.008817 09140 09470 09805

3-94531 .96096

.97633 .99144

II3-4 109.4 105.6 IO2.O

2733-97

10549

.02320

94.8

09245

.96591

108.2

10146

2.00629

98.6

2827.43

.01091

2.03782

91.66

.00956

3.98050

104.59

.01049

2.02088

95-30

2922.47

1128

.05216

88.68

0988

.99486

101.19

1085

•03524

92.21

3019.07

1165

.06628

85.84

IO2I

2.00898

97-95

II2O

.04936

89.25

3I][7-25

1203

.08019

83.14

1054

.02288

94-87

"57

.06326

86.45

3216.99

1241

.09386

80.56

1088

•03656

91.83

1194

.07694

8377

3318.31

.01280

2.10732

78.11

.01122

2.05003

89.12

.01231

2.09041

8l.2I

3421.19 3525-65

1320 1360

.12061 •13367

75-76 73-5i

1157 1192

.06329 •07635

86.44 83.88

1270 1308

.10367 .11673

78.76 76.43

3631.68

I4OI

.14655

71-36

1228

.08922

81.42

1348

.12960

74.20

3739-28

1443

.15924

69.30

1264

.10190

79.09

1388

.14228

72.06

384845

.01485

2.17174

67-34

.01302

2.11451

76.82

.01429

2.15489

70.00

39 59- i 9

1528

.18404

65.46

1339

74.69

1469

.16710

68.06

I 72

4071.50

1571

.19618

63-65

1377

.13887

72.63

'5"

•17925

66.19

4185.39

1615

.20817

61.92

HI5

.15085

70.66

J553

.19123

64.38

4300.84

1660

.22OOO

60.26

1454

.16267

68.76

J596

.20304

62.66

4417.86

.01705

2.23165

58.66

.01494

2.17432

66.95

.01639

2.21460

61.01

4536-46

I751

•243 i 7

57-13

J534

•18583

65.19

1684

.22621

59-40

77 78

4656.63 4778.36

1797 1844

•25453 .26574

55-65 54-23

IIII

.19718 .20839

63.50 61.89

1728 1773

•23756 .24877

57-87 56.39

4901.67

1892

.27681

52.87

1658

.21946

60.33

1819

•25974

54-99

5026.55

.01939

2.28769

51-56

.01700

2.23038

58.83

.01865

2.27076

53-6i

5 1 53-0°

i9&

.29848

50-29

T743

.24117

57-39

1912

•28155

52.29

5281.02

2038

.30914

49.07

1786

•25183

56.00

1960

.29221

5*-93

5410.61

2088

.31966

47.90

1830

.26236

54.66

2008

.30274

49-80

5541-77

2138

.33006

46.77

1874

.27276

53-36

2057

.3I3M

48.63

5674-50

.02189

2.34034

45-67

.01919

2.28304

52.11

.02106

2.32342

47-49

5808.80

2241

•35050

44.62

1964

.29320

50.91

2156

.33358

46-39

5944-68

2294

•36054

43.60

2OIO

.30324

49-75

2206

.34362

45-33

6082.12

2347

.37047

42.61

2057

•a'a1?

48.62

2257

•35355

44-3°

6221.14

2400

.38028

41.66

2IO4

•32298

47-54

2309

.36336

43-31

6361.73

•02455

2.38999

40.74

.02151

2.33269

46.49

.02360

2.37297

42.37

6503.88

2509

•39958

39.85

2199

.34228

45-47

2414

.38266

41.43

6647.61

2565

.40908

38-99

2248

•35178

44-49

2467

.39216

40-54

93 94

6792.91 6939-78

2621 2678

.41847 42775

38-15 37-35

2297 2347

.36116 .37046

43-54 42.61

2521 2575

.40154 .41084

39-67 38-83

7088.22

•02735

2.43694

36.56

.02397

2.37965

41.72

.02630

2.42003

38.02

7238.23

2793

.44604

35-8i

2448

.38874

40.86

2686

.42912

37-23

7389.81

2851

.45504

35-07

2499

.39775

40.02

2742

.43812

36.46

7542.96

2910

46395

34-36

2551

.40665

39.20

2799

.44703

35-72

7697.69

2970

.47277

33.67

2003

•4!547

38.42

2857

45585

35-oi

100

7853.98

.03030

2.48150

33-00

.02656

2.42420

37.65

.02915

2.46458

34.31

SMITHSONIAN TABLES.

62 TABLE 38.

METRIC UNITS.

Cross sections and weights of wires.

This table gives the cross section and the weight in metric units of copper, iron, and brass wires of the diameters given in the first column. For one tenth the diameter divide sections and weights by 100. For ten times the diameter multiply by 100, and so on.

Diam. in thou- sandths of a cm. 1

Area of cross section (jc0^0)s

Copper — Density 8.90.

Iron — Density 7.80.

Brass — Density 8.56.

J*j

Log.

5 J

1*3

Is!

Log.

Metres per Gramme.

Ill

&**

Log.

Metres per Gramme.

78.54

0.06990

2.84448

14.306

0.06126

2.78718

16.324

0.06723

2.82756

14.874

95-03

.08458

,-92725

11.823

.07412

.86996

13.492

•08135

.91034

12.293

12 13

113.10 132.73

.10065 .11813

1.00285 .07236

9-935 8.465

.08822 •!0353

_-94556 1.01506

n-335 9-659

.09681 .11362

_.98594 1-05544

10.330 8.801

153.94

.13701

.13674

7.299

.12008

•07945

8.328

'13*71

.11983

7.589

176.71

0-I573

1.19665

6.358

0.1378

7-I3936

7-255

0-1513

1.17974

6.611

2OI.O6

.1789

.25272

5.588

.1568

•19542

6.376

.1721

•2358o

5.810

226.98

.2020

•30538

4-951

.1770

.24808

5.648

•1943

.28846

5-H7

18 19

254-47 283.53

.2265 •2523

•35503 .40199

4415 3-963

.1985

.2212

•29773 •34469

5-038 4-522

.2178 •2427

•338ii •38507

4-591 4.120

314.16

0.2796

1.44654

3-577

0.2450

1.38925

4.081

0.2689

1.42963

3-7I9

346.36

.3083

.48892

.244

.27O2

.43162

3.701

.2965

.47200

•373

380.13

•52932

2.956

.2965

.47203

•373

•3254

.51241

•073

415.48

.3698

•56794

.704

•3241

.51064

.086

•3557

•55103

2.812

452.39

.4026

.60490

.484

•3529

•5476i

2.834

•3872

•58799

,582

490.87

0.4369

1.64036

2.289

0.3829

1.58306

2.612

0.4202

1.62344

2.380

530.93

4725

•67443

.116

.4141

.61713

415

4545

•65751

.200

572.56

.5096

.70721

1.962

.4466

.64992

•239

.4901

.69030

.040

6I575

.5480

.73880

.825

.4803

.68150

.082

•5271

.72188

1.897

660.52

•5879

.76928

.701

•5152

.71198

1.941

•5654

•75236

.769

706.86

0.6291

1.79872

1.590

o-55J4

^•74143

1.814

0.6051

1.78181

1-653

754-77

.6717

.82721

.489

.76991

.699

.6461

.81029

.548

804.25

.7158

.85478

•397

•6273

•79749

•594

.6884

•83787

453

855-30

.7612

.88151

•3*4

.6671

.82421

•499

.7321

.86459

907.92

.8o8l

.90744

.238

.7082

.85014

.412

.7772

.89052

.'287

962.11

0.856

7.93261

1.168

0.7504

7-87531

1-333

0.8236

1.91570

1.214

1017.88

.906

•95709

.104

•7939

•89979

.260

•8713

.94017

.148

1075.21

•957

.98088

•°45

.8387

•92359

.192

.9204

.96397

.087

1134.11

I.OI2

0.00504

0.988

.8866

•94775

.128

•9730

.98813

.028

1194.59

•063

.02661

.941

.9318

.96931

•073

1.0230

0.00969

0.978

1256.64

I.II8

0.04861

0.8941

0.980

1.99131

1.0200

1.076

0.03169

0.9296

4i 42

1320.25 I385-44

•175 •233

.07005 .09098

.8511 .8110

1.030 .081

0.01275 .03368

0.97II .9254

.130 .186

•05313 .07406

.8849 .8432

1452.20

.292

.11142

•7738

.133

.05412

.8828

•243

.09450

.8044

1 520.53

•353

•I3I39

•7389

.186

.07409

.8432

.302

.11447

.7683

1590.43

1415

0.15091

0.7065

1.241

0.09361

0.806 1

1.361

0-13399

0.7345

1661.90

479

.17000

.6761

.296

.11270

•7714

423

•15308

.7029

4£

1734-94

•544

.18868

.6476

•353

•13138

•7389

•485

.17176

•6734

1885.74

.611 .678

.20696 .22487

.6209 •5958

.411 .471

.14967 .16758

.7085 .6799

•549 .614

.19005 .20796

.6456 •6i95

1963.50

1-748

0.24242

0.5722

1-532

0.18513

0.6530

i. 68 1

0.22551

0-595°

51 52

2042.82 2123.72

.818 .890

.25962 .27649

•5500 .5291

•593 •657

.20232 .21919

.6276 .6037

•24371 •25957

•5705 •5501

2206.18

.964

.29303

•5093

.721

•23574

.5811

.888

.27612

•5295

2290.22

2.038

.30927

.4906

.786

•25197

•5598

.960

•29235

.5101

2375-83

2.114

0.32521

0.4729

I.853

0.26791

0-5396

2.034

0.30829

0.4917

SMITHSONIAN TABLES.

