^/^.
\
A SYNOPSIS
ELEMENTAIIY RESULTS
PUKE MATHEMATICS
CONTAINTNO
PROPOSTTTONS.'FORMUL^, AND METHODS OF ANALYSIS.
WITH
ABRIDGED DEMONSTRATIONS.
SlTPLEMFNTFI) ItY AN InDEX TO THE PaI'ERS ON Pi HE JIaTHEMATU S -WHUJI AKE TO BE FOVNn IN THE PHINCIPAL JoiBNALS AND TllANSACTlONS OF LEAUM I) Socill lEP,
poTH English and Foreion, of the present centuky.
G. S. CARR, M.A.
LONDON :
FRANCIS noi)(;soN, 80 farkin(tDon street, e.c
CAMBRIDGE: MACMILLAN & BOWES.
1886.
(AU rights reserved )
C3
neenng Librai:'^
LONDON : PRINTED BY C. F. HODGSON AND SON,
GOUGH SQUARE, FLEET STREET.
76. 6j6
N TVERSJT?
PREFACE TO PART I
Tin: work, of which tlio part now issued is a first instal- ment, has been compiled from notes made at various periods of the last fourteen years, and chiefly during* the engagements of teaching. !Many of the abbreviated methods and mnemonic rules are in the form in which I originally wrote them for my pupils.
The general object of the compilation is, as the title indicates, to present within a moderate compass the funda- mental theorems, formulas, and processes in the chief branches of pure and applied mathematics.
Tlie work is intended, in the first place, to follow and supplement the use of tl,ic ordinary text-books, and it is arranged witli tlie view of assisting tlie student in the task of revision of book-w^ork. To this end I have, in many cases, merely indicated the salient points of a demonstration, or merely referred to the theorems by which the proposition is proved. I am convinced that it is more beneficial to tlie student to recall demonstrations with such aids, than to read and re-read them. Let them be read once, but recalled often. The difference in the effect upon the mind between reading a mathematical demonstration, and originating cue wholly or
IV PEEFACE.
partly, is very great. It may be compared to tlic difference between the pleasure experienced, and interest aroused, when in the one case a traveller is passively conducted through the roads of a novel and unexplored country, and in the other case he discovers the roads for himself with the assistance of a map.
In the second place, I venture to hope that the work, when completed, may prove useful to advanced students as an aide-memoire and book of reference. The boundary of mathematical science forms, year by year, an ever widening circle, and the advantage of having at hand some condensed statement of results becomes more and more evident.
To the original investigator occupied with abstruse re- searches in some one of the many branches of mathematics, a work which gathers together synoptically the leading propo- sitions in all, may not therefore prove unacceptable. Abler hands than mine undoubtedly, might have undertaken the task of making such a digest ; but abler hands might also, perhaps, be more usefully emj^loycd, — and with this reflection I have the less hesitation in commencing the work myself. The design which I have indicated is somewhat comprehensive, and in relation to it the present essay may be regarded as tentative. The degree of success which it may meet with, and the suggestions or criticisms which it may call forth, will doubt- Ic: ■! have their effect on the subsequent portions of the work.
With respect to the abridgment of the demonstrations, I may remark, that while some diffuseness of explanation is not only allowable but very desirable in an initiatory treatise, conciseness is one of the chief reciuiremcnts in a work intended
PREFACE. V
for tlio piii-i)OSos of revision and rcfiTeiico only. In order, liowever, not to sacrifice clearness to conciseness, much moro la])our has been expended upon this part of the subject-matter of the book than will at first sip^ht be at all evident. The only ])alpal)le I'esult lK'in<^ a compression of the text, the result is so far a neji^ative one. The amount of compression attained is illustrated in the last section of the present part, in which moro than the number of propositions usually given in treatises on Geometrical Conies are contained, together with the figures and demonstrations, in the s})ace of twenty-foui* pages.
The foregoing remarks have a general application to the work as a whole. With the view, however, of making the earlier sections more acceptable to beginners, it will be found tliat, in those sections, important principles have sometimes been more fully elucidated and more illustrated by exam})les, than the plan of the work would admit of in subsequent di\isions.
A feature to which attention may be directed is the uni- form system of reference adopted throughout all the sections. AVithtlie object of facilitating such reference, the articles have been numbered progressively from the commencement in large Clarendon figures ; the breaks which will occasionally be found in these numbers having been purposely made, in order to leave room for the insertion of additional matter, if it should be re(piired in a future edition, without distui-bing tlie oi'iginal numbers and references. With the same object, demonstrations and examples have been made subordinate to enunciations and formidie, the former being [)rinted in small, the latter in bold
Vi rREFACE.
type. By tliese aids, tlic intordcpcndoncc of propositions is more reudily sliown, and it becomes easy to trace the connexion Ijetwecn theorems in different branches of mathematics, with- out the loss of time which would be incurred in turning to separate treatises on the subjects. The advantage thus gained will, however, become more apparent as the work proceeds.
The Algebra section was printed some years ago, and does not quite correspond mth the succeeding ones in some of the particulars named above. Under the pressure of other occupations, this section moreover was not properly revised before f>'oing to press. On that account the table of errata will be found to apply almost exclusively to errors in that section ; but I trust that the hst is e:s.haustive. Great pains liave been taken to secure the accm^acy of the rest of the volume. Any intimation of errors will be gladly received.
I have now to acknowledge some of the sources from which the present part has been compiled. In the Algebra, Theory of Equations, and Trigonometry sections, I am largely in- debted to Todhunter's well-known treatises, the accuracy and completeness of which it would be superfluous in me to dwell upon.
In the section entitled Elementary Geometry, I have added to simpler propositions a selection of theorems from Town- send's Modern Geometiy and Salmon's Conic Sections.
In Geometrical Conies, the line of demonstration followed agrees, in the main, with that adopted in Drew's treatise on the subject. I am inclined to think that the method of that author cannot be much improved. It is true that some im- portant properties of the ellipse, which are arrived at in
I'KKI'ACR. Vll
Drew's (\)nic Si^ctioiis tlirou^^h ccitnin iiitcnncdialc jn'oposi- tions, can be dcMliicod at once from tlie circle ])y tlic metliod of orthogonal projection. But the intcM-mediate propositions can- not on that account be dispenscnl w itli, for they are of value in th(Mns('lv(\^. ]\Ioreov(>r, tlie nietliod of projection applied to Ili(> hyperbola is not so successful; because a pi-oj)erty which lias first to bo proved true in the case of the equilateral hyperbola, might as will be proved at once for the general case. I have introduced the method of projection but spanngly, alwavs giving prefei'ence to a demonstration which admits of being n])])li(Ml in the same identical form to the ellipse and to the hyperbola. The remarkable analogy subsisting between the two curves is thus kept prominently before the reader.
The account of the C. G. S. system of units given in the preliminary section, has been compiled from a valuable con- tribution on the subject by Professor Everett, of Belfast, published by the Physical Society of London.* This abstract, and the tables of physical constants, might perhaps have found a more appropriate place in an after part of the work. I have, however, introduced them at the commencement, from a sense of the great importance of the rcfonu in the selection of units of measurement Avhich is embodied in the C G. S. system, and from a belief that the student cannot be too early familiarized with the same.
The Factor Table wliich folluAvs is, to its limited extent, a rei)rint of Burckluirdt's " Tnhlr.^ ilrs diviscurs,'' published in
* "Illustrations of the Centimetrc-Grammc-Sccoiid System of Units." London : Taylor and Francis. 1875.
Vlll riJEFACE.
1814-17, which give the least divisors of all numbers from 1 to 3,036,000. In a certain sense, it may be said that this is the only sort of purely mathematical table which is absolutely indispensable, because the information which it gives cannot be supplied by any process of direct calculation. The loga- rithm of a number, for instance, may be computed by a formula. Not so its prime factors. These can only bo arrived at through the tentative process of successive divisions by the prime numbers, an operation of a most deterrent kind when the subject of it is a liigh integer.
A table similar to and in continuation of Burckhardt's has recently been constructed for the fourth million by J.^Y. L. Glaisher, F.R.S., who I believe is also now engaged in com- pleting the fifth and sixth millions. The factors for the seventh, eighth, and ninth millions were calculated previously by Dase and Rosenberg, and pubhshed in 1862-05, and the tenth million is said to exist in manuscript. The history of the formation of these tables is both instructive and interesting.*
As, however, such tables are necessarily expensive to pur- chase, and not very accessible in any other way to the majority of persons, it seemed to me that a small portion of them would form a useful accompaniment to the present volume. I have, accordingly, introduced the first eleven pages of Burckh- ardt's tables, which give the least factors of tlie first 100,000 integers nearly. Each double page of the table here pi-iiUed is
* Sec " Factor Table for the Fourth JifiJltoii." By James Glaisher, F.R.S. London: Taylor and Francis. 1880. Also Camh. J'hil. Soc. Proc, Vol. TIL, Pt. IV., and Nature, No. 542, p. 402.
an exact rcpi'otluctiuii, iu all l)iit tlie tyi'e, of ;i Hiii^^e (|iiar(o
I)ai^n: of JJm-c'kluirdt's great work.
It may l)e noticed licro that Prof. Lebesquo constructed
ji tal)le to a))out tliis extent, on tlio ])lan of omlttinj^ tlio
n)ulti|)les of seven, and tlius re<lucins^^ tlie size of tlie tabic
))y about one-sixtli.* ]3ut a small calculation is re(|uired iu
using the table which counterbalances the advantage so gained.
The values of the Gamma-Function, pages 30 and 31 , have
been taken from Legendre's table in his ^'Excrciccs de Calcnl
Infnjral,'' Tome I. The table belongs to Part II. of tliis
Volume, but it is placed here for the convenience of having
all the numerical tables of Volume I. in the same section.
In addition to tlie autliors already named, tlie following
treatises have been consulted — Algebras, by AYood, Bourdon,
and Lefebure de Fourcy ; Snowball's Trigonometry ; Salmon's
Higher Algebra ; the Geometrical Exercises in Potts's Euclid ;
and Geometrical Conies by Taylor, Jackson, and Renshaw.
Articles 2G0, 431, o69, and very nearly all the examples,
are original. The latter have been framed with great care, in
order that they might illustrate the propositions as completely
as possible.
G. S. C.
Hadi.ry, ^Iiiii>i,i;.sKx ; AIu)/ 23, 18ba.
* "Tables divcrscs pour la decomposition des nombrca en leura HietLurs premiers." Par V. A. Lebesque. Paris. 18G-ii.
b
EIUiATA.
Art. 13,
„ 66,
Line 1,
„ 66,
„ 6,
,, 90,
„ 4,
„ 99,
„ 1,
„ 107,
„ 1,
„ 108,
„ 2,
,, 131,
„ 1,2
,, ,.
„ 5,
„ 133,
„ 3,0,
)) >>
„ 8,
M )>
„ 9,
„ „
„ 10,
„ 138,
„ 4,
„ 140,
„ 182,
», 6,
„ 191,
„ 4,
„ 220,
„ 6,
„ 221,
„ 4,
,,•237,
>, 11,
„ 238,
» 6,
„ 239,
„ 11,
„ 218,
„ 4»
„ 267,
„ 4,
„ 274,
„ 8,
,, 276,
„ 13,
,, ,,
„ 14,
,, 283,
„ 3,
„ 288,
„ 7,
„ 289,
„ 4,
„ 290,
9
„ 325,
p 17,
■ a^b^ read + a'b\
„ 333,
,, 361,
,, 481,
,, 614,
„ 617,
,, 614,
,, 651,
„ 704, ,, 729, Article 112 should
numerators 1, 1, 1 denominator r— 1 taken (190) 5
(-1)* 0.*; 4
204, 459 459
10.9.8 1.2.3
(-? + !)*
?(^^_j in nimiorator
(i'o3)
{^^ + !/ + ^Y
(1)
X- = 1
(a-- — 4x- + 8) on left side
(234)
(29)
(267)
in
p + 2
ip-l)
x= 1
n-\
U{r,n-\)
Jr{r+\, «-l)
^2
i'l /'.>/'•), last lino but one
C^)
3628 «-3 apiilying Descartes' rule
i'>
„ (11, 12) „ (i'lO) be as folbnv.s: —
1 i-_2y3 + y 2 ^ 1 + 2^3 + y2 ^ n_ (i + Vai-'-a" 11 + va
1,^, a».
w-1.
taken m at a time.
(360).
6.
(-IP.
3.r.
34.
102, 306.
9)1.
7.8.9.10.
{q + lf^ Notation of (96).
2
(IG4).
2{x + i/+zy.
square of (1).
x"=-l.
(x-—ix + 8)\
Deic.
(28).
(266).
2\U.
P + \.
u = b. n + \.
ll{>:,r-\). H{„,r). 1\
(^),
10284.
H-\.
Ihle.
id/.
Trunsjiosi
/I.
a — K.
(9, I(», 1).
(960).
2y3 4-^/2^(ll -4^/3) 73
TABLE OF CONTEXTS.
PAUT I. SECTION I.— MATHEMATICAL TABLES. Pap,
Introduction. The C. G. S. Systkm of Units —
Notation and Definitions of Units... ... ... ... 1
Physical Constants and Formulas ... ... ... ... 2
Table I. — English Measures and Equivalents in C. G. S. Units 4 II. — Pressure of Aqueous Vapour at different temperatures 4 III. — Wave lengths and Wave frequency for the principal'
lines of the Spectrum ... ... ... ... ... 4
lY. — The Principal Metals — Their Densities ; Coeffi- cients of Elasticity, Rigidity, and Tenacity ; Expan- sion by Heat ; Specific Heat ; Conductivity ; Rato of conduction of Sound ; Electro-magnetic Specific Resistance ... ... ... ... ... ... 5
V. — The Planets — Their Dimensions, ^Masses, Densities,
and Elements of Orbits ... ... ... ... 5
VI. — Powers and Logarithms of TT and e ... ... ... G
Vll.^Square and Cube Roots of the Integers 1 to .30 ... 6
VIII. — Common and Hyperbolic Logarithms of the
Prime numbers from 1 to 100... ... ... ... G
IX. — Factor Table —
Explanation of the Talilo .. ... ... ... 7
The Least Factors of all numbers from 1 to 00000... S
X. — Values OF THE Gamma-Function 30
SECTION IT.— ALriEBRA. x.^of
Ailiilo
Factors ... ... ... ... ... ... ... ... 1
Newton's Rule for expanding a Binniuial ... ... ... 12
Multiplication and Division ... ... ... ... ... 2S
Indices ... ... ... ... ... ... 20
Highest Common Factor 30
xii CONTENTS.
No. of Artu-lo
Lowest Common Multiple 33
Evolution —
Square Root and Culic Root ... ... ... ... 3.5
Useful Transformations ... ... ... ... ... 38
Quadratic Equations 45
Theory OP Quadratic Expressions SO
Equations IN ONE Unknown Quantity.— Exam rr,i:s 54
Maxima and Minima by a Quadi-atic lv[uation 58
Simultaneous Equations and Exam it, i:a 59
Ratio AND Proportion ... ... ... ••• ••• ••• ^^
The /.• Theorem 70
Duplicate and Triplicate Ratios ... ... ... ... ^2
Compound Ratios ... ... ... ... ... •■• '^
Variation ... ... ... ... ••• ••• •■• ••• 7G
Arithmetical Progrkssion ... ... ... ... ... •-• 79
Geometrical Progression ... ... ... ... ... ... 83
Haumonical Progression 87
Permutations AND Combinations ... ... ... 94
Surds ... ...
108
Simplification of -/a + \/fe and Va-^\/h 121
Simplification of Va-^ V~h 124
Binomial Theorem 125
Multinomial Theorem 137
Logarithms ... ... ••• •.• 142
Exponential Theorem 149
Continued Fractions and Convergents ... ... ... ... 160
General Theory of same ... ... ... 167
To convert a Series into a Continued Fraction ... ... 182
A Continued Fraction with Recurring Quotients ... 186
Indeterminate Equations ... ... ... ... ... ... 188
To reduce A Quadratic Surd to a Contini-kd Fkaction 195
To form high Convergents rapidly ... ... ... 197
General Theory ... ... ... ... ... ... 199
Equations —
Special cases in the Solution of Simultaneous Equations, .. 211
Method by Indeterminate Multipliers ... ... ... 21.S
^liscellancous Equations and Solutions ... ... ... "J 11
On Symmetrical E.xpressions ... ... ... ... iM'.l
Imaginahy Expressions ... ... ... ... ... ... 2"J;?
^IeT1K)D OF InDETEKMINATI; CoEl'FiiTlN IS ... ... ... ... 'I'.Vl
Method OK Proof BY I NDiCTioN ... ... ... ... ... 233
Parfial FuACTiONS. — F(tuR Cases 235
CONVEHGENCY AND J')lVEK(iENCY OF SeKIES ... ... ... ... 2:5;»
General Theorem of ^ (a:) ... ... ... ... ... 2li>
CONTENTS. XI U
N... <.f
Expansion of a FKAnmN ... ... ... ... ... ... -1-H
Ui;cLKi:iN(i Skhiks -•'''I
Tho General Term ... ... ... ... ... ... -">7
Case of Quailratic Fuctiir with Iiiia^Miiary Koots... ... 258
Lagrann^e's Rule ... ... ... ... ... ... 203
SlMMATIOX OF SkRIKS BY THE ^[Ernon OF DlFFIMiKNCE.S ... ... 2r,t
Interpolation of a term ... ... ... ... ... 2t;7
DiRKCT Fact(M{ial Skrik.s ... ... ... ... ... ... 2(58
Invkusk Factorial Serifs 270
Su.MMATiON IJY Paktial Fractions ... ... ... ... ... 272
CoMi'OsiTE Factorial Series ... ... ... ... ••• 271
Miscellaneous Series —
Sams of tho Powers of the Natural Numbers 27G
Suraof (i+(a + J)r+(a + 2(Z);-Hctc 27'J
Sum of n'' — n (7i — l)''4-&c. ... ... ... ... ... 2H.5
POLYOONAL Nu.MI!ERS ... ... ... ... ... 287
FuiURATE Numbers 289
Hypergeometrical Series 21>1
Proof that c" is incommensurable ... ... ... ... 21*5
Interest ... ... ... ••■ ••. ••• ••• ••• 2'JG
Annuities '^^^2
PROr.AniLITIES... ••• ••• ■•• ^*^^
Inequalities ... ... ... ••• 330
Arithmetic ^Ican > Geometric 'Mean ... ... ... 332
Arithmetic Mean of ?h*'' powers > m^^ power of A. ^l. ... 334
Scales OF Notation ... ... ... -.. ••• •^•i2
Theorem concerning Sam or Difference of Digits ... 3i7
Theory OF NuMRERS 3^9
Highest Power of a Prime ji contained in \m^ ... ... 305
Format's Theorem ... ... ... .... ... ... 3r)9
Wilson's Theorem ... ... ... ... ... ••■ 371
Divi.sors of a Number ... ... ... ... ... 374
S, (livisil.lebv2» + l 380
SECTION TIT.^THHOKY OF EQX'ATTON.S.
Factors OF AN EyuATiox ... ... ... ... -.. ••• '^'"^
To compute /(a) numcricallv 4i»3
Di.scriminati(m of Roots ... ... •.. ... ... ^O'J
Descartes' Rule OF SicNs 410
The DEiiiVED Functions of/ (.}•) ... ... .. -.. ■•• 424
To remove an assigned term ... ... ... ... 428
To transform an equation ... ... ... .. •■• 430
Ei^TAL Roots OF an Equation ... ... ... •• ••• 4.32
Pra-tical Rule •^^''
XIV
CONTEXTS.