TABLE 38 (continued).

METRIC UNITS.

Cross sections and weights of wires.

Diam. in thou- sandths of a cm. 1

Area of cross section d*,",)"

Copper — Density 8.90. ,

Iron — Density 7.80.

Brass— Density 8.56.

Log.

Metres per Gramme.

Log.

Metres per Gramme.

Log.

2375-83

2.114

0.32521

4729

1.853

0.26791

•5396

2.034

0.30829

.4917

2463.01

.192

.34086

.4562

.921

•28356

•5205

.108

•32394

4743

255I-76

.271

.35623

4403

.990

.29893

.5024

.184

•3393 *

4578

2642.08

•351

4253

2.061

.31404

.262

•35442

.4422

2733-97

433

^38618

.4112

.132

.32889

.4689

•340

•36927

4273

2827.43

2.516

0.40078

•3974

2.205

0-34349

4534

2.420

0.38387

.4132

2922.47

.601

4i5J4

•3845

.280

•35784

4387

•502

•39823

3019.07

.687

.42926

.3722

•355

.37196

.4246

41235

.3869

3117.25

•774

.44316

.3604

431

•38587

4113

.668

.42625

.3748

3216.99

.863

45684

•3493

•509

•39954

.3985

,760

.44092

•3623

33l8-3I

2-953

0.47031

•3386

2.588

0.41301

.3864

2.840

0.45339

•3521

3421.19

.48357

.3284

.669

.42627

•3747

•929

.46665

•3415

3525-65

.138

.49663

•3187

•750

43933

•3636

3.018

47971

•33*3

3631.68 3739-28

'•328

•5095° .52218

•3094 •3005

•833 .917

.45220 .46488

•3530 .3429

.109

.201

49258 •50526

.3217 •3I24

384845 4071.50

3.426

•524 .624

0-53479 •54700 •55915

.2919

.2838 .2759

3-003 .088 .176

0.47749 .48970 •50185

•3330 •3238 •3*49

3.295 485

0.51787 •53008 •54223

•3°35

4185.39

•725

.57H3

.2685

.265

•51383

•3063

•55421

.2791

4300.84

.828

.58294

.2612

•355

•52565

.2981

.682

•56603

.2716

4417-86

3-932

0.59460

•2543

3446

0-53731

.2902

3.782

0-57769

.2644

4536.46

4-037

.60611

.2477

.538

.54881

.2826

.883

.58919

•2575

4656.63

.144

.61746

.2413

.632

.56017

•2753

.986

.60056

.2509

78 79

4778.36 4901.67

.62867 •63974

•2351 .2292

.727 •823

.57137 .58244

.2683 .2615

4.090 •177

.61175 .62283

.2445 •2394

5026.55

4474

0.65066

.2235

3-921

0.59336

•2550

4.303

0.63375

.2324

S'SS-oo

.586

.66145

.2180

4.019

.60415

.2488

411

.64454

.2267

5281.02

.700

.67211

.2128

.119

.61481

.2428

•65519

.2212

5410.61

.815

.68264

.2077

.220

•62534

•2369

.631

.66572

•2159

5541-77

•932

.69304

.2027

-323

.63574

•2313

•744

.67612

.2108

5674-50

5.050

0.70332

.1980

4.426

0.64602

•2259

4.857

0.68640

.2059

87 88 89

5808.80 5944-68 6082.12 6221.14

.170 .291 413 •537

•71348 •72352 •73345 •74326

•1934 .1890

.1847 .1806

•531 -637

•744 .852

.65618 .66622 .67615 .68596

.2207

[2108 .2061

5'.2o6 •325

.69656 .70660

•71653 •72634

.2OI I .1965 .1921 .1878

6361.73

5.662

0.75297

.1766

4.962

0.69567

.2015

5.446

0.73605

.1836

6503.88

.788

.76256

.1728

5-O73

.70527

.1971

.567

•74565

.1796

6647.61

.916

.77206

.1690

.185

.71476

.1929

.690

•755*4

•1757

6792.91

6.046

.78144

.1654

.298

.72414

.1887

.815

•76452

.1720

6939.78

..176

•79074

.1619

413

•73344

.1847

.940

.77382

.1683

7088.22

6,309

0.79993

•1585

5529

0.74263

.1809

6.068

0.78301

.1648

7238.23

442

.80902

!646

«75I73

.1771

.196

.79211

.l6l4

7389-8i

•577

.81802

.1520

•764

•76073

•1735

•326

.80111

.1581

7542.96 7697.69

•713 .851

.82693 •83575

.1490 .1460

.884 6.004

.76964 .77846

.1670 .1665

457 .589

.81002 .81884

.1549 .1518

100

7853-98

6.990

0.84448

.I431

6.126

0.78718

.1632

6.723

0.82756

.1487

SMITHSONIAN TABLES.

TABLE 39. BRITISH AND METRIC UNITS.

Cross sections and weights of wires.

The cross section and the weight, in different units, of Aluminium wire of the diameters given in the first columni For one tenth the diameter divide sections and weights by too. For ten times the diameter multiply by 100, and so on.

Diameter.*

Area of cross section.*

Aluminium — Density 2.67.

Pounds Foot.

Log.

Feet per Pound.

Ounces per Foot.

Log.

Feet per Ounce.

Grammes per

Metre.*

Log.

Metres per Gramme.

78.54

.0000909

5.95862

1 1 000.

.001455

3.16274

687.5

.02097

2.32160

47.69

95-°3

01 1 00

4.04139

9091.

01760

.24551

602.4

•02537

•40437

39-41

12 13

113.10 132.73

01309 01536

.11699 .18630

7638. 6509.

02095 02458

.32111 .39062

477-4 406.8

.03020 .03544

•47997 .54948

33-"

28.22

1 53-94

01782

.25088

5612.

02851

45500

350.8

.04110

.61386

24.33

176.71

.0002045

4.31079

4889.

.003273

3.51491

305-6

.04718

2-67377

21.19

201.06

02327

.36685

4297.

03724

.57097

268.5

.05368

.72984

18.63

226.98

•02627

.41952

3876.

04204

.62364

237-9

.06060

•78250

16.50

254-47

02946

.46917

3395-

04713

.67329

212.2

.06794

•83215

14.72

283-53

03282

•S'613

3047.

05251

.72025

190.4

.07570

.87911

13.21

314.16

.0003636

4.56068

2750.

.005818

3.76480

I7I.9

.08388

2~.92366

11.922

346.36

04009

.60306

2494.

06415

.80718

155-9

.09248

.96604

10.813

380.13

04400

.64346

2273.

07040

•84758

142.0

.10149

1.00644

9-853

415.48

04809

.68208

2079.

07697

.88630

129.9

.11093

.04506

9.014

452.39

05237

.71904

1910.

08378

.92316

1194

.12079

.08202

8.279

490.87

.0005682

4-7545°

1760.

.00909

3-95862

110.00

.1311

1.11748

7.630

530-93

06147

.78867

1627.

0983

.99269

101.70

.1418

•I5I55

7.054

572.56

06628

•82135

1509.

1060

2-02547

94-3°

.1529

.18433

6.541

6i5-75

07127

.85293

1403.

1140

.05705

87.69

.1644

.21592

6.083

660.52

07646

.88341

1308.

1223

•08753

8i.75

.1764

.24640

5.670

706.86

.0008182

4.91286

1222.

.01309

2.11698

76.39

.1887

1.27584

5-299

754-77

08737

•94134

1145-

1398

.14546

71-54

•2015

•30433

4.962

804.25

09309

.96892

1074.

1489

•17304

66.89

•2147

.33*90

•657

855.30

09900

,-99565

IOIO.

'I?4

.19977

63- i 3

.2284

.35863

•379

907.92

10509

3.02158

952.

1681

•22570

59-47

.2424

•38456

•125

962.11

.OOIII4

3-04675

897.9

.01782

2.25087

56.12

•2569

1.40973

3.893

1017.88

1178

.07123

848.8

1885

•27535

53-05

.2718

.43421

.680

1075.21 1134.11

1245 I3l6

.09502 .11918

803.5 760.0

1991

2105

.29914 .32329

50.22 47-50

.2871 .3035

.45800 .48216

•483 •295

1194.59

1383

•14075

723.2

2212

•34487

45-20

.3190

•50373

•135

1256.64

.001455

3-16275

687.5

.02327

2.36687

42.97

•3355

1.52573

2.980

1320.25

1528

.18419

6544

2445

•38831

40.90

•3525

•54717

.837

I385-44

1604

.20512

623.6

2566

.40924

38.97

•3699

.56810

.704

1452.20

1681

•22556

594-9

2690

.42968

•3877

•58854

•579

1 520.53

1760

•24552

568.2

28l6

.44964

35.51

.4060

.6085!