No. of Article.
Limits OF THE Roots 448
Newton's Method 452
Rolle'fi Tlieorem 454
Newton's Method OF Divisors ... ... ... ... ... 459
RECiPROC-\Ti Equations 4GG
Binomial Equations 472
Solution of .^"±1 = 0 bj Do Moivre's Tlicorcm 480
Cubic Equ.viions ... ... ... ... ... ... ... 483
Cardan s Method 484
Trigonometrical Method ... ... ... ... ... 489
Biquadratic Equ.ations —
Descartes' Solution ... ... ... ... ... ... 492
Ferrai'i's Solution ... ... ... ... ... ... 496
Euler's Solution 499
Commensurable Roots 502
Incommensurable Roots —
Sturm's Theorem ... ... ... ... ... ••• 506
Fourier's Theoi-em ... ... ... ... ... ... 518
Lagrange's Method of Approximation ... ... ... 525
Newton's Method of Approximation ... ... ... 527
Fourier's Limitation to the same ... ... ... ... 528
Newton's Rule for the Limits of the Roots 5.30
Sylvester's Theorem .. . ... ... ... 532
Horner's Method 533
Symmetrical Functions of the Roots of an Equation —
^ms of the Powei'S of the Roots ... ... ... ... 534
Symmetrical Functions not Powers of the Roots... ... 538
The Equation whose Roots are the Squares of the
Differences of the Roots of a given Equation Sum of the m"' Powers of the Roots of a Quadratic
Equation Approximation to the Root of an Equation through the Sums of the Powers of the Roots
E-KPANsioN of an Implicit Function of a;
Determinants —
Definitions ...
General Tlieory
To raise the Order of a Dotonninant
Analysis of a Deterniinaut ...
Synthesis of a Detcrminiuit
Product of two Determinants of the 7i"' Onlei
Synimetrioal Determinants ..
Reciprocal Determinants ...
Partial and Ci>niplcnu'ntary DcierininnntH
545
548 551
55-i 556 564 568 569 570 574 575 576
CONTENTS. XV
\u. ..f
Aril. I.'.
Theorem of a Partial Ik-ciprocal Dclonuiiiunt ... ... .')77
Product of DiU'ercuce.s of /i Quantitius ... ... ... 578
Product of Squares of UiiTereiices of samo ... ... 579
Rational Algebraic Fraction expressed as a Ucleniiinant 5sl
Eli.mi.n.mio.n —
Solution of Linear Ecjuations ... ... ... ... r»s2
Orthogonal Transformation ... ... ... ... r)Sl.
Theorem of the ?i.—'2"' Power of a Deteniiinant... ... r).s5
Bezout's Method of Elimination ... ... ... ... 5SG
Sylvester's Dialytic Method ... ... ... ... 587
^lethod by Symmetrical Function.^ ... ... ... 5H8
Eliminatiox BY llu;iii:sT Cu.MMON Factou ... ... 5'J3
SECTION IV.— PLANE TRIGONOMETRY,
Angular Measuremext ... ... ... ... ... ... COO
Trigonometrical Ratios ... ... ... ... ... ... GOG
Formula) involving one Anglo ... ... ... ... G13
Formula; invoMng two Angles and Multii)lc Angles ... G27
Formula-" involving three Angles ... ... ... ... G7-1
Ratios OF 45°, G0=, 15°, 18°, <fcc GOO
Properties of THE Triangle 700
The s Formula) for sin ^^, &c. ... ... ... ... 70-i
The Triangle and Circle 709
Solution of Triangles —
Right-angled Triangles ... ... ... ... ... 718
Scalene Triangles. — Three cases ... ... ... ... 720
Examples on the same ... ... ... ... ... 859
Quadrilateral in a Circle 733
Bi.sector of the Side of a Triangle 738
Bisector of the Angle of a Triangle ... ... ... 71-2
Perpendicular on the Base of a Triangle ... ... ... 711i
Regular Polygon AND Cikcle 74G
Subsidiary Angles ... ... ... ... ... 749
Limits of Ratios 753
De Moivre's Theorem 75G
Expansion of cos ?i0, &c. in ])owcrs of sin 0 and cosO ... 758
Expansion of sinO and cos 0 in jiowers of 0 ... ... 7G4
Expansion of cos" 0 ami shi" U in cosines or sines of
multiples of (^ ... ... ... ... ... ... 772
Expansion of cos 7i0 and sin7j8 in powers of biuO ... 775
Expansion of cosn9 and sin «0 in powers of cos 0 ... 779
Expansion of cos nO in descending powers of cos 0 ... 780
Sin a -f- c sin (a + /3) + <tc., and similar series 783
XVI
CONTENTS.
No. of Artiol.;.
Gregory's Scries for 0 in powers of tauO .. ... ... 71>1
Formulas for tlio calculation of TT ... ... .. ... 792
Proof that TT is iucomracnsural)lc ... ... ... ... 795
Sina; = ?i sin (a; + a.). — Series for a; ... ... ... 790
Sum of sines or cosines of Angles in A. P. ... ... 800
Exi^ansion of the sine and cosine in Factors ... ... 807
Sin 1/0 and cosnf expanded in Factors ... ... ... 808
Sin(? and cos 0 in Factors involving (^ ... ... ... 815
e' — 2cos6 + e"' expanded in Factors ... ... ... 817
De Moivre's Property of the Circle ... ... ... 819
Cotes's Properties ... ... ... ... ... ... 821
Additional Formulae ... ... ... ... ... ... 823
Properties of a Right-angled Triangle ... ... ... 832
Properties of any Triangle... ... ... ... ... 835
Area of a Triangle ... ... ... ... ... ... 838
Relations between a Triangle and the Inscribed,
Escribed, and Circumscribed Circles ... ... 841
Other Relations between the Sides and Angles of a Triangle 850
Examples of the Solution of Triangles ... ... ... 859
SECTION v.— SPHERICAL TRIGONOMETRY.
Introductort Theorems —
Definitions
Polar Triangle Right-angled Triangles —
Napier's Rules Oblique-angled Triangles.
Formula) for cos a and cos A
The (S Formula) for siniJ, sinJa, <tc. ...
sin^ — sin JB _ sin 0
sin a sin b sin c
cos 6 cos (7 = cot a sin i— cot ^1 sin 0
Napier's Formula) ...
Gauss's Formulas Spherical Triangle and Circle —
Inscribed and Escribed Circles
Circumscril)cd Circles Si'UERiCAL Areas —
Spherical Excess
Area of Sphei'ical Polygon ...
Cagnoli's Theorem ...
Lhuillior's Theorem
870 871
881
882 884
894
895 89G 897
898 900
902 903 904 905
CONTKNTS. XV 11
N... o< ArtirUi.
PoLTiiEnnoNs ... ... ... ... ... ... ... i'06
The five RcgulaT Solids 1>07
Tlio Aiii^'lo hot woiMiAdjiuvnt Faces i'OO
Kadi i of Iii.siTil)L'd and Circiim.scriheil Spheres... ... 'JlO
SECTION VI.— ELEMENTARY GEOMETRY.
Miscellaneous Propositions —
Reflection of a point at a single surface ... ... ... 920
do. do. at Buccussive surfaces ... ... i.'2I
Relations between the sides of a triangle, the segments
of the base, and the line drawn from the vertex ... i'22
Equilateral triangle Yli'C; P.'P + Pi>" + i'0" 923
Sum of squares of sides of a quadrilateral ... ... 024
Locus of a point whose distances from given lines or
points are in a given ratio ... ... ... ... 92(3
To divide a triangle in a given ratio ... ... ... 930
Sides of triangle in given ratio. Locus of vertex ... 032
Harmonic division of base ... ... ... ...' 933
Triangle with Inscribed and Circum.scril}ud circles ... 935
TuE PRor.LE.Ms OF TUE Tangencies 037
Tangents and cliord of contact, fty =. u- ... . ... 0-48
To find any sub-multiple of a line ... ... ... 950
Triangle and three concurrent lines ; Three cases ... 951
Inscribed and inscribed circles ; /?, s — t/, &c. ... ... 953
Ni.N'E-PoixT Circle... ... ... ... ... ... ... 954)
CoNSTULCTiox OK Tkiano.les ... ... ... ... ... 900
Locus of a point from which the tangents to two circles
have a constant ratio .. . ... ... ... ... 003
CoLLiNEAR AND CoxcuKUENx Systems 007
Triangle of constant species circum.scribcd or inscribed
to a triangle ... ... ... ... ... ... \)77
Radical Axis —
Of two Circles l*8i
Of three Circles il'j7
Inveksiox —
Inversion of a point ... ... ... ... ... loOU
do. circle ... ... ... ... ... lii(,»0
do. right line ... ... ... ... ... Iiil2
Pole and Polar ... ... ... ... ... ... ... \u\C,
Coaxal Circles ... ... ... ... ... ... ... Iu21
Centres and Axes of Similitude —
Homologous and Anti-homuldgons pi )ini.s ... ... Iu37
do. do. chord.s ... ... Iu38
C
xvin
CONTENTS.
Constant product of anti-siniilitudc
Circle of similitude
Axes of similitude of three circles
Gergonne's Theorem
Anharmoxic Ratio and Pencil
HoMOGUAi'Hic Systems of Points Involution ... Projection ...
On Perspective Drawing ...
Orthogonal Projection
Projections of the Sphere
Additional Tueorems —
Squares of distances of P from equidistant points on a circle ...
Squares of perpendiculars on radii, etc. ...
Polygon n-ith inscribed and circumscribed circle of perpendiculars on sides, &c....
Sum
No. of Article.
1U43 1045 1046 1049 1052 1058 1066 1075 1083 1087 1090
1094 1095
1099
" SECTION VII.— GEOMETRICAL CONICS.
Sections of the Cone —
Defining property of Conic PS = ePM
Fundamental Equation
Projection from Circle and Rectangular Hyperbola Joint Properties of the Ellipse and Hypereola —
Definitions ...
CS:CA:CX
F8 ±PS'=AA'
CS" = AC ^ PC
SZ bisects ^ QSP
If PZ be a tangent PSZ is a right angle
Tangent makes equal angles with focal distances
Tangents of focal chord meet in directrix
CN.CT =AC'
as :PS =e
NG : NC =^ PC' : AC
Auxiliai-y Circle
J'N: QN = PC :AC
P^- : AN. NA' = L'C- : JO'-
Cn.a = PC'
sy.s'Y' = PC'
PP = A(!
To draw two tangents
Tangents subtend equal angles at the focus
1151 1156
1158
1160 1162 1103 1164 1166 1167 1168 1169 1170 1171 1172 117;i 1174 1 1 7t; 1177 1178 1179 1180 Ubl
TON TK NTS.
No. of Arti.l.'.
To draw two tanj^ents
1201
Asymptotic PHOi'F.nTir.s ok Titr TTviM;i;itor,\ —
RN^-IW = IK" 11^:^
rn.p>=n(" ... n-i-
('!■:= AC ll-<'
J'l> i.s i-Mnillcl (o the A.svni|.l..t.- 1I>^7
Qh'=r- ' "^^
rrj = n an.i (>v = <iV n^i^
Qn.Qx= Plr= UV'--(iV- ll'-'l
Arn.TK = cs' ^ ••• ii-'"-i
Joint Propkrtiks of Ellipre and HYi-r.Kr.oLA rksi'mkh. Cox-
jniATK DiAMETKKS —
QV-.FV.VF'^CD'' :Cr^ 11^.'^
rF.cn = AC .no nin.
rF.ra = nc" ami ff-fo'^zAC ii '■'•'»
PG.FG' = Cr)' ll'-'7
Diameter bisects parallel cliord.s ... ... ... ... 11'''^
Supplemental chord.s
Diameters arc mutually conjugate
CV.CT=r!F^ l-"-5
CN=dR, CB = pN l--^0.-S
CN^ ± CK' = A C\ FF- ^F}r- = FC- 1 '2' i7
CP± (72)^ = ^1 dfc 7>C''' I'-^H
FS.PS' = cn' 1-^-5
OQ . Oq \ OQ' . O'i = CD- : CF- 121t
SR:QL = e 1-'1'»
Director Circle ... ... ••• ■•• ••• ■•• 1-''
Properties of Parabola deduced from the Ellipse ... 121'.» The PAHAnoLA —
Defining property I',S' = P.!/- •.• 1'2"20
Latus Rectum = 4JN •• ^'-^'^
If FZ be a tangent, I'SZ is a right angle 1 -•^3
Tangent bisects Z ,ST.U and fe';^ .If l—'i
ST=SF = Sa l--'^
Tangents of a focal fliord intersect at right anglos in
directrix ... ... ... ... .•• •■• ■ ■ l'--*^
A^ = AT l--^27
NG = 2AS l^^*^
FN"- = iAS.AN l^^O
SA:SY:SF 1-'^^
SQ:SO:SQ' l--^-^
Z OSQ = OSQ' and QOQ' = { QSQ' 1 ■^■'-i-
DiAMF.TK.nS ... ... ... ... ... ■•• •■• l-'^'^
The diameter bi.sects itanilh'l chords ... ... ... I'-^^S
XX
CONTENTS.
QP=4P.9.Pr
0(2.0q :0(/ : Oq =rS iFS
Pai-abola two-thirds of circnmscribing parallelogram Methods OF Drawing A Cuxic
To find the axes and centre
To construct a conic from the conjugate diameters Circle OP Curvature
Chord of curvature = QV^ -^ PV ult
c • 1 1 . , CT)'' OD'- CL^
bemi-cliords ot curvature, — -— , rf^,=r, -77-r
C-P PJf AC
In Parabola, Focal chord of curvature = 4SP ...
do. Kadius of curvature = 2SP"' -^ ST
Common chords of a circle and conies are equally i
clined to the axis To find the centre of curvature ... MiSCKLLAM;OUS TlIKOKLMS
No. of Article.
1239 1242 1244 1245 1252 1253 1254 1258
1259
12G0 12G1
1263 1265 1267
INDEX TO TROPOSITIONS OF EUCLID
REFERRED TO IX THIS WORK.
Tho references to Euclid are made in Koinan and ^Vrabic numerals ; e.g. (VI. 19).
BOOK T.
I. 4. — Triaui^'los arc equal and similar if two sides and the included
an<^le of each are equal each to each. I. 5. — The angles at the base of an isosceles triangle are equal. 1. 0. — The converse of 5. I. 8. — Triangles are equal and similar if tlie tliroe sides of eacli arc
ecjual each to each. I. IT). — The exterior angle of a triangle is grojiter than the interior
and opposite. I. 20. — Two sides of a triangle are greater than the third. I. 26. — Triangles are equal and similar if two angles and one corres- ponding side of each are equal each to each. I. 27. — Two straight lines are parallel if tlicy make equal alternate
angles with a third line. I. 29. — The converse of 27. I. 32. — The exterior angle of a triangle is cqiial to tho two interior
and opposite; and tlic three angles of a triangle are equal
to two right angles.
C'riK. 1.— The interior angles of a ]-)olygon of n sides = («-2)7r.
C'oK. 2. — The exterior angles = 27r. I. 35 to 38. — Parallelograms or triangles upon tlie same or equal
bases and between tho same parallels are equal. I. la.— The conq)lements of the parallelograms about the diameter
of a parallelogram are c([ual. T. M . — Tlio square on the hypotenuse of a right-angled triangle is
equal to the scpiares <m the other sides. I. 48. — The converse of 47.
XXll INDEX TO PTtOrOSITIOXS OF EUCLID
BOOK II.
II. 4.— If a, h arc the two parts of a riglit line, {a + iy = a" -\-1nh-\-h-. If a right line be bisected, and also divided, internally or
externally, into two nnequal segments, then — II. 5 and 6. — The rectangle of the unequal segments is eqnal to the
difference of the squares on half the line, and on the line
between the points of section; or (a + i) (a-h) = a? — l/. II. 9 and 10. — The squares on the same unequal segments are together
double the squares on the other parts ; or
II. 11. — To divide a right line into two parts so that the rectangle of the whole line and one part may be equal to the square on the other part,
II. 12 and 13. — The square on the base of a triangle is equal to the sum of the squares on the two sides lolus or mimis (as the vertical angle is obtuse or acute), twice the rectangle under either of those sides, and the projection of the other upon it ; or a- = h- + c''-21jccosA (702).
BOOK III. III. 3. — If a diameter of a circle bisects a chord, it is perpendicular to
it : and conversely. III. 20. — The angle at the centre of a circle is twice the angle at the
circumference on the same arc. III. 21. — Angles in the same segment of a circle are equal. III. 22. — The opposite angles of a quadrilateral inscribed in a circle
are together equal to two right angles. ITT. 31. — The angle in a semicircle is a right angle. 111. 32. — The angle betAveen a tangent and a chord from the pcint of
contact is equal to the angle in the alternate segment. 111. 33 and 34. — To describe or to cut of ti segment of a circle which
shall contain a given angle. III. 35 and 30.— The rectangle of the segments of any chord of a
circle drawn through an inta'ual or external point is eiiual
to the square of the semi-chord perpendicular to the
diameter through the internal point, or to the square of the
tangent from the external point. III. 37.— The converse of 3G. If the rectangle be equal to the .scpiare,
tlic lino which meets the circle touches it.
iJKKRI.'KKn T(i IN THIS WoKK
lUJUK IV.
IV. 2. — To insoi-ilif a trian^k' of yivcii form in a i-irclc IV. 3. — To describu tlic same about a circle. IV. 4. — To inscribe a circle in a triangle. IV. 5. — To describe a circle about a triangle. IV. 10. — To construct two-iiftlis of a right angle. 1\'. 11. — To construct a regular pentagon.
VI
VI. 4.
VI. VI.
VI.
BOOK VI.
VI. 1. — Triangles and parallelograms of the same altitude arc
proportional to their bases. VI. 2. — A right line parallel to the side of a triangle cuts the other
sides proportionally ; and conversely. 3 and A. — The bisector of the interior or exterior vortical angle of
a triangle divides the base into segments proportional to
the sides. Eipiiangular triangles have their sides proportional honio-
logously. 5. — Tlie converse of -i. 0. — Two triangles are equiangular if they have two angles equal,
and the sides about them proportional. 7. — Two triangles are equiangular if they have two angles equal
and the sides about two other angles proportional, provided
that the third angles are both greater than, both less than,
or both equal to a right angle. 6. — A right-angled triangle is divided by the perpendicular from
the right angle upon the hypotenuse into triangles similar
to itself. 11 and l.'i. — Equal lyaralldoijrams, or trianjlcs which have two
angles equal, have the sides about those angles reciprocally
jiroportional ; and conversely, if the sides are in tliis i)ro-
jiDi-tion, the figures are eciual. ll*. — Similar triangles are in the duplicate latio of their homo- logous sides. 2". — Likewise similar jjiilyg'ons. 23. — E(|uiangular parallelogi-auis are in the ratio compoundetl of
the ratios of their sides. B. — The rectangle of the sides of a (riangle is ccjual to the s(]uare
of the bisector of the vertical angle i>lus the rectangle of
the segments of the base.
VI
VI
VI.
VI. VI.
VI.
XXIV INDEX TO PROPOSITIONS OF EUCLID.
VI. C. — The rectangle of the sides of a triangle is equal to tlie rect- angle under the perpendicular from the vertex on the base and the diameter of the circumscribing circle.
VI. D. — Ptolemy's Theorem. The rectangle of the diagonals of a quadrilateral inscribed in a circle is equal to both the rectangles under the opposite sides.
BOOK XI.
XI. 4. — A right line perpendicular to two others at their point of
intersection is perpendicular to their plane. XI. 5. — The converse of 4. If the first line is also perpendicular to a
fourth at the same point, that fourth line and the other
two are in the same plane. XI. 6. — Right lines perpendicular to tlie same plane are parallel. XI. 8. — If one of two parallel lines is perpendicular to a plane, the
other is also. XI. 20. — Any two of three plane angles containing a solid angle are
greater than the third. XI. 21. — The plane angles of any solid angle are together less than
four r'nAit ano'les.
TABLE OF CONTENTS.
PART II.
SECTION VIIT.— DIFFERENTIAL CALCULUS. No. of
Article.
Introduction ... ... ... ... ••• ■■• •■• 1400
Successive differentiation ... ... ... ... 1405
Infinitesimals. Differentials ... ... ... ... 1407
Differentiation.
Methods 1411-21
SoccEssivR Differentiation' —
Leibnitz's theorem ... ... ... ... ... 14(30
Derivatives of the ?ith order (see Index) ... ... 14G1-71
Partial Differentiation 1480
Theory of Operations ... ... ... ... ... 1483
Distributive, Commutative, and Index laws... ... 1488
Expansion of Explicit Functions —
Taylor's and Maclaurin's theorems ... ... ...1500,1507
Symbolic forms of the same ... ... ... ... 1520-3
f(x + h,y + k),&c 1512-4
Methods of expansion by indeterminate coefficients.
Four rules... ... ... ... •■• ••• 1527-31-
Method by Maclaurin's theorem 1524
Arbogast's method of expanding ^ (2) ... ... 15:3i>
Bernoulli's numbers ... ... ... ... • . 1. ">:'.".'
Expansions of ^ (j; + /0 — ^ (•'')• Stirling and Boole 151G-7
Expansions of Lmplicit Functions —
Lagrange's, Laplace's, and Burmann's theorems, 1552, 1550-03
Cayley's series for --— ... ... ... •• 1555
Abel's series for if>{x-\-a) ... ... ... .•• 1-'"-
Indeterminate Forms 1580
Jacodians ... ... ... ... ••• •• • 1^*^^
Modulus of transformation ... ... ... ... lOUt
XXVI
CONTENTS.
No. of Article.
QiAViics 1620
Euler's theorem .. ... ... ... ... 1621
Eliruinant, Discriminant, Iuvariant,Covariant, Hes.sian 1626-30
Theorems concerning discriminants ... ... ... 1635-45
Notation ^ = 6c-/, &c 1642
Invariants 1648-52
Cogredients and Emanents ... ... ... ... 1653-5
Implicit Functions —
One independent variable ... ... ... ... 1700
Two independent variables ... ... ... ... 1725
w independent variables ... ... ... ... 1737
Change of the Independent Variable ... ... ... 1760
Linear transformation ... ... ... ... ... 1794
Orthogonal transformation ... ... ... ... 1799
Contragredient and Contravariant ... 1813
Notation z„=p,&:,c.jq,r^s,t... ... ... ... 1815
Maxima and Minima —
One independent variable ... ... ... ... 1830
Two independent variables ... ... ... ... 1841
Three or more independent variables... .. ... 1852
Discriminating cubic ... ... ... ... ... 1849
Method of undetermined multipliers ... ... ... 1862
Continuous maxima and minima ... ... ... 1866
SECTION IX.— INTEGRAL CALCULUS.
Introduction 1900
Multiple Integrals 1905
Methods of Integration —
By Substitution, Parts, Division, Rationalization,
Partial fractions. Infinite sei'ies ... ... ... 1908-19
Standard Integrals ... ... ... ... ... ... 1921
Various Indefinite Integrals —
Circular functions ... ... ... ... ... 1954
Exponential and logarithmic functions ... ... 1998
Algebraic functions ... ... ... ... ... 2007
Integration by rationalization... ... ... ... 2110
Integrals reducible to Elliptic integrals 2121-47
Elliptic integrals approximated to ... ... ... 2127
Successive Integration ... ... ... ... ... 2148
Hyperbolic Functions cosh a-, sinh-i;, tanh.r ... ... 2180
Inverse relations ... ... ... ... ■• 2210
Geometrical meaning of tanh <S ... ... ... 2213
Logarithm of an imaginary quantity ... ... ... 2214
CONTENTS. XXVll
No. of Article.
DkFINITK iNlKfiRALS
Summation of series ... ... ... ... ... 2'J3U
'I'liroKKMS KESrECTINU LiMITS OF iMECiliATION ... ... 2233
Methops ok evaluating Definite iNTEiiBAi-s {Eiyitt rules) ... 2245
Differentiation under the sign of Integration ... ... 2253
Integration by tliis method ... ... ... ... 2258
Change in the order of integration ... ... ... 2261
Approximate Integration hy Beunoulm's Series ... ... 2262
The Integrals j;(Z,m) and r(») 2280
logr(l + M) in a converging series ... ... ... 22'.'4
Numerical calculation of r (x) ... ... ... 2317
Integration of Algebraic Forms ... ... ... ... 2341
Integration of Logarithmic and Exponential F(>i;.m.s ... 23i>l
Integration of Circular Forms ... ... ... ... 2451
Integration of Circular Logarithmic and Exponential Forms 2571 Miscellaneous Theorems —
Frullani's, Poisson'.'^, Abel's, Kummer's, and Cauchy's
formula) 2700-13
Finite Variation of a Parameter ... ... ... ... 2714
Fourier's formula ... ... ... ... ... 2726
The Function \P{.c) 2743
Summation of series by the function «/' (.t) ... ... 2757
4/ (.«) as a definite integral independent of \p{l) ... 27»)<>
Nu.merical Calculation of log r{x) 2771
Change of the Variables in a Definite JMl'ltiple I.ntegral 2774 Multiple Integrals- Expansions of Functions in Converging Seriks —
Derivatives of the nth order ... ... ... ... 2852
Miscellaneous expansions ... ... ... ... 2911
Legendre's function X„ ... ... ... ... 2936
Expansion of Functions in Trigonometrical Series ... 2955
Approximate Integration 2991
^lethods by Simp.son, Cotes, and Gauss ... ... 2992-7
SKCTION X.— CALCULUS OF VARIATIONS.
Functions of one Independent Variable ... ... ... 3028
Particular cases... ... ... ... ... ... 3033
Other exceptional cases ... •• •■• ■ 3045
Functions of two Dependent Variables ... ... ... 3051
Relative maxima and minima ... ... ... ... 3069
Geometrical applications ... ... ... ... 3070
XXVlll CONTENTS.
No. of Article.
Functions of two Independent Variables ... ... ... 3075
Geometrical applications ... ... ... ... 3078
Appendix —
Oil the general object of the Calculus of Variations... 3084
Successive variation ... ... ... ... ... 3087
Immediate integrability ... ... ... ... 3090
SECTION XL— DIFFERENTIAL EQUATIONS.
Generation of Differential Equations ... ... .. 3150
Definitions and Rules... ... ... ... 3158
Singular Solutions ... ... ... ... ... ... 3168
First Order Linear Equations ... ... ... ... 3184
Integrating factor for il/dc+iVfZi/ = 0 3192
Riccati's Equation ... ... ... ... ... 3214
First Order Non-linear Equations 3221
Solution by factors ... ... ... ... ... 3222
Solution by difFei'entiation ... ... ... ... 3236
Higher Order Linear Equations 3237
Linear Equations with Constant Coefficients ... 3238
Higher Order Non-linear Equations ... ... ... 3251
Depression of Order by Unity... ... ... ... 3262
Exact Differential Equations 3270
Miscellaneous Methods ... ... ... ... ... 3276
Approximate solution of Differential Equations by
Taylor's theoi'em ... ... ... ... ... 3289
Singular Solutions OF Higher Order Equations 3.301
Equations with more than two Variables... ... ... 3320
Simultaneous Equations with one Independent Variable... 3340
Partial Differential Equations 3380
Linear first order P. D. Equations ... ... ... 3381
Non-linear first order P. D. Equations ... ... 3399
Non-linear first order P. D. Equations with more
than two independent variables ... ... ... 3409
Second Order P. D. Equations 3420
Law of Reciprocity ... ... ... ... ... ... 3446
Symbolic Methods ... ... ... ... ... ... 3470
Solution OF Linear Differential Equations BY Series ... 3604
Solution by Definite Integrals ... ... ... ... 3617
P. D. Equations with more than two Independent Variables 3629
Differential Resolvents of Algebraic Eqlaitons ... 3631
CONTENTS. XXIX
Xo. of Article.
SECTION XII. — CALCULUS OF FINITE DIFFERENCES.
F0KMri,.K KOR FlKST AND uth UlKKKKKNCF.S '^7(^6
Expansion by factorials ... ... ... ••• li/.^O
Gcnemting functions ... ... ... ... ■•• 3732
The operations 1/, A, and (/.r ... ... ... ••• 373-5
Herscbel's theorem ... ... ... ... ••• 3/.)7
A theorem conjugate to Machiuriu's ... ... ... 3759
Interpoi-ation
37G2
Lagrange's interpolation formula ... ... ... 370H
MhXHANlCAL QUADRATURK ^^^72
Cotes's and Gauss's formula3 ... ... ... ... 3777
Laplace's formula ... ... ... ••• ... o/lH
Summation of Series '^781
Approximate Summation 3820
SECTION XIII.— PLANE COORDINATE GEOMETRY.
Systems of Coordinates —
Cartesian, Polar, Trilinear, Areal, Tangential, and
Intercept Coordinates 4001-28
ANALYTICAL CONICS IN CARTESIAN COORDINATES.
Lengths and Areas ^032
Transformation of Coordinates 4048
The Right Line 4(>
Equations of two or more right lines
General Methods
Poles and Polars 4124
The Circle ^^^G
Co-axal circles 4U»1
The Parabola '^^^^
The Ellipse and Hyperbola 4250
Right line and ellipse 4310
Polar equations of the conic 433t.
Conjugate diameters 4o4b
Determination of various angles 4375
The Hyperrola referred to its Asymptotes 4387
The rectangular hyperbola 439-
4110 4114
XXX
CONTENTS.
The General Equation
The ellipse and hyperbola
Invariants of the conic
The parabola
Method without transformation of the axes . . .
Rules for the analysis of the general equation
Right line and conic with the general equation
Intercept equation of a conic ...
Similar Conics
Circle of Curvature —
Contact of Conics CoNKOCAL Conics
No. of Article.
4400 4402 4417 4430 4445 4464 4487 4498 4522
4527 4550
ANALYTICAL CONICS IN TRILINEAR COORDINATES.
The Right Line
Equations of particular lines and coordinate ratios of particular points in the trigon Anharmonic Ratio
The complete quadrilateral The General Equation of a Conic
Director- Circle ... Particular Conics
Conic circumscribing the trigon
Inscribed conic of the trigon ...
Inscribed circle of the trigon ...
General equation of the circle...
Nine-point circle
Triplicate-ratio circle ...
Seven-point circle Conic and Self-conjugate Triangle...
On lines passing through imaginary points ... Carnot's, Pascal's, and Brianchon's Theorems The Conic referred to two Tangents and the Chord of Contact —
Related conics ...
Anharmonic Pencils of Conics
Construction of Conics
Newton's method of generating a conic Maclaurin's method of generating a conic ...
The Method or Reciprocal Polars...
Tangkntial Coordinates Abridged notation
4601
4628 4648 4652 4656 4693 4697 4724 4739 4747 4751 4754 47546 4754e 4755 4761 4778-83
4803 4809 4822 4829 4830 4844 4870 4907
CONTKNTS. XXXI
No. of Article.
51G7 r.l72
On Tin: 1m kksection ok two Conics—
Geonictricftl^mcanin^ of v/(- 1) ... ... 'I'-^l^'
The Methop of Pkojection '^•'-l
Invariants anp Covakiants 41K5G
To find the foci of the general conic ... .. ... 5008
THEORY OF PLANE CURVES.
Tangent and Normal -''l^'*^
Radius of Curvature and Evolute -"il-^^
Inverse Problem and Intrinsic Equation 51G0
Asymptotes
Asymptotic curves Singularities of Curves —
Concavity and Convexity ... ... ... ... 5174'
Points of inflexion, multiple points, &c. ... ... 5176-87
Contact of Curves ... ... ... ... ... ••• 5188
Envelopes 5192
Integrals of Curves and Areas ... ... ... ... 519G
Inverse Curves 5212
Pedal Curves ... ... ... ... ... ... ••• 5220
Roulettes ... ... ... ••• ••• ••• ••• 5229
Area, length, and radius of cui'vature... ... ... 5230-5
The envelope of a carried curve ... ... ••• 5239
Instantaneous centre ... ... ... ... •■• o243
Holditch's theorem ... ... ... ... ••• 5244
Trajectories ^24G
Curves of pursuit ... ... ••• 5247
Caustics 52-*8
Quetelet's theorem ... ... ... ••• ••• 5-49
Transcendental and other Cuhves —
The cycloid 5250
The companion to the cycloid... ... ... ... 5258
Prolate and curtate cycloids ... ... ... ... 52G0
Epitrochoids and hypotrochoids ... ... ... 52G
Epicycloids and hypocycloids... The Catenary ...
The Tractrix
The Syntractrix
The Logarithmic Curve 5284
The Equiangular Spiral 5288
The Spiral of Archimedes 5296
The Hyperbolic or Reciprocal Spiral 5302
GG 5273 5279
5282
XXXn CONTENTS.
No. of Article.
The Involute of tlie Circle ... ... ... ... -5306
TheCissoid 5309
The Cassinian or Oval of Cassini ... ... ... 5313
The Lemniscate ... ... ... ... ... 6317
The Conchoid 5320
The Lima9on ... ... ... ... ... .. 5327
The Versiera (or Witch of Agnesi) ... ... ... 5335
The Quadratrix... ... ... ... ... ... 5338
The Cartesian Oval 5341
The semi- cubical parabola ... ... ... ... 5359
The folium of Descartes ... ... ... ... 5360
Linkages AND LiNKwoRK ... ... ... ... ... 5400
Kempe's five-bar linkage. Eight cases 5401-5417
Reversor, Multiplicator, and Translator ... ... 5407
Peaucellier's linkage ... ... ... ... ... 5410
The six-bar invertor ... ... ... ... ... 5419
The eight-bar double invertor ... ... ... 5420
The Quadruplane or Versor Invertor ... ... 5422
The Pentograph or Proportionator ... ... ... 5423
The Isoklinostat or Angle-divider ... ... ... 5425
A linkage for drawing an Ellipse ... ... ... 5426
A linkage for drawing a Lima9on, and also a bicir-
cular quartic ... ... ... ... ... 5427
A linkage for solving a cubic equation ... ... 5429
On three-bar motion in a plane ... ... ... 5430
The Mechanical Integrator ... ... ... ... 5450
The Plauimeter 5452
SECTION XIV.— SOLID COORDINATE GEOMETRY.
Systems of Coordinates ... ... ... ... ... 5501
The Right Line 5507
The Plane 5545
Transformation of Cooi;niNATES ... ... ... ... 5574
The Sphere ... ... ... ... ... ... ... 5582
The Radical Plane 5585
Poles of similitude ... ... ... ... ... 5587
Cymxdrical and Conical Surfaces ... ... ... ... 5590
Circular Sections ... ... ... ... ... 5596
Ellipsoid, HvriiRUOLOiD, and Paraboloid ... ... ...5590-5621
CONTENTS.
N... of Article.
Centrai, Qcapiuc Surfack—
Tangent and diainotnil plaiu's... ... ... ... 5026
Eccentric values of the coordinates ... ... ... 5038
CoNKOCAL QuAi'urcs ... ... ... ... ... ... •''^•^0
Reciprocal and Enveloping Cones ... ... ... 56G4
Thk Genkral Equation of a Quadric 5073
Reciprocal Polars •'''"" !•
Theory of Tortuous Curves •'"^^•Jl
The Helix 575i;
General Theory of Surfaces —
General equation of a surface ... ... ... ... 5780
Tangent line and cone at a singular point ... ... 5783
The Indicatrix Conic ^^795
Eulei-'s and Meunier's theorems ... ... ... 5806-9
Curvature of a surface... ... ... -. ... 5826
Osculating plane of a line of curvatnri! 5835
Geodesics ... ... ... ... -.. ... ■.• 5837-48
Invariants ^856
Integrals for Volumes and Surfaces ... ... ... 5871
Guldin's rules ... ... ... ... ... ■•. 587i)
Centre of Mas.s 5884
Moments and Products of Inertia 5903
Momcntal ellipsoid
Momental ellipse
Integrals for moments of inertia ... ... ... 5978
Perimeters, Areas, Volumes, Centres of Mass, and Moments of Inertia of various Figures —
Rectangular lamina and Right Solid... ... ... 6015
The Circle <'019
The Right Cone 0043
Frustum of Cylinder ... ... ... ... ... 0048
The Sphere
5925-40 5953
6050
The Parabola G067
The Ellipse
0083
Fagnani's, Grinitli's, and li;uid)ert's tlu'orenis ...0088-0114
The Hyperbola ''llS
The Paraboloid Ol"'^<3
The Ellipsoid <>11--
Prolate and Oblate Spheroiils 0152-05
rKKFACK TO VAUT Jl
ArOLOGlES for the noii-completioii of tliis voliniie ;it ;iii earlier period are due to friends and enquirers. The hd)our involved in its production, and the pressure of other duties, must form the autlior's excuse.
In the compilation of Sections VIII. to XIV., the following works have been made use of : —
Treatises on theDifFerential and Integral Calculus, by liertratid, Hymer, Todhunter, Williamson, and Gregory's Examples on the same subjects; Salmon's Lessons on Higher Algebra.
Treatises on the Calculus of Variations, by Jellett and Tod- hunter ; Boole's Differeutial Equations and Supplement ; Carmichtiers Calculus of Operations ; Boole's Calculus of Finite Differences, edited by Moulton.
Salmon's Conic Sections; Ferrors's Trilinear Coordinates; Kompo on Linkages {Fruc. of Roij. Soc, Vol. 23) ; Frost aud Wolstenholme's Solid Geometry ; Salmon's Geometry of Three Dimensions.
Wolstenholme's Problems.
The Index which concludes the work, and which, it is hoped, will supply a felt want, deals with 890 volumes of o2 serial publications : of tliese publications, thirteen belong to Great Britain, one to Xew South Wales, two to America, four to France, five to Germany, three to Italy, two to Russia, and two to Sweden.
As the volumes only date from the year 180(j, the
XXXVl PREFACE.
important contributions of Euler to the " Transactions of the St. Petersburg Academy," in the last centur}^ are excluded. It was, however, unnecessary to include them, because a ver^- complete classified index to Euler's papers, as well as to those of David Bernoulli, Fuss, and others in the same Transactions, already exists.
The titles of this Index, and of the works of Euler therein referred to, are here appended, for the convenience of those who may wish to refer to the volumes.
Tableau general des publications de I'Academie Imperiale de St. Pek'i-sbourg depuis sa fondation. 1872. [B.M.C.:* 11. Ii. 2050, e.] I. Commentarii Academias Scientiarum Imperialis Petropolitanae.