•463

1590.43

.001841

3.26504

543-2

.02946

"2.46916

33-95

.4246

1.62803

2-355

1661.90

1924

.28413

5I9-8

3078

.48825

32-49

•4437

.64712

.254

1734-94

2008

.30281

498.0

3213

•50693

31.12

•4632

.66580

1809.56

2095

.32110

477-4

3351

.52522

29.84

.4832

.68408

.070

1885.74

2183

•33901

458.1

3492

•54313

28.63

•5035

.70199

1.986

1963.50

.002273

3-35656

440.0

.03636

2.56068

27.50

•5243

7.71954

1.907

Si 52

2042.82 2123-72

2365 2458

•37376 .39063

422.9 406.8

3783 3933

.57788

•59475

26.43 25.42

•5454 .5670

.73674 .75361

.833

2206.18

2554

.40717

394-2

4086

.61129

24.47

.5891

.77015

.698

2290.22

2651

.42341

377-2

4242

•62753

23-57

.6115

.78639

•635

-375.83

.002750

3-43934

363-6

.04400

2.64346

22.73

•6343

T-80233

I-576

* Columns 3-8, in thousandths of an inch ; 9-12, thousandths of a centimetre. SMITHSONIAN TABLES.

TABLE 39 (continued).

BRITISH AND METRIC UNITS.

Cross sections and weights of wires.

Area of cross section.*

Aluminium — Density 2.67.

Pounds Foot.

Log.

Feet

per Pound.

Ounces per Foot.

Log.

Feet per Ounce.

Grammes Metre.*

Log.

Metres per Gramme.

2375.83

.002750

343934

363.6

.04400

2.64346

22.73

0.6343

1.80233

1-576

2463.01

2851

45500

350^8

.04562

.65912

21.92

.606

.81798

.521

255I-76

2954

47037

338.6

.04726

.67449

21. l6

.6813

.83335

.468

2642.08

3058

.48547

327.0

.04893

.68959

20.44

•7054

.84846

.418

2733-97

3^5

.50032

316.0

•05063

.70444

'975

.7300

.86331

•370

2827.43

.003273

3.5I492

305.5

.05236

2.71904

19.10

07549

1.87790

1.325

6i 62

2922.47 3019.07

3383 3495

.52928 •54340

295.6 286.2

.05413

•05591

•73340 •74752

18.48 17.88

.89226 .90638

.282 .241

3117.25

3608

•55730

277.1

•05773

.76142

17.32

•8323

.92028

.201

3216.99

3724

.57098

268.5

.05958

•77510

16.78

8589

•93396

.I64

3318.31

.003841

3.58445

260.3

.06146

2.78857

16.27

0.8860

i"-94743

I.I29

3421.19

3960

•59771

252.5

•06336

.80183

1578

•9135

.96069

.095

67 68 69

3525-6J 3631.68 3739-28

4081 4204 4328

.61077 .62364 .63632

245.0 237.9 231.0

.06530 .06726 .06925

.81489

•82777 .84044

14$

14.44

•9413 .9697

.9984

.99930

.062 .031 .OO2

3848.45 3959.19

.004456 4583

3-64893 .66114

224.4 218.2

.07129 .07333

2.85305 .86526

14.03 13.64

1.028 •057

0.01191 .02412

0.9730 .9460

4071.50 4185.39 4300.84

4713 4845 4978

.67328 .68526 .69708

212.2 206.4 200-9

•07541 •07751 .07965

.87740 .88938 .90120

13.26 12.90 12.55

.087

.03627 .04825 .06006

.9199

4417.86

.005114

370874

195-5

.08182

2.91286

12.22

1.180

0.07172

0.8477

4536.46 4656.63 4778.36

5251 5390 5531

.72025 .73160 .74281

190.4

185.5 I80.8

.08402 .08624 .08850

•92437 •93572 .94693

11.90

11.60

11.30

.211

.0832.3 .09458 .10579

.8256 .8043 7838

4901.67

5674

•75387

176.2

.09078

•95799

11.02

.309

.11686

j 7641

5026.55

.005818

3-76480

I7I.9

•09309

2.96892

10.742

1.342

0.12778

07451

5965

•77559

167.6

.09544

.97971

10.479

.376

•13857

.7268

5281.02

6113

.78625

163.6

.09781

.99037

IO.224

4IO

•i 49-23

.7092

5410.61

6263

.79678

J597

.IOO2I

1.00090

9-979

•445

.15976

.6922

5541.77

6415

.80718

155-9

.10264

.01130

9-743

.480

.17016

.6757

5674-5°

.006568

3.81746

152.2

.IO5I

1.02158

9.515

!.5iS

0.18044

0.6600

5808.80

6724

.82762

148.7

.1076

•03174

9-295

•551

.19060

.6448

5944.68

6881

•83766

145-3

.IIOI

.04178

9.082

•587

.20064

.6300

6082.12

7040

.84758

142.0

.1126

•05170

8.878

.624

.21057

.6158

6221.14

7201

.85740

138.9

.1152

.06152

8.679

.661

.22038

.6020

6361-73

.007364

3.86710

135-8

.1178

1.07122

8.488

1.699

0.23009

0.5887

6503.88 6647.61

7528 7695

.87670 .88619

132.8 130.0

.1205 .1231

.08082 .09031

8.302

8.122

•737 •775

.23968 .24918

•5759 •5634

6792.91

7863

.89558

127.2

.09970

7-949

.814

.25856

-55H

6939.78

8033

.90487

124.5

.I2§5

.10899

7.780

•853

.26786

•5397

7088.22

.008205

3.91407

121.9

.1313

1.11819

7.617

1.893

0.27705

0.5284

7238.23

8378

.92316

119.4

•1341

.12728

7-459

•933

.28614

7389.81

8554

.93216

116.9

.1369

.13628

7-307

•973

.29514

.5068

7542.96

8731

.94107

"4-5

•J397

•I45I9

7-I58

2.014

•30405

•4965

7697.69

8910

.94989

1 1 2.2

.1426

.15401

•055

.31287

.4865

100

7853.98

.009091

3.95862

IIO.O

•1455

1.16274

6.875

2.097

0.32160

0.4769

* Columns 3-8, in thousandths of an inch; 9-12, thousandths of a centimetre. SMITHSONIAN TABLES.

TABLE 40.

SIZE, WEIGHT, AND ELECTRICAL

Size, Weight, and Electrical Constants of pure hard drawn Copper Wire of different numbers Size and Weight

Gauge Number.

Diameter in Inches.

Square of Diameter (Circular Inches).

Section in Sq. Inches.

Pounds

Log.

Feet per Pound.

OOOO

0.4600

0.21 l6

0.1662

0.6412

1.80701

1.560

000

.4096

.1678

.1318

.5085

.70631

1.967

.3648

.1331

.1045

.4033

.60560

2.480

•3249

•1055

.0829

•3«98

.50489

3.127

0.2893

0.08369

0-06573

0.2536

1.40419

3-943

.2576

.06637

.05213

.2OI I

.30348

4.972

.2294

.05263

.04134

•1595

.20277

6.270

.2043

.04174

•03278

.1265

.10206

7-905

.1819

.03310

.02000

.1003

.00136

9.969

0.1620 •1443

0.02625 .02082

0.02062 •01635

0.07955 .06309

2.90065 .79994

12.57 I5-85

.1285

.01651

.01297

.05003

.69924

19.99

.1144

.01309

.01028

.03968

.59853

25.20

.1019

.01038

.00815

.03146

.49782

3I-78

0.00074

0.008234

0.006467

0.02495

2.39711

40.08

.08081

.006530

.005129

.01979

.29641

50-54

.07196

.005178

.004067

.01569

.19570

63-72

.06408

.004107

.003225

.01244

.09499

80.35

•05707

.003257

.002558

.00987

3.99429

101.32

0.05082

0.002583

O.OO2O28

0.007827

3-89358

127.8

.04526

.002048

.001609

.006207

.79287

161.1

.04030

.001624

.001276

.004922

.69217

203.2

.03589

.001288

.OOIOI2

.003904

.59146

256.2

.03196

.001021

.000802

.003096

.49075

323-1

0.02846

O.OOOSlOI

0.0006363

0.002455

3-39004

408.2

•02535

.0006424

.0005046

.001947

•28934

5I3-6

.02257

.0005095

.0004001

.001544

.18863

647.7

.O2OIO

.0004040

.0003173

.001224

.08792

816.7

.01790

.0003204

.0002517

.000971

4.98722

1029.9

0.01594

0.0002541

0.0001996

O.O007700

4.88651

1298.

.01419 .01264

.0002015 .0001598

.OOOI 583 .0001255

.0006107 .0004843

.78580 .68510

1638. 2065.

.OII26

.0001267

.0000995

.0003841

.58439

2604.

.01003

.0001005

.0000789

.0003046

.48368

3283-

0.008928

O.OOOO797O

0.00006260

0.0002415

4.38297

4140.

.007950

.0000632!

.00004964

.0001915

.28227

5221.

33 34

.007080 .006304

.OOOO5OI3 .00003975

.00003937 .00003122

.0001519 .OOOI2O5

.18156 .08085

6583. 8301.

.005614

.00003152

.00002476

.0000955

5.98015

10468.

O.005000

O.OOOO25OO

O.OOOOIOJ53

0.00007576

5-87944

13200.

.004453

.00001983

.00001557

.00006008

.77873

16644.

.003965

.00001372

.00001235

.00004765

.67802

20988. *

.003531 .003145

.00001247 .00000989

.OOOOO979 .00000777

.00003778 .00002996

•57732 .47661

26465. 33372.

SMITHSONIAN TABLE*.

TABLE 40 (continued). CONSTANTS OF COPPER WIRE.

according to the American Brown and Sharp Gauge. Common Measure. Temperature 32° F. Density 8.90.