1726-1746; 14 vols. [B.M.C: 431,/.] II. Novi Commentarii A. S. I. P. 1747-75, 1750-77; 21 vols. [B. M. C. : 431,/. 15-17, g. 1-16, h. \, 2.]
III. Acta A. S. I. P. 1778-86; 12 vols. [B.M.C: 431, /i.;3-8; or
T.C. 8,a. 11.]
IV. Nova Acta A. S. I. P. 1787-1806; 15 vols. [B. M. C. : 431,
A. 9-15, LIS; or T.G. 8, a.23.] V. Leonliardi Euler Opera minora coUecta, vel Commentationes Aritli-
meticte collectse ; 2 vols. 1849. [B. M. rj. : 853 J-, ee.] VI. Opera posthuma mathematica et pliysica ; 2 vols. 1862. [B.M.C: 8534,/] VII. Opuscula analytica ; 1783-5; 2 vols. [B.M.C: 50,/. 15.] Analysis infinitorum. [B.M.C: 529,6.11.]
G. S. C.
Endslkigh Gakdkns, London, N.W., 1886.
British Museum Ctitiilogui
i
[Correct tons nhich mi important are marknl with an imtirink.)
•Art.
1,
Lino 7, fu
• volume
read weight.
„ 10,
f^amnie-million
,,
gramme-six.
6,
,, 5.
1-407 1490
•4971499.
,, 6,
■(•.(•.7o:i")S
,,
M447'-'90.
123,
2
2 v/5
,,
2^/15.
259,
2,
a- + /3-
,,
(a' + 0-)».
276,
„ 6,
3;(- + w — 1
,,
3«- +'in — \.
291,
„ 1,
S
,,
7-
292,
„ 3&4, „
a
,,
a.
322,
,, 0, ,,
45 and 13
,,
35 and 10.
361,
„ 7,
3528
,,
8684.
459,
„ 3&0, „
-6
,,
-16.
470,
„ 1,
.r,„
,,
X .
489,
„ 0,
„
3
555,
M 1-i.
a number of rows
,,
two columns.
593,
„ 11&12, „
li and i?
,,
El and i?...
604,
•2,
one-sixtieth
,,
one-ninetieth.
713,
M 2, ,.
II.
,,
III.
897,
,, •'', >
cosic
,,
sin Jc.
922ii.,
/>H2c-
,,
2J2 + C2,
949,
„ last, ,
D
,,
C.
1076,
M 2,
(Me "The projections
. . . are p
■irallel."
1158,
„ last, ,
1201
,,
1217.
1178,
M 1.
rs
,,
PS'.
1241,
,, 1,
parallel
>>
conjugate.
1413,
„ 3,
-dxdv
,,
+ dndv.
1491,
„ 3,
-
M
=
1849,
„ 1.
+
,,
-
1903,
footnote, ,
Iw
,.
/W-
1925,
supply dx.
1954-6,
supply X.
2030-2,
erroneous, because / in (1427) is necessarily an
integer
2035,
„ 1,
, ax
,,
a.
2140,
,, 1,
y applies to the whole denominator.
2136,
„ last, ,
, 2294
2293.
2354,
, -
+
2392,
xi'
X -1.
2465,
P. 2
T
3237,
,. 1.
, (»-l).r
("-I).
3751,
s„ppl,, ,
ii-
4678,
ilele 2 in the second term.
4680,
supply the factor 4 on
the left.
4692,
;-),
dele 2 in the second term.
4903,
,, :J,
mpplij the factor 4 on
the loft.
5154,
M •*.
, 3155
,
51 -.5.
5330,
,, 2.
, m
-
/>.
and refer to Fig. 129, Art. 5332 on the cardioid is wanting.
MATHEMATICAL TAJiLKS.
INTRODUCTION.
The Ccnthnctrv-Grammc-Second system of units.
Notation. — The decimal measures of length are the kilo- metre^ hectometre, decametre, metre, decimetre, centimetre, miUimetre. The same prefixes are used with the litre and gramme for measures of capacity aud volume; ^'^''f^
Also, 10' metres is deuouiinated a metre-><ei:en; 10"^ metres, a seventh-metre ; lO^f grammes, a gramme-fy'tcen ; and so on.
A o;rainme-imttwH is also called a megagramme ; and a milliontli-gramme, a microgramme ; and similarly "with other measures.
Definitions. — The C. G. S. system of units refers idl pliy- sical measurements to the Centimetre (cm.), tlie (Jramme (gm.), and the Second (sec.) as the units of length, mass, and time.
Tlie quadrant of a meridian is approximately a metre- seven. More exactly, one metre = 3-28U8Gyi feet = 89-370-i-j2 inches.
The Gramme is the Unit of mass, and the weight of a gramme is the Uiiit qftveight, being approximately the wciglit of a cubic centimetre of water; more exactly, 1 gm. = 15-432:U1) grs.
The jL/f;v' is a cubic decimetre: but one cubic centimetre is the C. G. S. Unit of volume.
1 litre = -035317 cubic feet = '22000(37 gallons.
The l)i/ne (dn.) is the Unit of force, and is the force wliicli, in one second generates in a gramme of matter a velocity of one centimetre per second.
The 7'i/v/ is the Unit of worlc and energg, and is the work done by a dyne in the distance of one centimetre.
The absolute Unit (f atmospltcric pressure is one meg:ulyno })er square centimetre = 71«'0G1 cm., or 2!>"."j14 in. of mercurial column at 0" at London, where ^ = y8ri7 dynes.
Elasticitij of Volume = l\ is the pressure per unit area upon a body divided by the cubic dilatation.
15
MATHEMATICAL TABLES.
Rigidity = n, is the shearing stress divided hj the angle of the shear.
Young's Modiihifi = M, is the longitudinal stress divided by the elongation produced, = 9nh -h (3/j + ??).
Tenacity is the tensile strength of the substance in dynes per square centimetre.
The Gramme-degree is the Unit of heat, and is the amount of heat required to raise by 1° C. the temperature of 1 gramme of water at or near 0°.
Thermal capacifi/ of a body is the increment of heat divided by the increment of temperature. When the incrc- raonts are small, this is the thermal capacity at the given temperatu"e.
Specific heat is the thermal capacity of unit mass of the body at the given temperature.
The Electrostatic iinit is the quantity of electricity which repels an equal quantity at the distance of 1 centimetre mth the force of 1 dyne.
The Electromagnetic unit of quantity = 3 X 1 0^'' electro- static units approximately.
The Unit of potential is the potential of unit quantity at unit distance.
The Ohm is the common electromagnetic unit of resistance, and is approximately = \(f G. G. S. units.
The Volt is the unit of electromotive force, and is = 10^ C. G. S. units of potential.
The Weber is the unit- of current, being the current due to an electromotive force of 1 Volt, with a resistance of 1 Ohm. It is = -j^ C. G. S. unit.
Resistance of aWiTe = Specific resistance X Length -r- Section.
Physical constants and Formulce.
In the lutitudc of London, cj = 3:2-1908t' feet per second.
= l>!^ri7 centimetres per second. In latitude X, at a liciglit h above the sea level,
g = (98O-0U56 — 2-.'')028 cos 2/\ — -OOOOO:]/;) centimetres per second. Seconds ])endaliim = (iJ9-85G2 — -2536 cos 2\ — -0000003 h) centimetres.
THE 7';.17i"i7f.— Semi-polar axis, 20,854890 feet* = G-3;.4ll x lO^centims. Mean semi-equatorial diameter, 20,9_'G202 „ * = 0 3782t x 10"
Quadrant oi" meridian, 39-377780 x 10' inches* = TOOOlOO x lO" metres.
Volume, r08279 cubic centimetre-nines. JIass (with a density 5g) = Six gramme- twenty-sevens nearly.
* These dimensions nro liikcn frjiu C'larko'a "Geodesy," 1880.
MATIIEMA TICAL TA BLES.
Velocity in orbit = 2033000 ccntims per sec. Ohii.iuity, -2:f 27' lo".»
Aiii^ular vclucily of rotation = 1 -=- 13713.
Precession, t>0"'20.* Prounession of Apse, U"-2.'). I':cCentricity, e = -01079.
Centrifugal Ibrce of rotation at tlio equator, ;>-3'.)12 dynes per «^nunnio.
Force of attraction upon moon, -2701. Force of sun's attraction, -0839.
Katio of (/ to centrilu^ral force of rotation, g : rw* = 2H0.
Sun's horizontal parallax, h"7 to '.»'.* Aberrat i<.n, 20"-ll to 20"70.*
Semi-diameter at earth's mean distance, 1»>' l"b2.*
Approximate mean distance, 1>2,UUOUUO miles, or l"'i8 centimetre-tliirleens.t
Tropical year, 3Go2422l6 days, or 31,550927 seconds.
Sidereal year, 305-250374 „ 31,558150 „
Anomalistic year, 305-259544 days. Sidereal day, 8010 !• seconds,
TJllJ M0UN.—Uas8 = Earth's ma.'^s X -011304 = 0-98 10^ granimes. Horizontal parallax. From 53' 50" to 01' 24".*
Sidereal revolution, 27d. 7h. 43m. 1 l-40s. Lunar month, 29d. 12h. l-lin. 2878. Greatest distance from the earth, 251700 miles, or 4U5 centinieire-tens. Least „ „ 225000 „ 303
Inclination of Orbit, 5° 9'. Annual regression of Nodes, 19° 20'.
Hulk. — {The yt'ar+l)-^19. The remainder is the Guhlen Number. {Tlie Uulih'ti Number— 1) X 11-^30. The remainder is the Ejiact.
GRAVITATION.— Attraction between masses ) mm clvues
m, m' at a distanco / j ~ ;-' x l-54o x It/ The mass which at unit distance (1cm.) attracts an eijual mass with unit force (1 dn.) is = v/(l-543x 10^; gm. = 3iV28 gm.
Tr.rr^/i!.— Density at 0°C., unity ; at 4°, 1 0000l3 (Kupffer).
Volume elasticity at 15°, 2-22 X iV".
Compression for 1 megadyue per sq. cm., 4-51x10-* (Amaury and Descamps).
The heat required to raise tlie temperature of a mass of water from 0° to i° is proportional to < + -00002/" + OiJ00003i* (Regnault).
G'yI-8'ii'iS'.— Expansion for 1° C, -003065 = 1-4-273.
Spccitic heat at constant pressure _ i.jAq Specific heat at constant volume Density of dry air at 0° with Bar. at 76 cm. = -0012932 gm. per cb. cm. (Regnault).
At unit pres. (a megadyne) Density = -0012759. Density at press, p = jtx 1-2759 X 10"'.
Density of saturated steam at t°, with j) taken) _ -7931 .09^;)
from Table 11., is approximately j (i -|- OUoOOO lO"*
SOUND. — Velocity = \/(elasticify of mediuvi -^ detusifij). Velocity in dry air at t° = 3o2 10 ^(1 -+--00300/) centimetres per second. Velocity in water at U' = 14;:U(J0 „ „
LIGHT. — A'elocity in a medium of absolute refrangibility /i = 3004 X lO"'-^^ (Coruu). If I' be the pressure in dynes per sq. cm., and / the temperature, ^i-l = 29(K; X lU-''i'-4-(l-|--OO30i;/) (Biot & Arago).
• These fliita iire from the "Nautical .Mmniiack" for 1S8:{.
t Inuisil ol' Vuuus, IbTt, " Aalruin. S c. XuI.lcs,'' Vols. 37, '<i8.
MATHEMATICAL TABLES.
Table I. Various Measures and their Equivalents in C. G. S. units.
Dimensions. 1 inch = 2-5400 cm.
1 foot = 30-4797 „
1 mile = 160933 „
1 nautical do. = 185230 „ 1 sq. inch = 6'451G sq. cm, 1 sq. foot = 929-01 „ ] Pq. yard = 83G1-13 „ 1 sq.mile = 2-59 X 10^°,, 1 cb. inch = 16387 cb. cm. 1 cb. foot = 28316 1 cb. yard = 761535 ., 1 gallon = 4541 „
= 277-274 cb. in. or the vo- lume of 10 lbs. of water at 62° Fall., Bar. 30 in.
1 grain
1 ounce
1 pound
1 ton
1 kilogramme
1 pound Avoir
1 pound Troy
Mas^.
= -06479895 gra. = 28-3495 = 453-5926 „ = 1,016047 „ = 2-20462125 lbs. = 7000 grains = 5760 „
^'cIoc■lfl/.
1 mile per hour = 44704 cm. per sec. 1 kilometre „ = 27'777 „
Pressure. 1 gm.persq.cm.= 981 dynes per sq. cm, 1 lb. pcrsci.foot = 479 „
1 lb. per sq. in. = 68971 76 centimetres-)
of mercury [ = 1,014,000 „
at 0° C. )
^^^^- P^^ ^q- ^"- = 70-307 = ^ gms. per sq. cm. -014223
Force of Gravity. upon 1 cramme = 981
1 grain = 6fi-^)Cj777
„ 1 oz. =2-7811x10*
„ 1 lb. = 4-4497 X 10-^
„ 1 cwt, =4-9837x10'
„ 1 ton = 9-9674 X 10»
WorJc (^ = 981), 1 gramme-centimetre = 981
dynes
erofs.
1 kilogram-metre 1 foot-grain 1 foot-pound 1 foot- ton
981 X 10-"^ 1-937 xlO\, 1-356x10" „ = 3-04 X W „
1 'hor.se po-wer' p. sec. = 7-46x10®
ITeat. 1 gramme-degree C, = 42 X 10" ergs, 1 pound-degree =191x10- „
1 pound-degree Fah. = 106 x lU* „
Table II.
Pressure of Aqueous Vapour in
dynes per scquare centim.
Teinj).
Press me.
Temp.
Pressure.
-20°
1236
40°
73200
-15°
1866
50°
122t;00
-lu°
2 7; 10
60°
li)8500
- 5°
4150
80°
472900
0°
6133
100°
1014000
5°
8710
120°
1988000
10°
12220
140°
3626000
15°
16930
160°
6210000
■ 20°
23190
180°
10060000
25°
31400
200°
15600000
30°
42050
Table III.
Values for the principal Lines of t he Spectrum in air at 1G0°C. with Bar. 76 cm.
Wave-length
No.ot'vibriitions
iu centini.<s.
per second.
A
7-604x10-''
3-950 X 10"
n
6-867 „
4-373 „
('
6-56201 „
4-577 „
1) (mean)
5-89212 „
5-097 „
J'J
5-26913 „
5-700
F
4-86072 „
6 179 „
G
4-30725 „
6-973 „
^1
3-96801 „
7-569 „
//,
3-93300 „
7-636
M. I THEM A TFCA L TA BLES.
Elect. Magn.
Specific
Resistance
at 0= C.
9158 2081 00100 10850 1521 1015
9827
13300 5000
Rjite of
Coniluction
of Sound
in cm. per sec.
Ci -f ' CO -1 -* -O Ol -O Ol ' ' CO
-c IN. Ol '-C r^ 'O CO o 01 ';0
CN Ah Ah Ol CO CO -rf< O "'o -f
Ill-
coo l-ioooco 'co 'coco '
Specific Heat be- tween 0 & 100 c.
"O o r^ ~ X' i^ ~
6? I?? i:^?. 1 IS 1 l^:i:
o o o o -7- 9' r
Linear Expansion between 0 & lOOC.
-000875 •001483
•002801
00100
•00175
•00103
■001111
■001258
001200
•00227
•00204
•00081
§ 1 O 1 1^ O O -lO -^ -+ -? CO O tj. _:o X) .o
'I|l^;|iiii'i';!r5iii
1 I 1 00 1 ->? 00 1 '.o CO r- i^ 1
1 =
3
or:;:;;: ;
I 1 1 1 I x,^ I ! ^
■^ CO 0 l^ XI 0 1
35 2
ir
0 E s :: = s r I 1 |x|^^c5„ci| Ico
CJ CO l^ -f< -^ CO 'C
"0 01 0 CO ri — . ^ 9> ^^^^^, 0
-0 CO — 1 lO t^ Ci 01
CO Ci 0 1 - -T^ I^ CO t^ -+ C5 Ci -t
01 ..0 CO -+ X rj< C'l 0 X Ol ^ 0
-^ 0 ^ Ah 6 X 60 l'-~ l'^ I^ t^ I-* 61
Platinum
Gold
^Mercury
Lead
Silver
Copper
Brass, drawn
Iron, cast
Iron,wrou<^'lit
Steel
Tin, cast
Zinc, ca.st
Glas.s, flint
•s
c
C
025 2-01 007 1-00 072 024 0-13 015 027
i
2: T' ^ V '^ 9 9 '.^ f ■
Diameter
in
Miles.
i CO {: r: ^ ?i .0 8 :ri
X C-- l^ CO CO X
II
X c; X -H '^ Ol -c w
01 OJ Ol Ol r-H
-^ 1 53 ?i ?i ?! ^' 2
a 6 .2 .-e -q
lis £°-2
III
1 X ol -3 X ?: in X ?i
Ol CO o CO i^ •-= :^
,'ri5x:^^i-."2
1 CO t^ C- CO -- O Ol I^
Greatest distance
from Sun.
Earth's mean
distance = 1.
-v ;f, ,^ ,^ ,^ ?, Zi 5
, '3 X o ..0 CO CO - X
l^r^o-3l^'o'ooi
t^ - ^ 5 =
a X _ u r S -
MATHEMATICAL TABLES.
Table
YI. — Functions of it and e.
TT = 31 1.15;i2G
TT- = ;)-.s(;;t(;<»44
7r» = 31'<tin;2761
-/tt = 1-7724539
logjoTT = t-4971499
ic),o-.7r = &m7m^
TT-' = -3183099
7r-2=: -1013212
TT-' = -0322515
200''-=-7r = 63''-6619772
180°-^7r = 57°-2957795
= 2002r.4"-8
e = 2-7182Hm3 e* = 7 389n:.(;Jl e-^ = 0-307s7!'4 e-2 = 01353353 lo2-„e = 0-43429448 low-, 10 = 2 -30258509
Table YII.
Table VIII,
No. 2
S([uaro root.
Cube root.
1-414213G
1-2599210
3
1-7320508
1-4422496
4
2-0000000
1-5874011
5
2-2360680
1-7099759
6
2-4494897
1-8171206
7
2-6457513
1-9129312
8
2-8284271
2-0000000
9
3-0000000
2-0800837
10
3-1622777
2-1544347
11
3-3166248
2-2239801
12
34641016
2-2894286
13
3-6055513
2-3513347
14
3-7416574
2-4101422
15
3-8729833
2-4662121
16
4-0000000
2-5198421
17
4-1231056
2-5712816
18
4-2426407
2-6207414
19
4-3588989
2-6684016
20
4-4721360
2-7144177
21
4-5825757
2-7589243
22
4-6904158
2-802o:;i»3
23
4-7958315
2-8438670
24
4-8989795
2-8841991
25
5-0000000
2-9240177
26
5-0990195
2 9624960
27
5-1961524
3-0000000
28
5-2915026
3-0365889
29
5-385164S
3-072316S
30
5-4772256
31072325
2
logio N.
log^.V.
-3010300
•69314718
3
-4771213
1-09861229
5
-6989700
1-60943791
7
-8450980
1-94591015
11
1-0413927
2-39789527
13
1-1139434
2-56494936
17
1-2304489
2-83321334
19
1-2787536
2-94443.^98
23
1-3617278
3-l;;540422
29
1-4623980
3-3672'.>5SH
31
1-4913617
3-4339S720
37
1-5682017
3-610111791
41
1-6127839
3-71357-207
43
1-6334685
3-761-2O01-J
47
1-6720979
3-8501 i-rt/.o
53
1-7242759
3 9 702; Ml 11
5i>
1-7708520
407753744
61
1-7853-298
4-110873S6
t)7
1-8260748
4-20460262
71
1-S5125S3
4-26267;t88
73
l-S(;;;3229
4-2:h)45:M.4
7'.^
1-8976271
4-36944785
83
1-9190781
4-41884061
89
1-9493900
4-48863637
97
1-9867717
4-57471098
101
2-0043214
4-61512052
103
2-01-28372
4-63472899
lor
2-0293838
4-67282883
Loi)
2-0374265
4-69134788
NoTi^. — The authorities for Table IV. are as follows: — Columns 2, 3, anil 4 (Mereury e.xcepted), Everett's experiments (Phil. Trans., 1867); (j is hero taken = 981-4. The densities in these cases are those of the specimens employed. Cols. 5 and 7, Kaid<;iiie. Col. 6, Watt's Diet, of Cliemistry, Col. 8, Dulong and Petit. Col. 10, Wertheim. Col. 11, Matthiesseu.
'J'al^le V. is abridged from Loomis's Astronomy.
The values iu Tabic III. are An^r^itrom's.
BURCKTIARDT'S FACTOR TABLES.
For all M.'.Mr.KKS FJiOM 1 to 9'JOOO.
Explanation. — Tlic tables give the least divisor of cvciy number from 1 up to 99000 : but numbers divisible by 2, 3, or 6 are not printed. All tlie digits of the number whoso divisor is sought, excepting the units and tens, will be found in one of the three rows of larger figures. The two remaining digits will be found in the left-haiid column. The least divisor will then be found in the column of the first named digits, and in the row of the units and tens.
If the number be prime, a cipher is printed in the place of its least divisor.
The numbers in the first left-hand column are not conse- cutive. Those are omitted which have 2, 3, or 5 for a divisor. Since 2"-. 3. 5"^ = 300, it follows that this column of numl)er3 will re-appear in the same order after each multiple of 300 is reached.
Mode of using TnE Tables. — If the number whose prime factors are required is divisible by 2 or 5, the fact is evident upon inspection, and the dinsion must be effected. The quotient then becomes the number whose factors are required. If this number, being within the range of the tables, is yet not given, if is dirisihle by 3. Di\'iding by 3, we refer to the tables again for the new quotient and its least factor, and so on.
Ex.\Mrr,ES. — Required the prime factors of 3101-55.
Dividinrr by 5, the quotient is G2031. This number is within the range of the tables. But it is not found printed. Therefore 3 is a divisor of it. Dividing by 3, the quotient is 20G77. The table gives 23 for the least factor of 2ftr)77. Dividing by 23, the quotient is SW.
The table gives 2i» for tlie least factor of H'.tO. Dividing by 20, the quo- tient is 31, a prime number. Therefore 31015-3 = 3.5.23.20.31.
Again, roipiired tiie divisors of 02881. The table gives 203 for the least divisor. Dividing by it, the quotient is 317. Referring to the tables lor 31 7, a cipher is found in tbe place of the least divisor, and this signifies that 317 is a prime nundjer.
Tlitrefore 02S81 = 203 X 317, the product of two primes.
It may be remarked that, to have resolved 02881 into these factors with- out the aid of tiio tables by the method of Art. 3G0, would have iuvolved fifty-nine fruitless trial divisions by prime uumbei-a.
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CO - i^
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t^ t^ — -1 t>.
CO c:^ o -. o
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r- CO t-
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C-- -gooo- 5 = 2 = °"
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to
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200 g;:;2«'-S gf^SS"*
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CI
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ocn t^—t^ — t^M cnOMO — —
0 22*-'^*' ^ S'^
^
t^orji-- OOC-; i^s-ci.-o t^-ri^os =r;t-oor-
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ao
CO
Oioont> 1^0-nt^ — noi-t^ t^c^i^oi^ o> n 0 0 0 n c - 0 = 0 -- cj- ^ « ^c;
r-r:t^ o»o -= ooooor^
OM — t^— ot-o — w4 r»a-. t^oo cct^t^— ot^« — n —
" - -« 2- 2- -=^^-
t-on ot^— ocoo — or^— no — 0 0 c» — •* —
CO CO
-ni--n nnOl^O 01-000 OiO-S>r^ OOOt~.C-. 31
0 — O OOOf»—— MSlOlOl^O
g
— nt^— 0 noc>o>t>. o-ot~o on-jci t^ — cncn
0-. = 0 t^-.OOr:r^ 0-.M03>re
CO
l^-r^t^M 00-1^0 ©on- 0 t^OiOOM OOt^OlOO
— t^ooo) t»r^r^OM r-ot^oo 3>oo — 0 000 — r»o -r — — c<— 0 — r» — - -
-00 0-00.. 2 S'^£=='S
co'^iot^-nc^ — t^ -1^0
j::
OnO-t- r-3-. -OCl nn-OO 3. l~MOI^ Ol-non-
«-r:i^ r:-oooco oooicot^
^ 2:: = 2z =•"2=5 '-2s = = = = '-s^ 2'"::s = =
_|t--c.p:o noot-o ot^oo- oowi^n o»oi-t-oo
°j;^n — m"^,-c« <o "e«
oo|°j:'"S= = = 5 = '' ='«;;«-2 5°S = 2 '"S^S^^ 0 1
S
-S = -2 *°°*S =«SS!2 ^ = 523 S^'^S^"
°^2 g^'^S^S «-»»2 = «
«S
0 r. - 0 n t^ 0 n - 0 - 0 '^ ^ 5 2°2°Z ° n ^ ° •" -
2:^f? S?2|j§«S S3555S5S
S?:=J^?: ???J?5S§g :?5SS5 S-S-2t-tf^ SSS?$?.2
30
Log T (n).
n
0
1
2 1 3
4
5
6
7
8
9
1 ("»('»
97497
95001 1 92512
90030 87655
85087
82627
80173
77727
1.01
9.9975287
72855
70430 68011
65600 63196
60799
58408
56025
53648
1.02
51279
48916
46561 44212
41870 139535
37207
34886
32572
30265
1.03
27964
25671
23384 21104
18831
16564
14305
12052
09806
07567
1.04
05334
03108
00889 ' 98677
96471
94273
92080
89895
87716
"5544
1.05
9.9883379
81220
79068 76922
74783
72651
70525
68406
66294
641 8S
LOG
62089
59996
57910 ' 55830
53757
51690
49630
47577
45530
43489
1.07
41469
39428
37407 35392
33384
31382
29387
27398
25415
23449
1.08
21469
19506
17549 1 15599
13655
11717
09785
07860
05941
'•4o29
1.09
02123
00223
98329 96442
94561
92686
90818
88956
87100
~525n
1.10
9.9783407
81570
79738
77914
76095
74283
72476
70676
68882
67095
1.11
65313
63538
61768 '60005
58248
56497
54753
53014
51281
49555
1.12
47834
46120
44411 142709
41013
39323
37638
35960
342y.s 1 32(;22
1.13
30962
29308
27659 26017
24381
22751
21126
19508
1789(1 1 iC.-JS'.t
1.14
14689
13094
11505 1 09922
08345
06774
05209
03650
02<j'.'t; H ■:,['.'
1.15
9.9699007
97471
95941194417
92898
91386
89879
88378
8(;s.-;; \ -:>:\:k\
1.16
83910
82432
.^0960 79493
78033
76578
75129
73686
7224s I 7< 'Sic
1.17
69390 1 67969
(;r,554 65145
63742
62344
60952
59566
5sls.-. !.-..;slu
1.18
55440
54076
52718 51366
50019
48677
47341
46011
44687
4336S
1.19
42054
40746
39444 38147
36856
35570
34290
33016
31747
30483
1.20
29225
27973
26725
25484
24248
23017
21792
20573
19358
18150
1.21
16946
15748
14556
13369
12188
11011
09841
08675
07515
06361
1.22
05212
04068
02930
01796
00669
99546
98430
97318
y6212
"5111
1.23
9.9594015
92925
91840
90760
89685
88616
87553
86494
85441
84393
1.24
83350
82313
81280
80253
79232
78215
77204
7619S
75197
74201
1.25
73211
72226
71246
70271
69301
68337
67377
66423
65474
64530
1.26
63592
62658
61730
60806
59888
58975
58067
57165
56267
55374
1.27
54487
53604
52727
51855
50988
50126
49268
48416
47570
4t;72s
1.28
45891
45059
44232
43410
42593
41782
40975
40173
39376
3^585
1.29
37798
37016
36239
35467
34700
33938
33181
32439
31682
30940
1.30
30203
29470
28743
28021
27303
26590
25883
25180
24482
23789
1.31
23100
22417
21739 21065
20396
19732
19073
18419
17770
17125
1.32
16485
15850
15220 14595
13975
13359
1274S
12142
11540
10944
1.33
10353
09766
09184 ' 08606
08034
07466
06903
06344
05791
05242
1.34
04698
04158
03624 ' 03094
02568
02048
01532
01021
00514
00012
1.35
9.9499515
990''>3
98535 ' 98052
97573
971 Of)
966)30
9(5166
95706
95251
1.36
94800
94355
9;5'.)13 93477
93044
9261 7
92194
;»1776
91362
',H);i53
1.37
90549
90149
S'J754 89363
KS977
Sh'5'.'5 SS218
87H46
87478
87115
1.38
86756
86402
86052 ' 85707
S5366
85030 S469S
S4371
84049
83731
1.39
83417
83108
82803 82503
82208
81916
81630
81348
81070
80797
1.40
80528
80263
80003
79748
79497
79250
79008
7S770
78537
78308
1.41
78084
77864
77648
77437
77230
77027
76829
766)36
7644i?
76261
1.42
76081
75:m)5
75733
75565
751.02
75243
75( »S9
74'.>39 74793
74.;52
1.43
74515
743K2
74254
74130
71(tlO
73Si»4 73783
736)76 73574
73476
1.44
73382
73292
73207
73125
73049
72d7() 7290S
72844 72784
72728
1.45
72G77
72630
72587
72549
72514
724H4 72459
72437 72419
72406
1.46
72397
72393
723',.2'
72396
72404
72416 72432
72452 72477
72506
1.47
72539
72576
7261 7
72662
72712
72766 72S24
72«8(') ' 72952
t:'-^ »22
1.48
73097
73175
7325H
73345
73436
73531 73630
73734 ! 73841
731 >53
1.49
74068
741 HK
74312
7444(.)
74572
74708 74848
74992 75141
75293
Note. — Tliis tabic is taken IVcmi Vol. 1 1, of Lcgendi-c's work, and nut from Vol. I., as slati d ill <lio Vro'iirc : tin' Tinmlic rs p;v«n in A'ol. 1. l)«'inu; innrcnrato in the srvfiiUi drciiniil (iI.k . In \',,] it. (hr values .arc ^ivon t<> twelve places of dotinmls. 1 lie (ij,'uri' Ian juihttd in tlir s^viuth place is
L(
g r(/0.
31
n
0
1
2
3 4 5
6
7
8
0
i.r,o
0.9475440
75610
75774
75013
7(;ii6
76202
76473
76658
76847
77040
].iA
77237
77438
77642
77851
78ot;4
78281
78502
78727
78056
70189
i:>-2
70426
70667
70012
8(»l(;i
H()414
80671
8O032
81106
81465
81738
\.'>-A
82015
82205
8258(1
K-1H6S
X3161
83457
83758
84062
84370
84682
{.'A
84'.»0S
8.")318
85t;42
8507<»
86302
8(5638
8f>077
87321
87668
KHOlO
I :.:.
8S374
88733
8o<.»0(;
8li4t;3
8'.t,s;54
0(»2O8
00587
0O060
01355
01745
I ru;
021311
02537
02038
03344
03753
04166
04583
05OO4
05420
05857
1.57
06280
06725
071 1;5
07600
0805r)
085( 8
08063
00422
008H5
00351
loH
0.05OOH22
01206
01774
02255
02741
03230
03723
04220
04720
05225
1.511
05733
06245
06760
07280
07803
08330
08860
09305
09033
10475
I. GO
1102O
11560
12122
12670
13240
13804
14372
14043
15510
16008
l.Ol
it;6s(i
17267
17857
18451
10O48
10650
20254
2os(;-j
21475
2201*1
l.tV2
22710
23333
2306O
24501
25225
25863
26504
27140
27708
28451
1.G8
20107
20767
30430
31007
31767
32442
33120
33801
34486
35175
1.04
35867
36563
37263
37066
38.;73
303H3
40007
40815
41536
422(JO
1.65
42080
43721
44456
45105
45038
46684
47434
48ls7
48044
40704
1.(36
50468
51236
52007
52782
5356(J
54342
55127
55016
5670S
575(J4
1.67
58;-{o3
50106
50013
60723
61536
t;2353
63174
6300S
6482(;
65(;56
1.68
66401
67320
68170
60015
60864
70716
71571
72430
73203
74150
1.60
75028
750U1
76777
77657
78540
70427
80317
81211
82108
83008
1.70
83912
84820
85731
86645
87563
88484
89409
90337
01268
02203
1.71
03141
04083
05028
05977
06020
07884
08843
OOsd.".
00771
Hi 740
1.72
9.9602712
03688
04667
05650
06636
07625
0861s
oor.u
10613
11616
1.73
12622
13632
14645
15661
16681
17704
18730
107t;n
20703
21S30
1.74
22860
23012
24050
260O0
27062
28118
20178
30-J41
3130S
32377
1.75
33451
34527
35607
36600
37776
38866
30050
41055
42155
43258
1.76
44364
45473
46586
47702
48821
40044
51070
52200
53331
54467
1.77
55606
56740
57804
50043
60105
61350
62500
63671
64836
66004
1.78
6717 G
68351
69529
70710
71805
73082
74274
75468
7666b
77866
1.79
79070
80277
81488
82701
83018
85138
86361
87588
88818
00051
1.80
91287
02526
93768
05014
96263
07515
08770
ij0020
"1201
^2555
1.81
9.9703823
05005
06360
07646
08927
10211
11408
12788
14082
15378
1.82
16678
17081
10287
20506
21908
23224
24542
25864
27180
28517
1.83
29848
31182
32520
33860
35204
36551
37000
30254
40610
41060
184
43331
44607
46065
47437
48812
50100
51571
52055
54342
55733
1.85
57126
58522
50022
61325
62730
r>4140
65551
6t;066
68384
♦ ".0805
1.86
71230
72657
74087
75521
76057
783^37
70830
81285
H2734
84186
1.87
85640
87008
88550
00023
01400
02060
04433
05010
0738'.*
08871
1.88
9.9800356
01844
03335
04830
06327
07827
00331
108;i7
12346
13859
1.89
15374
16893
18414
19939
21466
22996
24530
26066
27606
20148
1.00
30603
32242
33703
35348
36005
38465
40028
41505
43164
44736
1.01
46311 17-'.'M l'.'l-71
51055
526.42
54232
55825
57421
50020
60622
1.02
62226 1 •■,:;>:; 1 r,.-,-l.45
67058
('.8675
70204
71017
73542
75170
76802
1.03
78436 son 7;; si 713
83356
85002
86651
88302
80057
01614
03275
1.04
04038 ''.».;.;. 15 I its-j 74
99946
01621
O3200
O4080
06()C>3
08350
To039
1.05
9.9911732 13427
15125
16826
18530
20237
21047
2365it
25375
27003
1.96
28815
30539
32266
33995
35728
37464
30202
40043
42(".88
4-4435
1.97
46185
47937
40693
51451
53213
54077
56744
58513
t;o28(".
«;2062
1.98
63840
65621
67405
60102
70082
72774
74570
763f.s
78160
70072
1.99
81779
83588
85401
87216
80034
00854
02678
045O4
06333
08it;5
tho one nearest to the true value whether in oxcosa or defect. This table, and the table of Least Factors, have tai-h been subjected to two couiiilcte and ia- dependont rovisivua before linuUy printing ofl".
ALGEBRA.
FACTORS.
1 a'-h'= (n-b) («+6).
2 <r-lr = (a-b) {(r-\-(tb-irb-).
3 a'-\-i/' = (a-\-b) {(r-ab-\-b'). And generally,
4 «"-.ft" = (a—b) («"-^ + «"--/>+ ... + //-') iilwiiys.
5 a" — b" = {(i + b) {a"-' — a"--b+...—b"-') if n be even.
6 a"-\-b" = {a-\-b) («"-^-a»- -6+...+6"-') if n be odd.
8 Gr+^O G^*+&) (^<' + c) = .^•^^+(^f + /> + r') .r-^
ft / I /\-' ■' I .1 / I 7' 'i-(bc-{-('a-\-(ih).i -\-(ihc.
9 (r^ + o)- = (r-\-'2(W-\-b-. ' V I I / .
10 {(i-b)- = ir-'lab-\-}r.
11 (^/ + 6)'' =: a'-^\\irb^?uib-^-b^ = a'^b'-^i\ab {a-\-b).
12 {a-bf = a'-[\(rb-{-{\alr-lf' = d'-li'-Wab {a-b). Generally,
{a±by=a'±7(i'b-]-2\(vb-±'^'m'b'-\-:irui'b'±2](tW-^^
Newton's Rule ior forming- the coefficients : Miiltiphj (inn coefficient by the index of the leading qnantifi/, and divide bij the number of terma to that place to obtain the coefficient of the term next following . Tims 21xr)-^3 gives 35, the following coefficient in the example given above. See also (125).
To square a polynomial : Add to the square of each term twice the 2)roduct of that term and every term that folloivs it. Thus, {a-\-b-^r-]-f/y
= rt- + 2rt(ft + r-+f/) + //-+2/>(r+^/)+r- + 2rr/+^/-.
34 ALQEBUA.
13 a'-f a-6"+6' = {a--^ah-\-h-) {ir-ab-\-¥).
14 a'+b' = {a'+ab V2-\-I)') (a'-ab V2+6^).
15 („.+iy=,.Hi+2, (.+iy=.'+-L+3(,,.+i).
16 {a-\-b-^cY = (r-\-b--]-c'+2bc-\-2('a-\-2ab.
17 (a-^b + cf = a'-\-b'-[-c'-\-',^ {b'c^bc--\-c'a-\-ca'
■^a-b+a¥)-\-6abc.
Observe that in an algebraical equation the sign of any letter may be changed throughout, and thus a new formula obtained, it being borne in mind that an even power of a negative quantity is positive. For example, by changing the sign of c in (16), we obtain
{a-\-b-cf = a^ + h' + c'-2bc-2ca + 2ab.
18 a'+b'-c'+2(ib = {a-^b)--c' = («+6+c) {a+b-c)
^y (1).
19 (r-b'-r-\-2bc = a'-(b-cy = (a + b-c) {a-b-\-c).
20 a'-\-b'-\-c'-^abc = {a-^b+c) {a'-\-b'-\-c'-bc-ca-ab).
21 bc'+b'c-\-ca'+chi+ab'-\-(rb-{-a'-\-b'-\-e'
= {a-^b+c^((r+b'-^r).
22 bc'-\-b-c-\-c(r+c'a + ab''-^a-b + :\(ibc
= {a-\-b-\-c){bc-}-ca-\-ah).