Electrical Constants

Resistance and Conductivity.

Gauge Number.

Ohms per Foot.

Log.

Feet per Ohm.

Ohms per Pound.

Pounds per Ohm.

0.00x504629

S-6655I

2l6oi.

0.00007219

13852.

0000

.00005837 .00007361 .00009282

.76622 .86693 .96764

17131. 13586. 10774.

.00011479 .00018253 .00029023

8712.

5479- 3445-

000

00 0

0.0001170

4-06834

8544.

0.0004615

2166.8

.0001476

.16905

6775-

.0007338

1362.8

.000l86l

.26976

5373-

.0011668

857.0

.0002347

.37046

4261.

.0018552

539-0

.0002959

.47117

3379-

.0029499

339-0

0.0003731

4.57188

2680.

0.004690

213.22

.0004705 .0005933 .0007482

.67259

.77329 .87400

2125. 1685. r337.

.007458 .011859 .018857

134-08 84.32 53.03

.0009434

.97471

1060.

.029984

33-35

O.OOII90 .001500 .001892

3-0754I .17612 .27683

840.6 666.6 528.7

0.04768 .07581 .12054

20.973

12 13

.002385

•37753

419.2

.19166

5.218

.003000

.47824

332.5

.30476

3.281

0.003793

3.57895

263.7

0.4846

2.0636

.004783 .006031

.78036

209.1 165.8

.7705 1.2252

1.2979 0.8162

.007604 .009589

.88107 .98178

^•S 104.3

1.9481 3.0976

.3228

O.OI209 .01525

2.08248 .18319

82.70 65-59

4.925 7.832

0.20305 .12768

.01923

.28390

52.01

12.453

.08030

.02424

.38461

41.25

19.801

.05051

•03057

•48531

32.71

31.484

.03176

0.03855

2.58602

25.94

50.06

0.019976

.04861

.68673

20.57

79.60

.012563

.06130

.78743

16.31

126.57

.007901

.07729

.88814

12.94

2OI.2O

.004969

.09746

.98885

10.26

320.01

.003125

0.1229

.1550

1.08955 .19026

8.137 6.452

0.0019654 .0012359

•J954

.29097

5-"7

1286.5

.0007773

.2464

.39168

4.058

2045.6

.0004889

•3107

.49238

3.218

3252.6

.0003074

0.3918

.4941

7.59309 .69380

2.552 2.024

8224!

0.0001934 .0001216

.6230 .7856

•79450 .89521

1.605 1.273

13076. 20792.

.0000765 .0000481

38 39

.9906

.99592

1.009

33060.

.0000303

SMITHSONIAN TABLES.

TABLE 41 .

SIZE, WEIGHT, AND ELECTRICAL

Size, Weight, and Electrical Constants of pure hard drawn Copper Wire of different numbers Size and Weight.

Gauge Number.

Diameter in Centimetres.

Square of Diameter (Circular Cms.).

Section in Sq. Cms.

Grammes Metre.

Log.

Metres per Gramme.

0000 000

1.1684 .0405

1.3652

.0826

1.0722 0-8503

2.97966 .87896

0.001048 .001322

£25!

0.8586 .6809

.6743 •5348

600. 1 475-9

.77825 •67754

.001666 .002 1 OI

0.7348

0.5400

0.4241

377-4

2.57684

O.002649

.6^44

.4282

299-3

2^74

•47613 •37542

.003341 .004213

'.5189

.2693

.2115

188.2

.27472

.005312

.4621

.2136

.1677

149-3

.17401

.006699

0.4115

0.16936

0.13302

118.39

2.07330

0.00845

.3665 .3264

•'3431 .10651

.10549 .08366

93.88 74-45

1.97259 .87189

.01065 •01343

.2906

.08447

.06634

59-04

.77118

.01694

.2588

.06699

.05261

46.82

.67047

.02136

0.2305

0.05312

0.04172

37.13

I-56977

0.02693

.2053 .1828

.04213 •03341

•03309 .02624

29-45 23-35

•36835

•03396 .04282

14 15

.1628 .1450

.02649

.02IOI

.02081 .01650

18.52 14.69

.26764 .16694

.05400 .06809

0.12908

0.016663

0.013087

11.648

1.06623

0.0859

.H495

.013214

.010378

9-237

0.96552

.1083

.10237

.010479

.008231

7-325

.86482

•1365

.09116 .08118

.008330 .006591

.006527 .005176

5.809 4.607

•764" .66340

.1721 .2171

0.07229

0.005227

0.004105

3-653

0.56270

0.2737

.06438

.004145

•003255

2.898

.46199

.3450

23 24

•05733 .05106

.003287 .002607

.002582 .002047

2.298 1.822

.36128 •26057

•04545

.002067

.001624

I«445

•15987

.6920

0.04049

0.0016394

0.0012876

1.1459

0.05916

0.873

.03606

.0013001

.001 02 1 1

.9088

1.95845

1. 100

.03211

.0010310

.0008098

.7207

.85775

1.388

.02859

.0008176

.0006422

.5715

.75704

1.750

.02546

.0006484

.0005093

•4532

•65633

2.206

0.02268 .02019

O.OOO5I42 .0004078

0.0004039 .0003203

0-3594 .2850

I.55562 .45492

2.782 3.508

.01798

.0003234

.0002540

.2261

•35421

4.424

.01601 .01426

.0002565 .OOO2O34

.0002014 .0001597

•T793 .1422

•25350 .15280

5-578 7-034

0.01270

0.000l6l3

0.0001267

0.1127

1.05209

8.87

.01131

.OOOI 279

.0001005

.0894

2.95138

11.18

.01007

.OOOIOI4

.0000797

.0709

.85068

14.10 '

.00897

.0000804

.0000632

.0562

•74997

17.78

.00799

.0000638

.0000501

.0446

.64926

22.43

SMITHSONIAN TABLES.

TABLE 41 (.continued}.

CONSTANTS OF COPPER WIRE.

according to the American Brown and Sharp Gauge. Metric Measure. Temperature o° C. Density 8.90.

Electrical Constants.

Resistance and Conductivity.

Number.

Ohms Metre.

Log.

Metres per Ohm.

Ohms per Gramme.

Grammes Ohm.

0.0001519

4.18150

6584.

0.0000001592

6283000.

0000

.0001915

.28221

5221.

.0000002531

3951000.

ooo

.0002415

.38191

4141.

.0000004024

2485000.

.0003045

•48362

3284-

.0000006398

1563000.

0.0003840

4.58433

2604.

O.OOOOOIOI7

982900.

.0004842

.68503

2065.

.000001618

618200.

.0006106

78574

1638.

.000002572

388800.

.0007699 .0009709

.88645 .98715

1299. 1030.

.000004090 .000006504

244500. 153800.

4 5

0.001224

3.08786

816.9

O.OOOOIO34

96700.

.001544

.18857

647-8

.00001644

60820.

.001947

.28928

5^3-7

.00002615

38250.

.002455

.38998

407.4

.00004157

24050.

.003095

.49069

323.1

.00006610

I5I30.

0.003903

3.59140

256.2

O.OOOI05II

9514.

.004922

.69210

203.2

.00016712

598*

.006206

.79281

161.1

.00026574

3763.

.007826

.89352

127.8

.00042254

2367.

.009868

.99423

101.3

.00067187

1488.

0.01244

2.09493

80.37

0.0010683

936.1

.01569

.19564

63-73

.0016987

588.7

.01979

'29635

50-54

.0027010

370.2

.02495

.39705

40.08

.0042948

232.8

.03146

.49776

31-79

.0068290

146.4

0.03967

2.59847

25.21

0.010859

92.09

.05002

.69917

19.99

.017266

57.92

.06308

.79988

I5-85

.027454

36.42

•07954

^90059

12-57

.043653

22.91

.10030

1.00130

9-97

.069411

11.88

0.12647

T.IO2OO

7.907

O.II037

9.060

.15948

.20271

6.270

.17549

5.698

.201 10

•30342

4-973

.27904

3-584

.25358

.40412

3-943

.44369

2.254

.31976

.50483

3.127

.70550

1.417

0.4032

1.60554

2.480

I.I2I8

0.8914

.5084

.70624

1.967

1.7837

.5606

.6411

.80695

1.560

2.8362

.3526

.8085

.90766

1.237

4.5097

.2217

I.OI94

0.00837

0.981

7.1708

•1394

1-2855

0.10907

0.7779

11.376

0.08790

I.62IO

.20978

.6169

18.130

.05516

2.0440

.31049

.4892

28.828

.03469

2-5775

.41119

.3880

45.838

.02182

3.250I

.51190

.3076

72.885

.01372

SMITHSONIAN TABLES.

TABLES 42-43. WEIGHT OF SHEET METAL.

TABLE 42. - Weight of Sheet Metal. (Metric Measure.)

This table gives the weight in grammes of a plate one metre square and of the thickness stated in the

first column.

Thickness

in thou- sandths of

Iron.

Copper.

Brass.

.A 1 u m i n u m •

Platinum.

Gold.

Silver.

a cm.

78.0

89.0

85.6

26.7

215.0

193.0

105.0

156.0 234.0

178.0 267.0

171.2 256.8

£1

430.0 645-0

386.0 579-0

2IO.O 3I5-0

312.0

356.0

342.4

1 06.8

860.0

772.0

420.0

390.0

445-0

428.0

133-5

1075.0

965.0

525-0

468.0

534-0

5'3.6

160.2

1290.0

1158.0

630.0

546.0

623.0

599-2

186.9

1505.0

1351.0

735-0

624.0

712.0

684.8

213.6

1720.0

1544.0

840.0

702.0 780.0

801.0 890.0

770.4 856.0

240.3 267.0

1935-0 2150.0

1737.0 1930.0

945-0 1050.0

TABLE 43. -Weight of Sheet Metal. (British Measure.)

Thickness in Mils.

Iron.

Pounds per Sq. Foot.

.04058 .08116 .12173 .16231 .20289

•24347 .28405

-32463 .36520 .40578

Copper.

Pounds per Sq. Foot.

.04630 .09260

.1 ~

.15520 •23150

.27780 .32411

•37041 .41671 .46301

Brass.

Pounds per Sq. Foot.

.04454 .08908

.13363 .17817 .22271

.26725

•3" 79

.4 .44542

Aluminum.

Pounds per Sq. Foot.

.01389 .02778 .04167 •05556 .06945

•08334 .09723

.11112

.12501 .13890

Ounces per Sq. Foot.

.2222

•4445 .6667 .8890

I.III2

1-3335

1-5557 1.7780

2.OOO2 2.2224

Platinum.

Pounds per Sq. Foot.

.1119 .2237 .3356

-4474 •5593

.6711

1.0067

1.1185

Ounces per Sq. Foot.

1.790 3-579

7.158 8.948

10.738

12.527

'4-317 16.106 17-896

Thickness in Mils.

4 5

Gold.

Troy

Ounces per Sq. Foot.

1.4642 2.9285 4.3927 5.8570 7.3212

8.7854 10.2497

"•7139 13.1782 14.6424

Grains per Sq. Foot.

702.8

2108.5 2811.3 35M-2

4217.0 4919.8 5622.7

6323-5 7028.3

Silver.

Troy

Ounces per Sq. Foot.

0.7967 1-5933 2-: 3-U

4.7800 5-5767 6.3734 7.1700 7.9667

Grains per Sq. Foot.

382.4

764.8

1147.2

1529.6

1912.0

2294.4 2676.8 3059-2 3441.6 3824.0

SMITHSONIAN TABLES.

TABLE 44. STRENGTH OF MATERIALS.

The strength of most materials varies so that the following figures serve only as a rough indication of the strength of a

particular sample.

TABLE 44 (a). -Metals.

TABLE 44(1)). — Stones.*

Name of Metal.

Tensile strength in pounds per sq. in.

Material.

Size of test piece.

Resistance to crushing in pds. per sq.in.

Aluminum wire Brass wire Bronze wire, phosphor, hard- drawn Bronze wire, silicon, hard- drawn Bronze : Cu, 58.54 parts ; Zn, 38.70; Al, 0.21; with 2.55 parts of the alloy, Sn, 29.03, wrought iron, 58.06, ferro- manganese, 12.91 Copper wire, hard-drawn Gold wire Iron, cast " wire, hard-drawn " " annealed Lead, cast or drawn Palladium * Platinum * wire Silver * wire Steel " wire, maximum " Specially treated nickel- steel, approx. com p. 0.40 C ; 3.25 Ni ; treatment secret " piano wire, 0.033 in. diam. " piano wire, 0.051 in. diam. Tin, cast or drawn Zinc, cast " drawn

3OOOO-4OOOO 50000-150000

IIOOOO-I4OOOO 95000-II5000

60000-75000 60000-70000 2OOOO I3OOO-33OOO 8OOOO-I2OOOO 5OOOO-6OOOO 2600-3300 39000 5OOOO 42OOO 80000-330000 400000

250000

357000-390000 325000-337000 4000-5000 7000-13000 22OOO-3OOOO

Marble Tufa Brownstone Sandstone Granite Limestone

4 in. cubes

2 " "

4 in. cubes 4« - 4« «

7600-20700 77OO-Il6oo 7300-23600 2400-29300 9700-34000 6000-25000

* Data furnished by the U. S. Geological Survey. TABLE 44(0). -Brick.*

Kind of Brick.

Resistance to crushing in pds. per sq. in.

Tested flatwise.

Tested on edge.

Soft burned Medium burned Hard burned Vitrified Sand-lime

1800-4000 4000-6000 6000-8500 8500-25000 1800-4000

1600-3000 3000-4500 4500-6500 6500-20000

According to Boys, quartz fibres have a tensile strength of between 116000 and 167000 pounds per square inch.

Brick piers laid up in i part Portland cement, 3 of sand, have from 20 to 40 per cent the crushing strength of the brick.

* Authority of Wertheim.

* Data furnished by the U. S. Geological Survey.

TABLE 44 (d).- Concretes.*

Coarse material. " Aggregate."

Proportions by volume. Cement : sand : aggregate.

Size of test piece.

Resistance to crushing in pds. per sq. in.

Sandstone Cinders Limestone Conglomerate Trap

1 : 5 : 14 to : I : 5 1:3:6 " : I : 3 1:4:8 " : 2 : 4 I :6 : 12 " : 2 :4 1:3:9 " : 2 : 4

12 in. cube

12 " " 12 " " 12 " " 12 " «

1550-3860 790-2050 1200-2840 1080-3830 820-2960

* Data furnished by the U. S. Geological Survey. SMITHSONIAN TABLES.

72 TABLE 45.

STRENGTH OF MATERIALS.

Average Results of Timber Tests.

The test pieces were SMALL and SELECTED. Endwise compression tests of some of the first lot, made when green and containing over 40 per cent moisture, showed a diminishing in strength of 50 to 75 per cent.

See also Table 46. A particular sample may vary greatly from these data, which can indicate only in a general way the relative values of a kind of timber. Note that the data below are from selected samples and therefore probably high.

The upper lot are from the U. S. Forestry circular No. 15 ; the lower from the tests made for the loth U. S. Census.

TRANSVERSE TESTS.

COMPRESSION.

SHEAR- ING.

NAME OF SPECIES.

Modulus of rupture. Ib./sq. in.

Modulus of elasticity. Ibs./sq. in.

| to grain. Ibs./sq. in.

J_ to grain. Ibs./sq. in.

Along the gram. Ibs./sq. in.

Long-leaf pine

12,600

2,070,000

8,000

I26o

835

Cuban pine

13,600

2,370,000

8,700

1200

770

Short-leaf pine Loblolly pine

10,100 11,300

1,680,000 2,050,000

6,500 7,400

1050 1150

770 800

White pine

7,900

1,390,000

5,400

700

4OO

Red pine

9,100

I,62O,OOO

6,700

IOOO

500

Spruce pine

IO,OOO

1,640,000

7,300

I2OO

Boo

Bald cypress

7,900

1,290,000

6,OOO

800

500

White cedar Douglass spruce

6,300 7,900

9IO,OOO 1,680,000

5,200

5,700

700 800

400 500

White oak

I3,IOO

2,090,000

8,500

2200

IOOO

Overcup oak Post oak

11,300 12,300

I,62O,OOO 2,030,000

7,300

7,100

lOXX) 3000

IOOO IIOO

Cow oak

11,500

I,6lO,000

7,400

I9OO

900

Red oak

11,400

1,970,000

7,200

2300

IIOO

Texan oak

13,100

I,86o,OOO

8,100

2OOO

900

Yellow oak

IO,8OO

1,740,000

7,300

I800

IIOO

Water oak

12,400

2,000,000

7,800

2000

IIOO

Willow oak

10,400

1,750,000

7,200

IOOO

900

Spanish oak

I2,OOO

1,930,000

7,700

I800

900

Shagbark hickory Mockernut hickory

16,000 15,200

2,390,000 2,320,000

9,500

10,100

2700 3100

IIOO IIOO

Water hickory

12,500

2,o8o,000

8,400

2400

IOOO

Bitternut hickory

15,000

2,280,000

9,600

22OO

IOOO

Nutmeg hickory

12,500

1,940,000

8,800

2700

IIOO

Pecan hickory

15,300

2,530,000

9,100

2800

1200

Pignut hickory

18,700

2,730,000

10,900

3200

1200

White elm

10,300

1,540,000

6,500

I2OO

800

Cedar elm

13,50°

1,700,000

8,000

2100

1300

White ash

10,500

1,640,000

7,200

1900

IIOO

Green ash

11,600

2,050,000

8,000

I7OO

IOOO

Sweet gum

9,5oo

1,700,000

7,100

1400

800

Poplar

9,400

1,330,000

5,000

1120

Basswood

8,340

1,172,000

5,190

880

Ironwood

7.540

1,158,000

5,275

2000

Sugar maple White maple

16,500 14,640

2,250,000 I,8oo,000

8,800 6,850

3000 2580

Box elder

873,000

4,580

1580

Black walnut

11,900

1,560,000

8,000

2680

Sycamore

7,000

790,000

6,400

2700

Hemlock

9,480

1,138,000

5,400

IIOO

Red fir

13,270

1,870,000

7,780

1750

Tamarack

13,150

1,917,000

7,400

1480

Red cedar

1 1, 800

938,000

6,300

2000

Cottonwood

10,440

1,450,000

5,ooo

IIOO

Beech

16,200

1,730,000

6,770

2840

SMITHSONIAN TABLES.

TABLE 46.

UNIT STRESSES FOR STRUCTURAL TIMBER EXPRESSED IN POUNDS PER SQUARE INCH.