23 bc'-\-b'c + m''-\-c'a-\-ab'+(rb + 2abr={b-{-c)(c-\-((){a-j-b)
24 b(r + b'c + cd' + c'n + ab- + (rb — 2a be — ({' — h' — r '
= {b^c-a) {c^(i-b) {(i^b-c).
25 bc^-b^c + ca'-c^a-irab'-irb = (b-c) (c-a) (a-b).
26 2b'c'-\-2c'a'-Jr2a'b~-a*-b'-c'
= {(i + b-]-c) ib-\-c-a) (c + a-b) (a-\-b-c).
27 .rH2.t%+2.r/ + // = (,,■+//) C^■-• + .^// + /^).
Generally for the division of {x + //)" — {x" -\- //") by .r- + xi/ -\- y- see (545).
MULTIPLIOATIOS AXD J >I VISION.
35
MrLTirLTCATTON AXD DTVTSTOX,
15YTHE MKTIKil) OF DETACH HI) roKFFICI KNTS.
28 Ex. 1
(a*-SaV + 2ab' + b*) x (a'-2a6»-26').
1+0-3+2+1 l+U-2-2
1+0-8+2+1
-2-0+6-4-2 -2-0+6-4-2
1+0-5+0+7+2-6-2
Result a^-5aV- + 7a%* + 2o:'b'-Gab"-2b'
Ex. 2: (x^-bx' + 7x' + 2x'-6x-2)-r-(x*-Sx' + 2x + l).
1+0-3+2+1) 1+0-5+0+7+2-6-2(1+0-2-2 -1-0+3-2-1
0-2-2+6+2-6 +2+0-6+4+2
-2+0+6-4-2 +2+0-6+4+2
■ Result a;»-2.i— 2.
S(/ntlift}r Dici.sinn . Ex. 3: EmployiTig the ]a.st example, the work stands thus, 1+0-5+0+7+2-6-2 0+0+0+0 +3+0-6-6 -2+0+4+4 -1+0+2+2
-0 +3
-1
1+0-2-2
Re.sult
[See also (248).
Note that, in all operations with detached coefficients, the result mn.st he written out in successive powers of the quantity which stood in its successive powers in the original cxpre.-^sion.
36 ALGEBRA.
INDICES.
29 Multiplication: a}x c(^ = a}'^^ = a^, or ^^a^;
a'
" X a" =
1 1 m + n
a'" "= a'"" ,
or
Va'-".
Division :
a'
4 . 4
L JL
' -i-a'" =
or or
Involution :
(a*)i =
= a«'<4 = a*,
or
l/a.
Evolution :
ya»=:
a«^^=aA
or
Va\
a~
a
= 1
HIGHEST COMMON
FACTOR.
30 Rule. — To find the highest common factor of two ex- pressions : Divide the one 'which is of the highest dimension hy the other, rejecting first any factor of either exjrression which is not also a factor of the other. Operate in the same manner ujjon the remainder and the divisor , and continue the process until there is no remainder. The last divisor ivill he the highest common factor required.
31 Example.— To find the H. C. F. of
3.«^- lO.f^ + 15.1- + 8 and ^ - 2x' - 6.«» + k/' + 13,c + 6.
1- 2- 6+ 4 + 13+ 6 3 + 0-10+ 0 + 15+ 8 3
3
3_ 6-18 + 12 + 39 + 18 -3- 4+ 6 + 12+ 5
2 ) -10-12 + 24 + 44 + 18
- 5- G + 12 + 22+ 9 3
-15-18 + 36 + 66 + 27 + 15 + 20-.30-60-25
-3 + 6 + 18-
-12-
-39-
■18
2)6 +
8-
-12-
-24-
10
3 + -3-
4- 9-
- 6-
- 9-
-12- - 3
■ 5
+
5_15_15_
5 + 15 + 15 +
5 5
2)2-1- G+ 6+ 2 Result H. C. F. = .'c» + 3.r + 3.« + l.
1+ o+ 3+ 1
EVOLUTIOS. 37
32 Otherwise. — To form the H. C F. of two or more
algebraical expressions : Sfpanite the e.rprcHtiions into their simplcd fdctorx. The 11. C. F. will be the product of the factors co)nmoiL to all the exjyrGSsionSy taken in the loivest powers that orcur.
LOWEST COMMON MULTIPLE.
33 The L. G. M. of two quantities is equal to their product dicided hi/ the E. G. F.
34 Otherwise. — To form the L. C. M. of two or more algebraical expressions : Separate them into their simplest factors. The L. G. M. will he the product of all the factors
that occur, taken in the highest powers that occur.
Example.— The H. C. F. of a\h-xfchl and aXh-xfc'e is a=(6-.r)V and the L. C. M. is a'{b — xY'c'de.
EVOLUTION.
To extract the Square Root of
., 3a \/a S\/a , 41a , ,
"'-—^ 2- + 16-+'-
Arranging accoi'ding to powers of a, and reducing to one denominator, the
16a2-24;a'-|-41a-24a5 +16
expression becomes
16
35 Detaching the coefficients, the work is as follows :- 16-24 + 41 -2-4 + 10 (4-3 + 4 16
8-3 -3
-24 + 41 24- 9
8-6 + 4 32-24 + 16
' -32 + 24-16
D li. 4a — 3o* + 4 T / , 1
Result — ' — =a — ^v/a+l
38 ALGEBRA.
To extract the Cube Root of 37 Sx' - 36a;'' ^ij + 66x'y - 63xhj ^y + 33,cy - 9^- ^y + y\
The terms here contain the successive powers of .r and \/y ; therefore, detaching the coefficients, the work will be as follows: —
I. II. III.
6-3) 12 8-36 + 66-63+33-9 + 1(2-3 + 1
-6)
-18+ 9^
1
12-18+ 9 + 9j
1
12-36 + 27 6-
-9 + 1
-s
6-9 + 1 12-18+ 9 f -36 + 66-63 + 33-9 + 1
+ 36-54 + 27
12-36 + 33-9 + 1
12-36 + 83-9 + 1 -12 + 36-33 + 9-1
Result 'la?—3x^y + y.
Explanation. — The cube root of 8 is 2, the first term of the result.
Place 3x2 = 6 in the first column I., 3x2^ = 12 in column II., and 2*= 8 in III., changing its sign for subtraction.
— 36-f-12 = — 3, the second term of the result.
Put -3 in I.; (6-3) X (-3) gives -18 + 9 for II.
(12 — 18 + 9) X 3 (changing sign) gives 36 — 54 + 27 for III. Then add.
Put twice ( — 3), the term last found, in I., and the square of it in II. Add the two last rows in I., and the three last in II.
12-T-12 gives 1, the thu'd term of the result.
Put 1 in col. I., (6-9 + 1) xl gives 6-9 + 1 for col. IT.
(12 — 36 + 33- 9 + 1) X 1 gives the same for III. Change the signs, and add, and the work is finished.
The foregoing process is bub a slight variation of Horner's rule for solving an equation of any degree. See (533).
Transformations frequently required.
38 If^=^, then ;^ = ^^ [68.
39 If .'■+.'/ = «^_^,^^ S.r = \{u+b)
and cV — ?/ =^hy \a)=. \[a — 0)
40 i^^-^yr+i^v-^jY = 2 (.r+y^).
41 {,v-\-i,y-{.v-i/y = i.vij.
EQUATIONS. 39
42 0^'+.y)' = (.r-//)-'+4r//.
43 {^-f/Y = (.r+//)'-4f//.
44 Examples.
2 y g^ - b' + ^6' - x' _ 3 y/g^ - //- + y/r- - ^Z'^
v/c^-o^ y/c^-rf^
9(r-a;^) =4(r-tZ-),
,[38.
a^ = y^c'+w
To simplify a compound fraction, as
' .,+ 1
a^ — ab + h- a- + ah + li- 1 1
a*— a6 + 6* a* + ab + b''
multiply the numerator and denominator by the L. C. M. of all the smaller denominators.
Result (a^ + ab + b') + (a'-ab + b')^a- + lr
(a- + ab + b')-(a--ab \-b') ab
QUADKATIC EQUATIONS.
'2(1
46 If «cr-+2^>( -fr = (I ; that is, if the coefficient of ,r be
an even number, .i' = .
47 Method of solution without the formula.
Ex.: 2.r— 7« + 3 = U.
7 3
Divide by 2, x'— -—x+ - = 0.
2 <j
40 ALGEBRA.
n w ^1 2 7 , /7\- 49 3 25
Complete the square, x^ x-\- { = = —- .
2 \ 4 / 16 2 16
Take square root, re— — = ± — ,
4 4
.r = ^^ = 3 or - "4 2*
48 Rule for "completing the square" of an expression like
33^ — fci' : Add the square of half the coefficient of x.
49 The solution of the foregoing equation, employing formula (45), is
_fc±v/fe2_4^^ 7^y49_24 7±5 o 1
" = 2a = 4 = -^ = ^ "' 2-
THEORY OF QUADRATIC EXPRESSIONS.
If a, /3 be the roots of the equation ax--\-hx-\-c = 0, then
50 a.v'-]-kv-{-c = a {a—a){.v-S).
51 Sum of roots a+/8 = — -.
a
52 Product of roots a/3 = -.
a
Condition for the existence of equal roots —
53 b^—4<ac must vanish.
54 The solution of equations in one unknoAvn quantity may sometimes be simplified by changing the quantity sought.
Ex.(l): 2.+ «»L-l+ l^Lte =14 (1).
Sx + 1 dx' + bx—l
6.^- + 5.g-l ^ 6(3j; + 1) ^ j^ 3a; + l 6ar + bx — l
-^^^^ (^)-
EUi'ATloXS. 41
thus y + - = i^-
y
y having been dcUrininetl from this quadratic, x is afterwards found from (2).
55 Ex.2: a;H -, +X+-- =4.
(..!)%(.. l) = c.
Pat x-\ = ij, and solve the qnadi-atic in y.
X
56 l'^>i- '^ ■■ x' + x+^^2x' + x + 2 = -iy + 1
2x- + X + S v/2.«* + x + 2 = 2, lx^ + x + 2 + 3y/2.x^ + x + 2 = 4.
Put v2x*-\-x + 2 = y, and solve the quadratic
57 Ex.^: ^^"+3|v.= ¥-'
'i , 2 ? Iti '■' + 3 ■'■ = 3 ■
2.1
A quadratic in y =z x^ .
58 Tojind Md.vitnd (ind MininKt rahn'.s hi/ menus of a Q 1 1 (I (h'd t ic Ju/ uatiou.
Ex.— Given ;/ = 3.r + G.c + 7,
to find what value of x will make y a maximum or mitiimnm. Solve the quadratic equation
3.c' + 6a; + 7-y = 0.
Tl,>,s ^^-3±y3y-12 ,.45
o
In order that .r may be a real quantity, we must have '^y not less than 12 ; therefore 4 is a minimum value of //, and the value of x which makes y a minimum is — 1.
O
oe
42 ALGEBBA.
SIMULTANEOUS EQUATIONS.
General solution icith tu-o unknotvn quantities. Given
59 (ti^v-\-b,ij = Cil ^^. — c,b, — cA ^ ^ c,a,-c,a, a.a-[-b,i/=eJ' ' a^h.—a,h^ h^a.—b^a^
General solution with three unknown quantities.
60 Griven chA^-{-b,y-^c,z = (U\
a..v-\-biy-^c.z = dj
_ d,(h,c,-hc.?i + d, {b,e,-b,Cs) + (h (brC,-b,e,) ^ «i {b-2Cz—bsCo)-{-ao {hc^—b,Cs)-\-a3 {biC. — b.eyY niid symmetrical forms for y and z.
Methods of solving simultaneous equations bettceen two unknown quantities x and y.
61 I. By substitution. — Find one unhioivn in terms of the other from one of the tivo equations, and substitute this value in the remaining equation. Then solve the resulting equation.
Ex.: .r + 52/ = 23 (1)]
77/ = 28 {-I)]'
From (2), y = 4-. Substitute in (1) ; thus
.i- + 20 = 23, .r=3.
62 IL By the method of Multipliers.
Ex.: '6x + 5y = 36 (1) \
2x-:hj= 5 (2)V
Eliniinitc .»■ l)y multiplyincr oq. (1) by 2, and (2) by 3; thus 6x + l0y = 72, 6x— % = 15, I9y = 57, by subtraction, .'/= 3;
,T = 7, by substitution in Lt[. (2).
I'Uil'ATJONS. 4'.\
63 Til. />// clKm^,in^: thr (/unnfitirs .sou;iIif.
Ek. 1: x-y= 2 (1))
.r-2/- + a- + v/ = :iO (-1))'
Let .(• + // = ", x — if = i\
Substitute tliese valiius in (1) and (2),
uv + u = 30 )
n = 10 ; x-\-i/ = 10,
From which x = 6 and // = -i.
64 Ex.2: 2 -Ltl + 10 ^l^JL = 9 (1)
^* X — 1J x + y
z' + 7>r = 0i (2)
Substitute ;: for ^^^' in (1) ; x-y
,. 2.-^1^ = 0;
2-^-92+10 = 0. From which z = ~ or 2,
— !^ = 2 or — .
x-y I
7 From which x — '.iy or — »/.
Substitute in (2) ; tlius .'/ = 2 and x = 6, or ^ ~ 77? ^"'^ '*^~ ~^'^'
65 Ex.3: 3.j; + 5^= a-y (1) )
2x + 7y = 3.vy (2))-
Divide each (juantity by xy ;
^+ ^ =1 (^))
y « f
- + ^=3 (I, •
V ■*' I
Multiply (o) by 2,- and (I) by 3, and by subtraction y i.s eliminated.
44 ALGEBRA.
66 IV. % substituting y = tx, tfhen the equations are
homogeneous in the terms tvhich contain a' and y.
Ex.1: 52x^ + 7.ry = 52/^ (1)7
^x-^ = n (2)5"
From (1), h^x' + ltx' = 6fx' (3) j
and, from (2), bx-Stx = 17 (4))'
(3) gives 52 + 7t = 6t\
a quadratic equation from whicli t must be found, and its value substituted
in (4).
X is thus determined ; and then y from y = tx.
67 Ex.2: 2x'' + xy + '3y' = l6 (1) |
3y-2x= 4 (2)3'
From (1), by putting y = tx,
x'(2 + t + 5t') = 16 (3)) .
from (2), a. (3^-2)= 4 (4)3 '
squaring, a;' (9^^-12^ + 4) = 16 ;
9t'-12t + 4^ = 2 + t + Sf, a quadratic equation for t.
t beino- found from tliis, equat'on (4) will determine x ; and finally y — tx.
RATIO AND PROPORTION.
68 \i a\h v. e \ d\ then ad = be, and — = — ;
a-\-b __c-\-d ^ a — b_e—d^ a-^b _e-\-d ~~b d ' b ~ d ' a — b c—d
69 " T = 17 = 7 = '^" ' """ T - i+</+/+&c.-
General theorem.
70 If ^ = 4 = 4 := kv. = k say, then
b d J
. ^ ^pa'' + f/c"-]-re" + Szc.} I lpb"-\-qd"-\-rf"-^&G.))
where /), q, r, Sic. arc any quantities ^vhatever. Proved as in (71).
72/1770 AND ritOPOHTlON. 45
71 Rile. — To verify any equation between such proportional
quantities: Suhsfifufe for d, r, c, (Jv., their eqiiicalenta kh, Inly kj\ ^'c. respect Ivcl ij, in the given equation.
Ex. — If a '.h '.: c : d, to show that
y/g — 6 _ \/a— \/b ^c — d Vc— -/d Fni kb for a, and kd for c ; thus
^/a^> ^/kb-b s/by/k-i s/b
x/c-d -^kd-d Vds/k-l ^d Va-K/h ^ Vkh-Vb _ v/6 ( x/k-1) ^ v^ Vc-Vd s/kd-Vd ~VdWk-\) -/d' Identical results being obtained, the proposed equation must be true.
72 li a : b : c I d '. e &c., forming a continued proportion, then a : c :: cr : fr, the duplicate ratio of a I b,
a : di: a^ : b^, the triplicate ratio of a I b, and so on.
Also \^a : ^^h is the subduplicate ratio of a : 6, a' : h^ is the sesquiplicate ratio oi a : h.
73 The fraction -^ is made to approach nearer to unity in
value, by adding the same quantity to the numerator and denominator. Thus
-— !— IS nearer to 1 than — is. 6 + aj f)
74 Def. — The ratio compounded of the ratios a : b and c : d is the ratio ac : Id.
75 li a : b :: c : d , and a' : b' :: c' : d' ; then, by compound- ing ratios, aa : bl/ :: cc' : dd'.
VARIATION.
76 If rt oc c and Ij a c, then (a + b) cc c and \/ab a c.
77 If ^ Gc^ 7 i.u LI 1 "' '^
• • ^ - [ , then ac cc bd and — oc — .
and coed) c d
78 If <t cc^) ^ve may assume a = hib, where m is some constant.
46 ALGEBEA.
ARITHMETICAL PROGRESSION.
General form of a series in A. P.
79 a, a-\-(I, a + 2f/, « + ;W, a-\-{n — l)d.
a = first term,
d = common difference,
/ = last of n terms,
s = sum of n terms ; then
80 I =a-[-{n-l)d.
81 * = (« + /) I .
82 s={2a-\-{n^l)d}^.
Proof. — By writing (79) in reversed order, and adding both series together.
GEOMETRICAL PROGRESSION.
General form of a series in G. P.
83 a, ar, ar, ar^, «r" "^
a = first term,
r = common ratio,
I = last of n terms,
s = sum of n terms ; then
84 l = ar"-\
85 s = a or a
r — 1 1 — r
If r be less than 1, and n be infinite,
86 s= -i^, since r" = 0.
I— r
Proof. — (85) is obtained by multi]>lyiiig (83) by r, and siil)(racting one series from the other.
PEIiMUTATlONS AND COMIHSATIOXS. 47
HARMONICAL PROGRESSION.
, -, — , -y, &G. are in Aritb. Prog.,
87 cf, b, Cy d, &c. are in Harm. Prog, when the reciprocals
i_ 1_ 1 1
a b r a
88 Or when a : b :: a -b : b — c is tlie rehition subsisting between any three consecutive terras.
89 «^'' term of the series = r^ ; -. [87, 80.
{n-l)a-{n-2)
90 Approximate sum of n terms of the Harm. Prog. , &c., wlien d is small compared with a,
ft + rf' a + 2d' a-^Sd
_{a-{-(l)"-a"
1 2
Proof. — By takintr instead the G.P. , + 7— —77-0 + ; — r^TK + ••• •
91 Arithmetic mean between a and h = — ^^.
92 Geometric do. = \/ab.
93 Harmonic do. = — —r.
<t-\-h
The three means are in continued proportion.
PERMUTATIONS AND COMBINATIONS.
94 Tlie nui'iber of permutations of v things taken all at a thne = n{u-\){n^'>) ...\\.'lA = n\ or ;i"".
Proof hy IxnucriON. — Assume the foniiiila to ho true for n things. Now take ?i + l things. After eaeh of those the remaining n things may bo arranged in n ! ways, making in all nX n\ [that is (»t + l) !J permutations of w + 1 things; therefore, &c. See also (23."^) for the mode of proof by Induction.
4S ALGEBRA.
95 The number of permutations of n things taken r at a time is denoted by P {n, r).
P {n, r) = n (w-1) {n-2) ... (w-r+l) = n(^>.