Recommended by the Committee on Wooden Bridges and Trestles, American Railway Engineering Association, 1909.

BENDING.

SHEARING.

KIND OF TIMBER.

Extreme fibre stress.

Modulus of elasticity.

Parallel to grain.

Longitudinal shear in beams.

Average ultimate.

Safe stress.

Average.

Average ultimate.

Safe stress.

Average ultimate.

Safe stress.

Douglass fir

6lOO

1200

,510,000

690

170

270

Long-leaf pine

6500

1300

,010,000

720

1 80

300

1 20

Short-leaf pine

5600

IIOO

,480,000

710

170

330

130

White pine

4400

900

,130,000

400

100

180

Spruce Norway pine

4800 420O

1000 800

,310,000 ,190,000

600 590

ISO 130

170

100

Tamarack

4600

900

,220,000

670

170

260

100

Western hemlock

5800

IIOO

,480,000

630

160

270*

100

Redwood

5000

900

8OO,OOO

300

Bald cypress Red cedar

4800 4200

1,150,000 860,000

500

120

;

—

White oak

5700

IIOO

1,150,000

840

210

270

COMPRESSION.

J.4

KIND OF TIMBER.

Perpendicular to grain.

Parallel to grain.

l.si

Formulas for safe

stress in long

0 &

U ** V

columns over 15

.9 cP

Elastic

Safe

Average

Safe

o «*«

diameters. t

3'£

limit.

stress.

ultimate.

stress.

*'§ ^

K <»

Douglass fir Long-leaf pine Short-leaf pine

630 520 340

3JO

260 170

3600 3800 3400

1200 1300 IIOO

900 830

I200(l-L/6o.D)

i3oo(i-L/6o.D) noo(i-L/6o.D)

10 10 10

White pine

290

150

3000

1000

7 So

iooo(i-L/6o.D)

Spruce Norway pine Tamarack

370

180

150 220

3200 2600* 3200*

IIOO 800 IOOO

830 600 75°

noo(i-L/6o.D) 8oo(i-L/6o.D) iooo(i-L/6o.D)

Western hemlock

440

220

35°°

I2OO

900

i20o(i-L/6o.D)

Redwood

400

I5°

3300

900

680

9oo(i-L/6o.D)

Bald cypress Red cedar White oak

340 470 920

170 230 450

3500

IIOO

900 1300

830 680 980

iioo(i-L/6o.D) 90o(i-L/6o.D) i30o(i-L/6o.D)

These unit stresses are for a green condition of the timber and are to be used without increasing the live- load stresses for impact.

SMITHSONIAN TABLES.

* Partially air-dry.

t L= length in inches. D = least side in inches.

TABLES 47-47A. ELASTIC MODULI,

TABLE 47. - Rigidity Modulus.

If to the four consecutive faces of a cube a tangential stress is applied, opposite in direction on adjacent sides, the modulus of rigidity is obtained by dividing the numerical value of the tangential stress per unit area (kg. per sq. mm.) by the number representing the change of angles on the non-stressed faces, measured in radians.

Substance.

Rigidity Modulus.

Refer-

ence.

Substance.

Rigidity Modulus.

Refer- ence.

335°

25»0 3550 3715 3700 1240 4060 2450 4780 4213 445° 4664 2850

3950 5210 6706

7975 6940 8108

7505 1710 7820 4359

5 10 ii 5 5 5

10 19 5 14

5 15

14 5 5 ii

Quartz fibre . .

2888 2380 2960 2650 2566 2816 8290

7458 8070 7872 173° 1543 3880 3820 6630 6220

2350 2730 1770 1280 1190 2290

2O 21

5 10 16 ii 16 15 5 ii

5 19

J? 16

23 23 23 23

« «<

Brass

Silver

" cast, ooCu-f 12 Sn . Bismuth, slowly cooled . . Bronze, cast, 88 Cu + 12 Sn . Cadmium, cast

" hard-drawn .... Steel

" cast . .

" cast, coarse gr. . . .

Tin, cast ....

Gold

Zinc . '

Iron, cast .......

Glass

Clay rock

Granite . .....

Magnesium, cast .... Nickel

Marble

Slate ........

Phosphor bronze ....

References 1-16, see Table 48. 21 Boys, Philos. Mag. (5) 30, 1890. 17 Gratz, Wied. Ann. 28, 1886. 22 Thomson, Lord Kelvin. 18 Savart, Pogg. Ann. 16, 1829. 23 Gray and Milne. 19 Kiewiet, Diss. Gottingen, 1886. 24 Adams-Coker, Carnegie Publ. No. 46, 20 Threlfall, Philos. Mag. (5) 30, 1890. 1906.

TABLE 47a. —Variation of the Rigidity Modulus with the Temperature. nt = n0 (i — of — &f2 — 7/3), where / = temperature Centigrade.

Substance.

aio«

Authority.

Brass . . .

2652 3200 3972

^ 8108

6940 6632 2566 8290

2158

455 2716

57J 206 483 in

387 187

48 36

5° 38 59

32 47 — ii

—8 ii —9

Pisati, Nuovo Cimento, 5, 34, 1879. Kohlrausch-Loomis, Pogg. Ann. 141. Pisati, loc. cit. K and L, loc. cit. Pisati, loc. cit. K and L, loc. cit. Pisati, loc. cit.

•^*

. , .

Platinum . .

. . .

Silver . . .

Steel

nt* = «w [i — o (t— 15)] ; Horton, Philos. Trans

.204 A, 1905.

Copper Copper (com- mercial) Iron Steel

4-37*

3.80 8.26 8-45

a =.00039

.00038 .00029 .00026

Platinum Gold Silver Aluminum

6.46* 2-55

a = .00012 .00031 .00048 .00148

Tin i. Lead o. Cadmium 2. Quartz 3.

eo* o = . 00416 80 .00164 31 .0058

DO .OOOI2

SMITHSONIAN TABLES.

* Modulus of rigidity in io11 dynes per sq. cm.

Young's Modulus

TABLE 48. ELASTIC MODULI.

Young's Modulus.

Intensity of longitudinal stress (kg. per sq. mm.)t Elongation per unit length

Substance.

Temp.

i •

Aluminum ....

Lead, drawn .... " annealed . . . Bronze .....

20 12.3 IS IS

IS IS o

15.6

IS IS 12-9

0 20 19-5 IS 0

ii-S

7200 7462 1803

1727 9194 7070 11697 20869 20794 20310 21740 11713 15750 19385 20500 8131

12450 10520 12140 12550 13220

8543 9810

IO22O

993° 10450 12094 "550 13300 20300 22790

3 3 4

3 3

8 4 9

3 3

3 3 7 i

9 3 7 ii 10 9 4 ii

12 II

Nickel-steel, 5^% ni. . " . u 25% «« . Palladium, annealed . Phosphor-bronze . . Platinum, drawn . . annealed .

IS 13.2

15 15

IS iS-S

IS IS

9709

I2OIO 17044 I55I8 10020 15989

7357 7140 18810 17280 19550 19560 21136

2III2 21700 20705 20910 2O6OO 3190

8734 4148 1700 ( 6000

I to

(8000

i t0 ( 2500 6316

8^85

13 13

3 ii

3 3

2 I

3 3 3 3 3 3 4 4 9 13 13 13 5 3 3 13

24 24 24

Cadmium . . . .

Delta metal .... Iron, drawn .... annealed . . .

" drawn . . Silver, drawn .... " annealed . . Steel wire, drawn . . " " annealed . Steel, cast, drawn . . " annealed . Bessemer . . . puddle ....

cast

drawn .... " drawn .... Gold, drawn .... " annealed . . . " drawn .... Copper, drawn . . . " annealed . . " drawn . . . " drawn . . . " electr. h'd d'n Brass, drawn ....

very soft . . . half soft . . . " hard .... Bismuth

Zinc, drawn .... Tin, drawn .... " cast

Glass .,..,.

" drawn ....

German silver ... h'd d'n

Nickel ......

Marbles

Granites

Basic intrusives . . . Rocks : See Nagaoka, Philos. Mag. 1900.

" hard drawn . .

i Slotte, Acta Soc. Fenn. 26, 1899; 29, 1900". 10 Baumeister, Wied. Ann. 18, 1883. 2 Meyer, Wied. Ann. 59, 1896. n Searle, Philos. Mag. (5) 49, 1900. 3 Wertheim, Ann. chim. phys. (3 12, 1844. 12 Cantone, Wied. Beibl. 14, 1890. 4 Pscheidl, Wien. Ber. II, 79, 1879. I3 Mercadier, C. R. 113, 1891. 5 Voigt, Wied. Ann. 48, 1893. J4 Katzenelsohn, Diss. Berlin, 1887. 6 Amagat, C. R. 108, 1889. 15 Wertheim, Pogg. Ann. 78, 1849. 7 Kohlrausch, Loomis, Pogg. Ann. 141, 1871. 16 Pisati, Nuovo Cimento, 5, 34, 1879. 8 Thomas, Drude Ann. i, 1900. References 17-19, see Table 47. 9 Gray, etc., Proc. Roy. Soc. 67, 1900.

Compiled partly from Landolt-Bornstein's Physikalisch-Chemische Tabellen. SMITHSONIAN TABLES.

7 TABLES 49-52.

COMPRESSIBILITY, HARDNESS, CONTRACTION OF ELEMENTS.