Proof. — By (94) ; for («—r) things are left out of each pei-mutation ; therefore P (n, r) = nl -i- {n—r)l .
Observe that r = the number of factors.
96 The number of combinations of 7i things taken r at a time is denoted by G (n, r).
r r,, r) - n{n-l)in-2) ...(n-r-j-l) _ n^^^ ^ ' ^ ~ 1.2.3. ..r = 7T
= C {n, n — r).
r\ {n — r)
For every combination of r things admits of r ! permuta- tions; therefore G {n, r) = P{ii, r) -^ r!
97 G {n, r) is greatest when r = ^u or i{n + \), according as n is even or odd.
98 The number of homogeneous products of r dimensions of n things is denoted by H(y, r).
^ ^ ' * ^ 1.2...r = V\ •
When r is > n, this reduces to
(>-+l)(>'+2)...(/^ + >— 1)
99
(V-I)!
PrOOK. — Jl{n, r) is equal to the number of terms in the pi-otluct of the expansions by the Bin. Th. of the n expressions (1— a.i')~\ (1 — Z/.j)"\ (1 — cr)"', &c.
Pnt a=-h = c =■ &c. = 1. The number will be the coenTicIeut of x'^ in (1-a:)-". (128, 129.)
ri'lUil UTA TIONS AND COM I'.LXA TIOXS. VJ
100 The niimbor of perimitations of n tliin<j:s tjiken all to- gether, when a of them are alike, h of them alike, c alike, &c.
a ! ^1 c! ... &c.
For, if the a things were all different, they would form a! permutations where there is now but one. So of b, c, &c.
101 The number of combinations of n things r at a time, in which any^ of them will always be found, is
= C(n-p, r-p).
For, if the p things be set on one side, we have to add to them r—j) things taken from the remaining n—2) things in every possible way.
102 Theorem: C(n-\, /— 1) + C(;t-1, r) = C{u, >•)•
Peoof by Induction ; or as follows : Put one out of n letters aside; there are G{u — l,r) combinations of the re- maining 71 — 1 letters r at a time. To complete the total C(n, /•), we must place with the excluded letter all the com- binations of the remaining n—l letters /*— 1 at a time.
103 If there be one set of P things, another of Q things, another of ii things, and so on ; the number of combinations formed by taking one out of each set is = FQIl ... &;c., the product of the numbers in the several sets.
For one of the P things will form Q combinations with the Q, things. A second of the P things will form Q more combinations ; and so on. In all, PQ combinations of two things. Similarly there will be PQE combinations of three things; and so on. This principle is very important.
104 On the same principle, if p, 7, r, &c. things bo taken out of each set respectively, the number of combinations will be the ])roduct of the iiuniberR of the separate combinations ;
that is, = C{rp) . ('{Qr/) . C{Rr) ... Sec.
60 ALGEBRA.
105 The number of combinations of n things taken m at a time, when p of the n things are alike, q of them alike, r of them alike, &c., will be the sum of all the combinations of each possible form of m dimensions, and this is equal to the coefficient of x'" in the expansion of
(l^-.^' + ,T2+•••+.^'')(l+c^' + cT'-h...+a?'')(l^-■T^-.^''+■•.+.^'•)••••
106 The total number of possible combinations under the same circumstances, when the n things are taken in all ways, 1, 2, 3 ... 7i at a time,
= (p+l){g+l){r+l)...-l.
107 The number of permutations when they are taken m at a time in all possible ways will be equal to the product of m ! and the coefficient of x'" in the expansion of
&c.
SURDS.
108 To reduce >/2808. Decompose the number into its prime factors by (360) ; thus,
V28iJ8 = y2\ 3M3 = 6 Vl3,
^a'" 6'" c^ = a'» b'^" c = fV' h' c' h c- = a' h" c' Vh6'
109 To briug 5^3 to an entire surd.
5y;3 = vo'. 3 = yi875, a;» y^ z' = a;- y^ z^" = V^z^.
110 To rationnllse fractions hnvinf:; .surds in their drnnminators.
j_^ y?. 1 ^ ^49 ^ y4o
SUIiVS. 61
111 .J3^o=^^r-'<"^^'^°''
since (9 - ^80) (9 + ^80) = 81 - 80, by ( 1 ) .
^^^ l+2y8-v/2 (l + 2y3)'--J 11 + 4/3
^(1 + 2/3+^2) (11-4/3) 73
1^3 ?/3-v/2 3*--i*"
Put 3* = a-, 2' = ij, and take G the L.C.M. of the deriominafors 2 and 3, tlien
„ . 1 3' + 3«2' + 352* + 3'2» + 3«2« + 2*
thereiore — -= z
3i-2* 3^-2'
= 3 y9 + 3 y72 + 6 + 2 yG48 + 4 y3 + 4 v/2.
114. . Here the result will be the same as in the last exainplo
^^^ y3+/2
if the signs of the even terms be changed. [See 5.
115 A surd cannot bo partly rational ; that is, y/a cannot be equal to >^''h + c. rrovcd by squaring.
116 'J he product of two unlike squares is irrational;
^7 X y/'^ = ^/2], an iri-atioual (piantity.
117 The sum or difference of two unlike surds cannot produce a single surd; that is, \/a-\-x/h cannot be equal to \/c. 15y S(jnaring.
118 If " -\- \/m = J'i-^'^n; then a = h and w = n.
Theorems (115) to (118) are i)roved indirectly.
119 If ^/a+ W>= ^.c-\- s^ij,
then
By squaring and by (HH).
52 ALGEBRA.
120 To express in two terms \/7 + 2V6.
Let v/7 + 2v/6= ^x+^tj; then x + y = 7 by squaring and by (118),
and X-2J = ^7'-{2^6y = a/49-24 = 5, by (119) ;
.•. ic = 6 and y = 1.
Result ye+i.
General formula for the same —
121 \/a±^b=\/i{a-{-x/a'-b)±\/i{a-x/a'-b).
Observe that no simplification is effected unless a' — b is a perfect square.
122 To simplify v/a+ Vb.
Assume \/a-\- Vb = x-{- \^y.
Let c = y/a^—b.
Then x must be found by trial from the cubic equation
4cr^— SccV = a, and 7/ = cv'^—c.
No simplification is effected unless a^—h is a perfect cube.
Ex.1: V7 + 5^2 = x+y7j.
c= ^49- 50= -1. 4.(;* + 3.t; = 7 ; .-. x=\, y Result 1 + v/2.
Ex. 2: y9v/a — 11n/2 = v/-T+ v/y, two different surds.
Cubing, 9 v/3 - 11 v/2 = a; v/.x' + 3.« y?/ + Sy ^x ^y^y,
.-. 9v/3 = (.T + %)ya;-) . .^^o^
liy2 = (:3x- + 2/)y2/) ' ^ ^
.-. .r = ."{ and ?/ = 2.
lUSOMlAL THEOREM. 53
123 To simplify v/(12 + 4y3 + '4yr) + 2yi5). Assume v/(12 + 4v/3 + 4v/5 + 2yi5) = ^x^ ^y-\- ^z. Square, and equate corresponding surds.
Result v/3+yi+-/5.
124 To express \/A + B in the form of two surds, wliere A and B are one or both quadratic surds and n is odd. Take (/ such that q (A^—B-) may be a perfect n^^ power, say />", by (361). Take s and t the nearest integers to V'y (^4 + /?)'' and Vq{A--B)\ then
2Vq
Example: To reduce y89y8 + lU9y2. Here A =89^3, B = 109^/2,
A''-B' = l; .-. p=l and q = I.
vq (A + By = 9+f \ f being a proper fraction ;
^qiA-By=l-f\' .-.8=9,1=1.
Result i(^9 + l + 2±v/9 + l-2) = y3+v/2.
BINOMIAL THEOREM.
125 (n+by =
126 General or (/•4-l)^" term,
r!
127 or , ''[, ,a"-n/
if n be a positive integer. If b be negative, the signs of the even terms will be changed.
54 ALGEBRA.
If n be negative tlie expansion reduces to
128 {a+br^ =
129 General term,
v\
Elder's proof.— Let the expansion of (1 +.'«)", as in (125), be called /(7i). Then it may be proved by Induction that the
equation f{'>n)Xf{n) =f{m + n) (1)
is true when m and n are integers, and therefore universally true ; because the form of an algebraical product is not altered by changing the letters involved into fractional or negative quantities. Hence
/(m + ?i+j9 + &c.) =fim)Xf{n)Xf(p), &c.
Put 7n = 71= 2^ = &c. to Jc terms, each equal —, and the
theorem is proved for a fractional index.
Again, put —n for m in (1) ; thus, whatever n may be, f{-^i)Xf{n)=f{0) = l, which proves the theorem for a negative index.
130 For the greatest term in the expansion of (a-^-by, take
... -, ^ c {n^-l)b {n-l)b
r = the mtegral part of ^^ — -—f- or ^^ f— ,
° '■ a-{-b a — o
according as n is positive or negative.
But if b be greater than o, and n negative or fractional, the terms increase without limit.
Required the 40th term (.f ( 1 —
Examples.
Hero r = 39 ; a = 1 ; b = - '- ; n= 12. By (127), ilio term will he
_42! / _ 2..\-_ _ ^-^ ■ iU-i^. (2.,-y« (^(3^ y!3i)!\ :W . 1-2.3 \sl ^ '
lilNOMIM. TllbUiUKM.
Roqniied the Slst term of (a — .r)"*. Here r = 30 ; h=-x\ 7i = — 4. By (129), the term is
^4.5.6...80.:U.:V2.:i1 ,>. ^_ 31 .82. 33 «•'" ^~^^ 1.2. 3... 30 " ( -^^ - 1.2.3 -a" ''^ ^^
131 IleqairoJ the greatest term ia the expansion of — — when a
— = (l-|-.r)"'. Here n = ^^ a = 1, h = x in the formula
(n-l)fc_5x|>_ 231 . a -6 1-H-
thert'foro r = 23, by (130), and the greatest term
_ , ,.o3 5.6.7...27/14\'»^ 24 .25 .26 .27 lU^ ~^ ^ 1.2.3... 23\17/ 1.2.3.4 \17/'
132 Find tlie fir.st negative term in the expansion of (2a + 3&)'*'.
We must take r the first integer which makes n — r-\-\ negative; there- fore r>Jt + l = V +1 = 6| 5 therefore r = 7. The term will be
17 14 11 8 8 2. C 1"\ ,
(2(7)-»(36/ by (126;
17.14.11. 85^2^ 1 J/_ 7! ■" (2«)5'
133 Required the ooeffioient of .?;" in the expansion of i- — -^ j .
g±M=(2.3..V(2-a.r'=C^)'(i-^)-'
the three terms last written being tliose which produce .r'*' after niultiplyii by the factor (l-|-3a;-|- Jx*) ; for we have
33(|)%3„.x3^(^;:-)%lx35(|)^'
giving for the coeflicit nt of J'" iu the result
The coefficient of j" will in like jnanjicr be Ibi i !] /' ".
56 ALGEBRA.
134 To write the coeflBcient of x^'"*^ in the expansion of (x- ^)
The general term is
(2jt + l-r)!r! x"- (2«+l-r)!r!
Equate 4n—4r + 2 to 3m+l, thus
. _4n — Sm + l
Substitute this value of r in the general term; the required coefficient becomes (2n+l)\
The value of r shows that there is no term in x^'"*^ unless — — "^."^ is an
[i(4n + Sm + 3)]\ [i{4n-Sm + iy
ae of r sho' integer. **
135 An approximate value of (1+a?)", when x is small, is l-\-7iXy by (125), neglecting x^ and higher powers of x.
136 Ex. — An approximation to \/y99 by Bin. Th. (125) is obtained from the first two or three terms of the expansion of
(1000-1)* = 10-1 . 1000-5 = 10- 3^^ = mi- nearly.
MULTINOMIAL THEOREM.
The general term in the expansion of (a-{-hx-\-cx--\-&G.y is
j3, «(»-!) (»-2)...(p+l) ^^„ j,^,^^, ,^„«r..,.. ql rl si ...
where j;-f-r/ + r + 5+&c. = n,
and the number of terms p, q, r, &c. corresponds to the number of terms in the given nndtinomial.
]> is integral, fractional, or negative, according as n is one or tlie other.
If n be an integer, (137) may bo written
138 , ]'■ , , a' h" (••■ </, . . . .r''+2''+3.,
pi (/ . r\ .V I
I neduccd fi-om Mio Tliii. Thoor.
MUI/VISOMIA L TIIKOUEM.
57
Kx. 1. — To write f lie enofficient of a'fec" in tlieoxpanfiinii of {n +{> -f r + i/)" Hero put )/= 1(». ,. = 1, i> = 'A, q=\, *- = r., s = () in (IMS).
Kesi.lt
10! :^! 5
= 7.8.0.10.
Ex. 2. — To obtiiin till' CDcflSc-icnt of .r* in tlio cxiciusion ol (l-2,« + 3.i'»-4a;')'. Here, compariiii,' with (l;{7), we liave ii = \, h = —2, c = :i, </ = — 4,
q + '2r + ;is- = 8,
1
0
1
2
0
2
0
2
0
1
2
1
0
0
4
0
Tlie nuniljer.s 1, 0, 1, 2 are particular values of p, q, r, s respectively, which satisfy the two equations given above.
0, 2, 0, 2 are another set of values which also satisfy those equations ; and the four rows of numbers constitute all the solutions. In forming these rows always try the highest possible numbers on the right first.
Now substitute each set of values of p, q, r, s in formula (138) succes- sively, as under :
||l'(-2)",S'(-4)- = 57(3 2^1«(-2r3«(-4)^ = 384
4!
Z|i«(_o).3-.(_4^
8(54
ir.(_oy.;).(_4^o^ SI Result 1!>(I.-,
Ex. 3.— Required the coefficient of v* in (H-2.-C— 4'c'-2a.-»)~*. Hero (/ = 1, /i = 2, r = — 4, i1 = — 2, ?; = — ', ; inid the two ciinations arc p + q + '■ + s = — \, 'J + 2/- + :rs- = 4,
— S.
1
0
1
2'
0
2
0
1 2
2
1
0
! -^
4
0|0|
58 ALGEBRA.
Employing formula (137), the remainder of the work stands as follows :
2M-^})(-l)'"'^"^-*''(-'>"= "
iT(-i)(-|-)(-|)l-^^=(-4)'(-)»= 15
Result 22f
139 The number of terms in the expansion of the multi- nomial {a-\-h-{-r--\- to n terms)'' is the same as the number of homogeneous products of n things of /' dimensions. See (97) and (98).
The greatest coefficient in the expansion of {a-{-J>-\-<^-{- to m terms)", n being an integer, is
Proof. — By making the denoniiuator in (138) as small as possible. The notation is explained in (96).
LOGAEITHMS.
142 log,, ^^' = •^' signifies that a' = N, or
Def. — TJie logarithm of a nmiiber is the power to irliirJt tin base must he raised to produce that number.
143 log.« = l, log 1 = 0.
144 log MN = log 3/4-log N.
log—- = log .1/ — log N.
log (3/)" = «log3/.
log:;/ j/ = J- log J/. [li'^
EXPONENT! A L Til IKlU EM. 59
145 '"f^"' ^ n;;77/
Tliat is — 'riir lixjiirilhiii of a innnhrr In miij Imsr is niiml fo (he loijnvitJnn- of flu' nuMibcr dlrulitl hi/ fhc /ni/(irifhiii of the hdur, the two last named logaritlnns being taken to any tlie same l^ase at ])leasui'e.
Pi;(iOl'. — Let, log,. u = .)' and \i)<y,.l,= i/\ ihvn a = c-'', b = r". Eliiiiiiuitc <■.
c = a" = h"; .'. a = b", that is, \o<r,^a = -' ■
y y. e. d.
146 l<>K/.<^ = ,— ^- •'>'< '•=--" ii' (1 ^5)-
147 ,og,„.v = ;2gby(l4n).
is called the modulus of the common system of logarithms ; that is, the factor which will convert loi>arithms of nnml^ers calculated to the base e into the corros]~)ondino; loo-nrit1ims to the base 10. See (154).
EXPONENTIAL THEOREM.
149 *' = 1 + r.r + '-^ + '^ + etc.,
where c = {a-])-\ ('(-\Y + \ (n-\y-Scc.
PiJOOF. n^ = fl + (a — l)j^ Ivxpand tliis Ijy JJinomial Thcoiviii, ami collect the coedicients of .c ; thus c is obtained. A.ssnme r.,, c„ Arc, as the coefficients of the succeeding poweis of ;r, and with this assumption write ont the expansions of a"", a", and a'"*". Form the product of the first two series, which product must be equivalent to the third. Therefore equate the coefficient of .c in this product with that in the expansion of a'*". In the identity so olitained, equate the coefficients of the successive powers of y to determine Cj, f,, &c.
60 ALGEBRA.
Let e be that value of a which makes c = 1 , then
150 ,.' = i + .,.+ |l4.i_4.&c.
151 ''^^ + ^ + ^ + ^ + ^^-
= '2-718281828... [See (2;»5).
Proof. — B}^ making x = 1 in (1-^0).
152 By making a; = 1 in (149) and ,t = c in (150), we obtain
a = e"" ; that is, c = log^ a. Therefore by (149) 154 \og,n = {a-l)-i(a-iy-hi(a-iy-&G.
155 l«g(l+.r)= .,.-£- + :;^_±-+&c.
156 \og{l-.v) = -.v~:^-^-'^-&c. [154
— »» 4
157 .-. l..sl±£=2J,, + :^ + 4^+&c.^-.
Put for ,r in (157); tluis,
158 l..g», = ■> )^ +|(!^y'+i(^f+&c.( .
(m-\-l ,\\m-\-\' ;)\m + l' )
Put ^ ^ for ,r in (157); thus, 2/^ + 1 ^
159 lot,^ (// + !) -loi^M*
CUNTIS ri:i> FRA < "I'lOXS.
<;i
CONTINUED FRACTIONS AND CONVERGENT S.
160 'l^> liii'l foiiV('rt<( Ml ts to 3-M.i:)9 = ill tlu" rule for II. ('. F.
;n4ir,<>
roceea hh
100000 99113
31415!) 300000
887 854
83 29
4 4
14159
887
5289 4435
854 66
15
25
194 165
28 !
The contiinKMl fraction is
3 + 1 7 + 1
15 + &C.
or, as it is more conve- niently written,
1 1
3 +
7+ 15 +
cV-C.
The convergents are formed as follows : — 3 7 15 1 25 1 7 4
3 22 333 355 9208 95(53 76149 814159
1 ' 7' 106' 113' 2931' 3044' 24239' lOOOOO'
161 KuLE. — AVrite the ([uotients in a row, and the first two convergents at sight (in the example 3 and 3+y). Multiply the numerator of any convergent by the next quotient, and add the previous numerator. The result is tlie numerator of the next convergent. Proceed in the same wa} to determine the denominator. The last convergent should be the original fraction in its lowest terras.
162
Furtnuld fin' fhrniinii- tlir f(nirrr::rnf.s.
If -?^, ^^^!i^, ''" are any consecutivt' converu-ents, and qn-2 7«-i V» . .
^»-2» <'«-!' ",> f^"' coi'r(^S|)on(liiiii: (|iiotienls: then
62 ALGEBBA.
The /i"' convergent is therefore
7» (lnqn-\-\rqn-l
The true value of the continued fraction will be expressed
by
163 p^anPn-r-^Pn^.^
(f„q»-i-\-qn->
in which ct'^ is the complete quotient or value of the continued fraction commencing with a^.