TABLE 49. — Compressibility of the More Important Solid Elements.

Arranged in order of the increasing atomic weights. The numbers give the mean elastic change of volume for one megabar (0.987 atm.) between 100 and 500 megabars, multiplied by io5.

Lithium 8.8

Potassium 31.5

Selenium n.8

Iodine 13.

Carbon 0.5

Calcium 5.5

Bromine 51.8

Caesium 61.

Sodium 15.4

Chromium 0.7

Rubidium 40.

Platinum 0.21

Magnesium 2.7

Manganese 0.7

Molybdium 0.26

Gold 0.47

Aluminum 1.3

Iron 0.40

Palladium 0.38

Mercury 3.71

Silicon 0.16

Nickel 0.27

Silver 0.84

Thallium 2.6

Red phosphorus 9.0

Copper 0.54

Cadmium i 9

Lead 2.2

Sulphur 12.5

Zinc 1.5

Tin 1.6

Bismuth 2.8

Chlorine 95.

Arsenic 4.3

Antimony 2.2

Stull, Zeitschr. Phys. Chem. 61, 1907. TABLE 60.— Hardness.

Agate 7.

Brass 3-4.

Iridosmium 7.

Sulphur I'S"2-5

Alabaster 1.7

Calimine 5.

Iron 4-5.

Stibnite 2.

Alum 2-2.5

Calcite 3.

Kaolin i.

Serpentine 3-4.

Aluminum 2. Amber 2-2.5

Copper 2.5-3. Corundum 9.

Loess (o°) 0.3 Magnetite 6.

Silver 2.5-3. Steel 5-8.5

Andalusite 7.5

Diamond io.

Marble 3-4.

Talc i.

Anthracite 2.2

Dolomite 3-5-4-

Meerschaum 2-3.

Tin 1.5

Antimony 3.3 Apatite 5.

Feldspar 6. Flint 7.

Mica 2.8 Opal 4-6.

Topaz 8. Tourmaline 7.3

Aragonite 3.5

Fluorite 4.

Orthoclase 6.

Wax (o°) 0.2

Arsenic 3.5

Galena 2.5

Palladium 4.8

Wood's metal 3.

Asbestos 5.

Garnet 7.

Phosphorbronze 4.

Asphalt 1-2.

Glass 4.5-6.5

Platinum 4.3

Augite 6.

Gold 2.5-3.

Plat-iridium 6.5

Barite 3.3 Beryl 7.8

Graphite 0.5-1. Gypsum 1.6-2.

Pyrite 6.3 Quartz 7.

Bell-metal 4.

Hematite 6.

Rock-salt 2.

Bismuth 2.5

Hornblende 5.5

Ross' metal 2.5-3.0

Boric acid 3.

Iridium 6.

Silver chloride 1.3

From Landolt-Bornstein-Meyerhoffer Tables : Auerbachs, Winklemann, Handb. der Phys. 1891. TABLE 51.— Relative Hardness of the Elements.

1 0.0

6-S

3-°

2-5

1.8

0.6

9-5

5-0

3-°

2-3

1.8

°-S

9.0

4-8

2.9

2.0

'•5

o-S

7.0

4-5

2.7

2.0

I-S

0.4

7.0

4-3

2.5

2.0

l-S

o-3

6.5

3-5

2-5

2.0

1.2

0.2

Rydberg, Zeitschr. Phys. Chem. 33, 1900. TABLE 52.— Ratio, p, of Transverse Contraction to Longitudinal Extension under Tensile Stress.

(Poisson's Ratio.)

Metal

0-45

0.42

0-39

0.38

o-3S

o-34

o-33

0-33

0.31

0.30

0.28

From data from Physikalisch-Technischen Reichsanstalt, 1907. t

p for : marbles, 0.27 ; granites, 0.24 ; basic-intrusives, 0.26 ; glass, 0.23. Adams-Coker, 1906. SMITHSONIAN TABLES.

TABLE 53. 77

" ELASTICITY OF CRYSTALS.*

The formulae were deduced from experiments made on rectangular prismatic bars cut from the crystal. These bars were subjected to cross bending and twisting and the corresponding Elastic Moduli deduced. The symbols o /3 y, at /3t yt and 03 /32 yz represent the direction cosines of the length, the greater and the less transverse dimensions of the prism with reference to the principal axis of the crystal. E is the modulus for extension or compression, and T is the modulus for torsional rigidity. The moduli are in grammes per square centimetre.

Barite. Tnio ^- = 16.13* 4 i8.5i0* + 10.427*+ 2(38.79/8V 4 I5-2I7V 4- 8.88a202)

T(->10

S|r = 69.52* + 1 17.660* +|i 16.467* + 2(20.16$ V 4 85.2972a2 + 1 Beryl (Emerald).

io10 - .. Oj Oa of the specimen make with the

-^- = 15.00—3.675 cos^2 — 17-536 cos2^ cos% [ principal axis of the crystal.

Fluor spar.

TCllO

!£• = 13.05 -6.26 (*+

^ = 58.04 - 50.08 O V + T2-2

Pyrites.

= 18.60 - 17.95 (/8V + T2*2

Rock salt.

^ = 3348 - 9.66 (a4+j8* + 74)

TrtlO

^ = 1 54-58 - 77.28 (/8V + T2-2 +«W

Sylvine.

lg-0 = 75.i_48.2(*+0* + 74)

Tnio

IP^- = 306.0 - 192.8 ( 0V + 72«2 + aW

Topaz.

^ = 4.34i«* + 3-460/5* + 3.7717* + 2 (3-879W + 2.8567^ + 2.

T010

^ =14.88* + 16.54/8*+ i6.4574430.89/3V440.8972a2443-5i«2/8a Quartz.

TCvlO

I|P = 12.734 (i -72)24 16.693 (I -T2)?2 4 9-70574-8.46o/37 (&-&

TOio

i- = 19.665 + 9.060732 + 22.9847V ~ 16.920 [(7/3rf f*7i) (3««i ~ flBi) - ^272)]

* These formulae are taken from Voigt's papers (Wied. Ann. volt. 31, 34, and 35). SMITHSONIAN TABLES.

78 TABLE 54.

ELASTICITY OF CRYSTALS.

Some particular values of the Elastic Moduli are here given. Under E are given moduli for extension or compression in the directions indicated by the subscripts and explained in the notes, and under T the moduli for torsional rigidities round the axes similarly indicated.

(a) REGULAR SYSTEM.*

Substance.

E«

T«

Authority.

Fluor spar . , » Pyrites

Rock salt ....

«,

Sylvinc . . . . .

1473 X io6 3530 X io6 419X106 403 X io6 401 X io6

1008 X io6 2530 X io6 349Xio6 339 X 106

2OQ X IO

910 X io6 2310 X io6 303 X io6

345 X io6 1075 X io6 I29X io6

Voigt.t («

Koch4 ii

172 X IO6

196 X io6

get V IO6

Voigt

Sodium chloride . Potash alum . . . Chrome alum . . Iron alum ....

405 X io6 181 X io6 161 X io6 186 X io6

319 X io6 199 X io6 177 X io6

—

Koch.

Beckenkamp.§ « «

(6) RHOMBIC SYSTEM.||

Substance.

E«

Authority.

Barite . Topaz .

620 X io6 2304 X io6

540 X io6 2890 X io6

959X106 2652 X io6

376 X io6 2670 X io6

702 X io6 2893 X io6

740 X io* 3180 X io6

Voigt.

Substance.

T12 = T21

T13 = T31

T23 = T32

Authority.

Barite ........

283 X io6 1336X106

293 X io6 1353X106

121 X I06

U04X io6

Voigt. «

Topaz

In the MONOCLINIC SYSTEM, Coromilas (Zeit. fur Kryst. vol. i ) gives

G sum 1 Emax ~ ^7 X io6 at 21.9° to the principal axis. I £^ = 313X106 at 75-4°

EmM == 22I3 X lo6 m the PrmciPal ax^s-

Emm = 1554 X io6 at 45° to the principal axis.

Mica

In the HEXAGONAL SYSTEM, Voigt gives measurements on a beryl crystal (emerald). The subscripts indicate inclination in degrees of the axis of stress to the principal axis of the crystal.

E0= 2165X108, £45=1796X106, E90 = 2312X106, TO = 667 X io6, Tgo = 883 X io6. The smallest cross dimension of the prism experimented on (see Table 82), was in the principal axis for this last case.

In the RHOMBOHEDRIC SYSTEM, Voigt has measured quartz. The subscripts have the same meaning as in the hexagonal system.

Eo = 1030 X io6, E_ 45 = 1 305 X io6, E+ 45 = 850 X io6, Ego = 7§5 X io6,

To = 508 X io6, T90 = 348 X io6. Baumgarten 1" gives for calcspar

Eo = 501 X io6, E_ 45 = 441 X io6, E + 45 = 77 2 X io6, E9o = 79° X io6.

* In this system the subscript a indicates that compression or extension takes place along the crystalline axis, and distortion round the axis. The subscripts b and c correspond to directions equally inclined to two and normal to the third and equally inclined to all three axes respectively.

t Voigt, "Wied. Ann." vol. 31, 34-35; 36, 642.

i Koch, ''Wied. Ann." vol. 18.

§ Beckenkamp, " Zeit. fur Kryst." vol. io.

|| The subscripts i, 2, 3 indicate that the three principal axes are the axes of stress ; 4, 5, 6 that the axes of stress are in the three principal planes at angles of 45° to the corresponding axes.

f Baumgarten, " Pogg. Ann." vol. 152.

SMITHSONIAN TABLES.

TABLES 55-57. COMPRESSIBILITY OF GASES.

TABLE 56.— Relative Volumes at Various Pressures and Temperatures, the volume at 0 C and at 1 atmo- sphere feeing taken as 1 000 000.

Oxygen.

Air.

Nitrogen.

Hydrogen.

Atm.

0°

99°-5

i99°-5

0°

99°.4

200°.4

0°

99°-5

1990.6

0°

99°-3

200°. 5

100

9265

9730

9910

200

4570

700O

9095

505o

7360

9430

5'95

7445

9532

5690

7567

9420

300

3208

4843

6283

3<>58

5*70

6622

3786

53oi

6715

4030

5286

6520

400

2629

3«30

4900

3036

4170

5240

3142

4265

5331

3207

4147

5°75

500

23I2

3244

4100

2680

3565

4422

2780

30.S.S

4515

2713

3462

4210

600

2II5

2867

3570

2450

3180

3883

2543

3258

3973

23*7

3006

3627

700

1979

26lO

3202

2288

2904

35°2

2374

2980

3S89

2149

2680

3212

800

1879

2417

2929

2168

2699

3219

2240

2775

33°°

1972

2444

2900

900

1800

2268

2718

2070

2544

3000

2149

2616

3<>85

1832

2244

2657

1000

1735

2I5I

1992

2415

2828

2068

1720

2093

Amagat : C. R. in, 1890 ; Ann. chim. phys. (6) 29, 1893.

TABLE 56. - Ethylene. pv at o° C and i atm. = i.

Atm.

0°

10°

,00

30°

40°

60°

80°

100°

i37°-5

i98°.s

0.562

0.684

—

0.508

—

5°

0.176

O.42O

0.629

0.731

0.814

0-954

1.077

1.192

1.374

1.652

—

0.240

0.598

—

0.229

0.561

—

O.227

0.524

—

100

0.310

0-33 *

0.360

0.403

0.471

0.668

0.847

1.005

•247

1.580

150

0.441

0.459

0.485

0.5IS

O-551

0.649

0.776

0.924

.178

1.540

200

0.565

0.585

0.610

0.638

0.669

0-744

0.838

0.946

.174

1.537

300

0.8o6

0.827

0.852

0.878

0.908

0.972

1.048

1.133

.310

1.628

500

1.256

1.280

1.308

'•337

i-367

1.431

1.500

1-578

.721

1.985

1000

2.289

2.321

2-354

2-387

2.422

2-493

2.566

2.643

2.798

Amagat, C. R. in, 1890; 116, 1893. TABLE 67. — Ethylene.

Pressure in

Relative values of PV at —

metres of

mercury.

i6°.3

20°.3

30°. r

4o°.o

5o°.o

6o°.o

70°.o

79°-9

89°.9

I00°.0

1950

2055

2220

2410

2580

2715

2865

2970

3090

3225

810

900

IIOO

1535

1875

2100

2310

2500

2680

2860

1065

"95

1325

1510

1710

1930

2l6o

2375

2565

120

1325

1370

1440

1540

1660

1780

1950

2115

2305

2470

*5°

1590

1625

1690

1785

1880

1990

2I25

2250

2390

2540

1 80

1855

1890

1945

2035

2130

2225

2340

2450

2565

2700

2IO

2110

2145

2200

2285

2375

2470

2565

2680

2790

2910

240

2360

2395

2450

2540

2625

2720

28lO

2910

3015

3125

270

26lO

2640

27IO

2790

2875

2965

3060

3150

3240

3345

300

2860

2890

2960

3040

3I25

3215

33°0

3380

3470

3560

320

3035

3065

3^5

3200

3285

3375

3470

3545

3625

3710

Amagat, Ann. ohim. phys. (6) 22, 1881.

SMITHSONIAN TABLES.

TABLES 58-60. COMPRESSIBILITY OF GASES.

TABLE 58. — Carbon Dioxide.

Relative values of PV at —

Pressure in

mercury.

l8°.2

35°-i

400.2

5Q°.o

6o°.o

7o°.o

8o°.o

90°.o

100° .0

3°

liquid

2360

2460

2590

273°

2870

3120

3225

—

1725

1900

2145

2330

2525

2845

2980

625

75°

825

I2OO

1650

2225

2440

2635

110

825

930

980

IO9O

1275

1550

1845

2105

2325

140

1 020

1120

1175

1250

1360

1525

1715

1950

2160

170

I2IO

I3IO

1360

143°

1520

1645

1780

1975

2135

200

1405

1500

1550

1615

1705

1810

1930

2075

2215

230

1590

1690

1730

I800

1890

1990

2090

22IO

2340

260

1770

1870

1920

1985

2070

2166

2265

2375

2490

290

1950

2O6O

2IOO

2170

2260

2340

2440

2550

2655

320

2135

2240

2280

2360

2440

2525

2620

2725

2830

' Relative values of pv ; pv at o° C. and i atm. = i.

0°

10°

20°

30° 40°

60° 80°

.000

137° 198° 258°

50 100

150

0.105 0.202 0.295

0.114 0.213

0.680 0.229 0.326

0.775 0-75°

0.255 0.309 0.346 0.377

0.984 1.096

0.66 1 0.873 0.485 0.68 1

1. 206 1.030

0.878

1.380 1.259 1.582 1.847

1.159 1.530 1.818

300

0-559

0.578

0-599

0.623 0.649

0.710 0.790

0.890

1.108 I-493 1.820

500

0.891

0.913

0.938

0.963 0.990

1.054 1.124

I.2OI

1.362 1.678

IOOO

1.656

1.685

1.716

1.748 1.780

1.848 1.921

1.999

Amagat, C. R. m, 1890; Ann chim. phys. (6) 29, 1893 ; 22, 1881.

TABLE 59. — Compressibility of Gases.

Gas.

p.v. (\ atm.).

i d(p.v.) p.v. dp = a.

a t = O

Density.

Density. Very small pressure.

pov0(i atm.).

P = 76<"»

1.00038

— .00076

11.2°

— .00094

32-

32.

H2 NI

0.99974 I.OOOI5 1.00026

+ .00052 — .00030 — .00052

10.7 14.9 13.8

+ .00053

— .00056 — .0008 1

2.015 (i 6°)

28.005 28.OOO

2.0173 28.016 28.003

CO2

1.00279

— .00558

15.0

— .00668

44.268

44.014

N20

1.00327

— .00654

II.O

— .00747

44.285

43.996

Air

I.OOO26

— .00046

1 1.4

—

NH3

1.00632

Rayleigh, Zeitschr. Phys. Chem. 52, 1905.

TABLE 60. - Compressibility ol Air and Oxygen between 18° and 22° 0.

Pressures in metres of mercury, pv, relative.

Air

24.07 26968

34-90 26908

45-24 26791

55-30

26789

64.00

26778

72.16

26792

84.22 26840

101.47 27041

214.54

; 29585

3°4-Q4 32488

P pv

24.07 26843

3//9

26614

26185

64.07

26050

72.15

25858

84.19 25745

101.06 25639

214.52 26536

303-03 28756

Amagat, C. R. 1879.

SMITHSONIAN TABLES.

TABLES 61-62.

RELATION BETWEEN PRESSURE, TEMPERATURE AND VOLUME OF SULPHUR DIOXIDE AND AMMONIA.*

TABLE 61.— Sulphur Dioxide.

Original volume looooo under one atmosphere of pressure and the temperature of the experi- ments as indicated at the top of the different columns.

Pressure in Atmos.

Corresponding Volume for Ex- periments at Temperature —

Volume.

Pressure in Atmospheres for Experiments at Temperature —

S8°.o

99°.6

l83°.2

58°.o

99°-6

i83°.2

8560

9440

6360

7800

lOOOO

9.60

4040

6420

9000

9.60

10-35

S310 4405

8000

10.40

11.85

—

4030

—

7000

"•55

I3-05

—

28 32

3345 2780 2305

3180

2640

6000 5000

12.30 13-15

14.70 16.70

—

I93S

2260

4000

14.00

20.15

1450

2040 1640

1375

3500 3000

14.40

23.00 26.40

29.10

—

1130

2500

—

30.15

33-25

—

930

2000

35-20

40-95

9°

100

79° 680

1500

39.60

55-20

120

545

1000

76.00

140 160

430 325

500

••

—

117.20

TABLE 62. — Ammonia.

Original volume 100000 under one atmosphere of pressure and the temperature of the experiments as indicated at the top of the different columns.

*8g

5 a

Corresponding Volume for Ex- periments at Temperature —

Volume.

Pressure in Atmospheres for Experiments at Temperature —

46°.6

99°-6

,830.6

3o°.a

46°.6

99°.6

.830.0

95°0

10000

8.85

9.50

12.5

7245

7635

9000

9.60

10.45

15 2O

iO_0U

4645 3560

4875 3835

8000 7OOO

10.40 11.05

11.50 13.00

I2.OO 13.60

2875

3185

6000

11.80

14-75

15-55

35 40

45 50

2440 2080

1795 I490

2680

2345 2035

1775

5OOO 4000 3500

12.00

16.60

18.35 18.30

18.60 22.70 25.40

19.50 24.00 27.20

—

1250

1590

3000

—

29.20

3i-5o

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