164 Pnqn-i—Pn-iqn = ± 1 alternately, by (162).
The convergents are alternately greater and less than the original fraction, and are always in their lowest terms.
165 The difference between F^ and the true value of the
continued fraction is
< and >
quqn+i qniqn+qn+i)
and this difference therefore diminishes as n increases.
Pkoof.— By taking tlie difference, ^« - Y"^'"^^"- (163)
Also F is nearer the true value than any otlier fraction witli a less denominator.
166 l'\iFn+\ is greater or less than F'^ according as F„ is greater or less than i^,,+i.
Grucral Theory of Vontlnucd Fnfrfion.s.
167 b'irst class of continued [ Second class of contimu d fraction.
fraction. 1/ _ ^1 ^'2 ^h
<'i — ''■•— ''.{ — &c. ill, hi, itc. are taken as positive (piaiitities.
CONTINI'JJJ) Fh'AcriONS. {]^
'. '■'' , Sec. ;ii"(' tcfiiKMl (•(nii/ioiKii Is of the coiit iiiiicd IViK'-
(ion. It' tlir compoiicnt s he iiitiiiilc in iiiiiiihrr, the coiit iiiiicil IVactiou is said lo bo iuliniU-.
Let the successive conver<ients bo tloiiotod l)v
^'i . 1'-i _f'\ ^1 . Pa _ ^^i ^^2 ki .
; and so oi
168 1 lit' law of formation of the convergents is I'^or /', I For r,
Pn = f. P„ 1 + '^, />« .' { Pn = ftn pn 1 " '>„ />« .'
•/« = <f„ Un - I + f'n 7« - -1 ( V" — ^^< qn-l—f^H Hn-l
[Proved by Induction.
The relation between the successive differences of the converg-ents is, by (108),
169 l^_lK^h„..U„U!K_1K^\
7»+i Un qn^\ \q. v.-i
Take the — sign for 7'', and th(> + ^ov I '.
171 The odd convergents for i^, ^, ^'•\ &c., continually
Vi V.t
decrease, and the oven convero-onts, '-, ^-, S:c., continually
'/•' qi increase. i^^'O
Every odd convn'oHMit is greater, and ovorv even con- vergent is less, than all following convergents. (lGi>)
172 l^i.i'. — If the diffei-onco between consecutive conver- gents diminishes without limit, the infinite contiiuied fraction is said to l)e dcjiiilfr. if the same difference tends to a fixed value greater than zero, the infinite continued fraction is hi- drfinifr ; the odd convergents tending to ..nc value, and the even converu'ents to another.
64 ALGEBRA.
173 F is definite if the ratio of every (|uotioiit to tlie next component is greater than a fixed quantity.
Proof. — Apply (1(39) successively.
174 F is incommensurable when tlie romponents arc* all proper fractions and infinite in number.
Proof. — Indirectly, and by (168).
175 if a be never less than /> + l, the convero'ents of V are all positive proper fractions, increasing in magnitude, Pn and (/„ also increasing with 7^. By (167; and (168).
176 If, ill this case, V be infinite, it is also definite, being = 1, if a always =h-\-\ while h is less than 1, (175); and being less than 1, if a is ever greater than h-{-\. By (ISO).
177 V is incommensurable when it is less than 1 , and the components are all proper fractions and infinite in number.
180 If in the continued fraction V (167), we have a„ = h„ -\- 1 always; then, by (168),
'p,^=^ hy-\-hih.2-\-h]^h.2,b-i-{- ... to n terms, and q^z= p^^-{-\.
181 Ifj iu the continued fraction F, a,^ and h,i are constant and equal, say, to a and h respectively ; then ^,;„ and (/„ are respectively e(pml to the coefiicients of x!'"'^ in the expansions
/. h T a-\-hv
of - , 2 and .,.
1 — ax — 0 J? 1 — ^^c — hxr'
Proof. — p,i and q^ are the w*'' tei-uis of two recurring seiies. See (IGS) and (251).
182 /''> v(»n'('i't (I Scries info a dnitiuucil Fractiini.
The series i + ^ + :!l + ... + —
is ('(jual to a continued fraction 1^ 0^'^'^), with ;/ -|- 1 com- poneiits ; th(^ first, second, and //-t-l"' coinj)on('iits being
1 ir,v u'i ,.r
u iii-{-i(.r «,, + "" -i<^'
[Proved by Induction.
COSTISUFA) FUACTIOXS. 05
183 Tlio scries
1 r r'- .r"
-+ — + -^+...+
r rr, rr^r. f'''\i'- ... ''»
is o(|ual to ;i coiitiimod fraction T (1C»7), with // + 1 coiiipo- ncnts, the first, second, and // + !"' components beini^-
J_ l\r !JI_1±, [Proved l)v In.lucfinn.
184 'Hie sio-n of w may be changed in eitlicr of the state- ments in (182) or (18;3). '
185 Also, if any of these series are convergent and iidinite, the continued fractions become infinite.
186 To find fhr rahir of a rnntinurd fraction with rrrnrrinii: qnitfirnf.s.
Let tlie continued fraction be
where ?/ = -- -
so tliat there are m recurring quotients. Form the |/"' con- vergent for X, and tlie m^^' for //. 'Vhvu, by substituting the complete quotients a„-\-i/ for a„, and a,,,.,,, -f-// for */„,,„ in (lC),s), two equations are obtaincMl of the forms
from which, by (eliminating //, a (iii;i(lr;it ic (miumI ion foi' (h-- terminimr ;'' is obtained.
1R7 Tf !^^ ^h—
i)i' a colli iinicd I nii't ion, ;ni<
El I^
7i' (/"
66 ALGEBRA.
tlie correspoiKliiii^ first vi convorgents ; then '" "\ developed by (1<J8), ]iroduces tlie continued fraction
1 hn />„_! h, b.
(In + (ln-l-\- (ln--2-\- '" + fh + «1
tlie quotients being the same but in reversed order. INDETERMINATE EQUATIONS.
188 Given aa-\-hii = c
free from fractions, and a, /3 integral values of x and // wiiich satisfy the equation, the complete integral solution is given by
.r= a — ht
y = fi+at
where t is any integer.
Example. — Given Src + -h/ = 1 1 '2.
Then X = 20, y = 4< are valnrs ;
x = 20-:U\
y= .l. + r,M •
The v.alues of x and y may be exhiliited as niuler:
t = -2 -I 0 I 2 .S -i 5 6 7 x= 26 28 20 17 14 11 8 5 2-1 7/=-G -1 -4 !) 11- 1!) 24 29 84 89 For solutions in positive integers / must lie between \" = 6j; and — t ;
tliat is, t must be 0, 1, 2, 8, 4, 5, or 6, giving 7 positive integral solutions.
189 If the equation be
(u—hjl = c
tlie solutions are given by
,v— a-\-ht
INDETEnMIXA TE FA^J'A TIONS.
EXAMPLK : 4c -8// = li>.
Here X = 10, // = 7 satisfy ilie equaticjii ;
' ~ fiinii.Nli :ill tilt' solutions.
,/ = 7+ U *
The simultiinoous vahu'S of /, .r, und // will he as follows : —
t=-l) -i -:i --2 -I 0 1 -J :;
., = -5 -2 1 1- 7 10 l:{ 10 I'.'
,j = -rA -9 -5-1 3 7 11 1.-) 19
The number of positive integral solutions is infinite, ami the least positive integral values of x and ij are given by the limiting value of /, viz.,
t>-\- and t>-\-'
that is, t mast be —1, 0, 1, 2, 3, or greater.
190 It" two values, a and /3, cannot readily be found by inspection, as, for example, in tlie equation
17.t' + 13// = 14900,
dlridr In/ f/ic huisf roi'ffirient, and equate the re iiiaiiiliKj frac- tions to t, an intc/jer; thus
*+■■'+ if ="«+!;, '"■
4a— 2 = l-.it. Repeat the process ; thus
4 4
Pat
^ + •2 = 4
H.
u
=
1,
t
=
2,
X
=
18/ + 2 _ 4
7 =
f ;
and _y + .,; + /= 114(5, by (1),
7/ = 114(5-7-2 = ll;w The general solution will be
.,' = 7-l.S^ II = 1137 + 17/, Or, changing the sign of / for convenicncp,
.(• = 7 + 13/, y = 1137-17/.
68 ALGfEBBA.
Here the number of solutions in positive integers is equal to the number of
^ ■ X. , 7 , 1137
integers lymg between — and -— - ;
or ~ Tq ^^^ ^^Tf ; t^^t is, 67.
191 Otherwise. — Two values of x and y may be found in
the following manner : —
17 33
Find the nearest converging fraction to y^. [By (160).
This is — . By (1G4) we have
17x3-13x4 = -1. Multiply by 14900, and change the signs;
17 (-44700) + 13 (59600) = 14900 a = -44700
which shews that we may take , , ^^^
^ ( /5 = 59600
and the general solution may be written
x = -44700 + 13/,
y= 59600-17^. This method has the disadvantage of producing high values of a and y8.
192 The values of x and //, in positive integers, which satisfy the equation ax + bi/ = c, form two Arithmetic Pro- gressions, of which h and a are respectively the common differences. See examples (188) and (189).
193 Abbreviation of the method in (169).
Example : ll.i;— 18;/ = 63.
Put X = 92, and divide by 9 ; then proceed as before.
194 To ohld'ni iiifriinil s(thitioN.s' nf (H-\-f)t/-\-rz = (I. Write the equation thus
ax -{-III/ = (J — cz. Put successive integers for ;:, and solve for .r, // in encli cnse
ItEDUGTION OF A QtlADJiATfC srUD. GO
TO Iv'KDlTCF] A QUA1>HATI(^ SLIHI) TO A CONTINUED FRACTION.
195 EXAMI'I-K :
^29= 5+v/29-r, = 5-h '^
,29 + 5'
y29 + 5_ ^, v/29-:5^ ^ 5
4 '^^ . 4 "^^ ,'29 + 3'
5 ~ ^ 5 ^^29 + 2' v/29 + 2_ . , v/29-3_ , 4
" 5 ~ "^ 5 ^ ■^V^29 + :>'
/29 + 3_ g , v/29-5 _ 2, •
—4 - ^ + "—4—- ^ + v^29+.y
^/29 + 5 = 1U+ V 29 -5 = 10 +
v/29 + 5'
Tlio (iiiotients 5, 2, 1, 1,2, 10 arc the gTcatest integers contained in the quantities in the first cohimu. The quotients now recur, ami the surd \/29 is equivalent to tlie continued fraction
1_ 1_ 1_ 1 J 1_ 1_ 1 ]
5+ 2+1+1+ 2+ 10+ 2+ 1+ 1+ 2 + c^c.
The convcrgents to v/29, formed as in (IGO), will be
5 11 10 27 70 727 1524 2251 3775 9801 T' 2' 3' 5' 13' 135' 283' 418' 701' 1820'
196 Note that the last quotient 10 is the greatest antl twice the first, that the ><i'ron(l is the first of the recurring ones, and that the recurring quotients, excluding the last, consist of pairs of equal terms, <[iu)tients e(|ui-distant from the first and last being ecjual. These properties are universal. (See 204 -210).
To for)) I liiii'h ro)H'C)'i»'r)ifs rnpUllji.
197 Suppose m. the number of recurring (piotients, or any
70 ALGEBRA.
multiple of that number, and let the m^^' convergent to \/Q be represented by F„,; then the 2??^"' convergent is given by the
formula F„„ = i .Jf;,+ -^^- by (203) and (210).
198 i'or example, in approximating to \/29 above, there ai-e five recurring quotients. Take m = 2x5 = 10 ; therefore, by
i^,y = ^-— — , the 10^'^ convergent. 1820
Therefore F,, = {||^^ + 29x^
1820 ) ^ 192119201 )801 ) 35675640^
the 20^'' convergent to \/29 ; and the labour of calculating the interveninf convergents is saved.
GENERAL THEORY.
199 'J'he process of (174) may bo exhibited as follows :-
= a.
^±^ = ..+
''Q + r„,-
200 Tl
1 1 1
v/ g = f / , + a, -h a, + a^-irScv.
Tli(! (juotients '/,, rr,, a.^, iScc. are the integral parts oF the fi'ac- tions on tlie left.
Ri'iDUCTios OF A (,n' A i>i:.\'r h' srii'it. 71
201 'I''"' •'•I'intions coinicct iii^- tlic rnniiiiiiii^- (|ii;iiit it ii'S iii-c
r, r= (I. )'.,—(■ >:5 —
r., = ^/,. ,r
,-.= ^=:^
Tlie ?/*'' r()iivorn:(Mit to \/(? will bo
202 ^ = ^iiLZ!zL-_Lii!±l2_ [By Tndnctioii.
The tnu> value of v'^^ i^ ^^''^^^t tliis becomes wIh-ii we
substitute for (/„ the complete quoticDt ^ ' ", of wliicli <i„ is only the integral part. This gives
By tlie relations (1 '.)<)) to (203) tlie following theorcius ai'*^ demonstrated : —
204 All the (juantities (/, r, and r ai'e positive integers.
205 'I'Ik' greatest c is r.,, and c, = a^.
206 No >i or /• can be greater than 2'/,.
207 n /■„ = 1, then r„ = a,.
208 I'^or all values of n groat(>r tlian 1, rt—r„ is < >•„.
209 'Hi'- number of (juotients cannot be greater than 2a'l The last (luotient is 2(i,, and after that the terms repeat.
The first complete quoti(>nt that is repeated is ^ \ '\ and (7o, 7-0, r., commence each cycle of re}»eated terms.
72 ALGEBRA.
210 I^et </,„, r,„, r,„ be the last terms of the first cycle ; then <*»n-ij '''m-ii Cm-\ ^^^'c respcctivcly equal to rto, r.,, c-2', «,«-2j ''m-2j c„,_2 are equal to a.^, r.^, r.^, and so on. rp^ (187).
EQUATIONS.
Special Cases in the Snlntion of Simidtaneous Equations.
211 First, witli two unknown quantities.
a^e-\-hyii = i\\ ^ _ cA—Cobi _ ^1^2 — ^2^1
If the denominators vanish, w^e have
^ = '\ and X = cc, ?/ = 00 ; «2 b.,
unless at the same time the numerators vanish, for then a._h,_c,, 0. 0.
a, ~ h ~ c./ 0 ' '^ ~ 0 '
and the equations are not Indepmdent, one being produced by multiplying the other by some constant.
212 Next, with three unknown quantities. See (60) for the ecpiations.
If (l^, (L, (I,; all vanisli, divide each equation by .v, and we
have three equations for finding the two ratios -'- and • , two
only of which equations are necessary, any one being dedu- cible from the otlier two if the three be consistent.
213 ^« solve simultaneous equulions bt/ huleterntnuitr Multipliers.
Ex. — Take the equations
,/' + 2// + ;lv + lip = 27, '5.'c + r,//+ 7,v+ //• = -l-S, bx + 8y-\- 10,v - 2/r = 05, 7x + 6y + 5,^ + iw = 53.
MISCELLANEOUS KQJWTIONS. 7:\
Multiply the first by A, the second by /?, the third l)y C, leaviiiLi;- one C(inution nnnniltiplied ; and then add the results.
Thus (J+3//4-5r7-{-7)./' + (2.l-h'")/>'4-8^' + (;)//
+ (;5J +7/>' + l()C'-f-5) .v-[-(4J + /;-26'^-4) w
= 27-l + 18Zy + GoO + 5;].
To determine either of the unknowns, for instance .f, equate the coefficients of the other three separately to zero, and from the three equations find A, B, G. Then
^ 27J+48/y + G50 + o:] * A + W-\-hG-\r7 '
MISCELLANEOUS EQUATIONS AND SOLUTIONS.
214 ^''±1 = 0.
Divide by x^, and throw into factors, by (2) or (o). See also (480).
215 .r'-7d-i) = i).
X = —1 is a root, by inspection; therefore ''+1 is a factor. Divide by x-\-l, and solve the resulting (quadratic.
216 aH n; I' = !'>''>.
x*-\-lC).>- = l-j.'),/' = (')<■) X 7,/',
•' + 2 -""^ 2'
.i;" = /i''
= 7.
Rrr-E. — Divide the absolute* terra (here 455) into two factors, if possible, such that one of them, minus the scpian* of the other, equals the coefficient of x. ^^ee (483) for i^cMicral solution of a cubic equation.
I.
74 ALGEBRA,
217 .r*-i/ = 145(>0, .r-v = 8.
'*"t^^ .f = «+v and 2/ = z—v.
Eliujiiiate c, and obtain a cubic in 2, which solve as in (216).
218 .i^-/ = 3093, ci— 1/ = 3.
Divide the first equation by the second, and subtract from tlie result the fourth power of x—y. Eliminate {x^-\-if)j and obtain a quadratic in xij.
219 On forming Symmetrical Expressions.
Take, for example, the equation
(y-c){z-h) = aK
'Vo form the remaining equations symmetrical with this, write the corresponding letters in vertical columns, obser\dng the circular order in which a is followed by h, h by c, and c by a. So with X, ;?/, and z. Thus the equations become
0/-rj {z-h) = a\
[z-a) Gr-e) = b\
{.v-h){y-a) = c\
To solve these equations, substitute
x = h + c,-\-x\ y = c-\-a + y', :: = a -{- b -\- ::' ;
and, ]nulti})lying out, and eliminating // and ;:, we obtain
^^ho{b + r)-a(lr-hr) hc — ca — ab
niid tlicrefore, by symmetiy, the values of y and ;:, by the niK> just given.
220 // + .^' + //- = ^r (1),
:^-^-.r-\-x.v = fr (2),
'♦■H/ + .*//-r^ (3);
••• :5(//.r + .v.' + ,o/)-=r :l/n--\-2r',r-\-2'rlr-a'-b'-c* (4).
iMAniXAUY i<jxi'i:j:ssi()xs. 75
Now add (1), (2), and (3), and av(> o])tain
From (4) and (5), (,r + // -|- ;:) is obtained, and then (1), (2), and (o) are readily solved.
221 .,.-^^;/~ = «'^ (I),
jr-z^=fr (2),
z^-.n/=f- 05).
Mnltiply (2) by (:>), and subtract tlie square of (1). Result X (3./'//^ - Jf -if- ::'') = h'<- -n\
X _ y ^
/A.2_ft^ (-V--/.* a'b''-r' ^ ^^^'
Obtain X" by proportion as a fraction witli numerator = x^ — yz = a^.
222 .v=n,-\-bz (1),
,^ = az^c.i (2),
z = Lv-^(a/ (3).
Eliminate a between (2) and (3), and substitute tlie value of X from equation (1).
XieSUlC -r-" — r^ '„ j: ;,-
IMAGINARY EXPRESSIONS.
223 'Hic following are conventions : —
That v/(-'f-) is equivalent to a^^{—li); that a y,/{ — \) vanishes wlien a vanishes; that the symbol a, y(— 1) is sub- ject to the ordinary rules of Algebra. \'(— 1) is denoted l)y /.
76 ALOEBEA.
224 If a + ?'/3 = 7 -f- i^ ; then a = y and (i = B.
225 « + //3 and a — /'fS are conjugate expressions ; tlieir pro- duct = a- 4-/3-.
226 The sum and ])roduct of two conjugate expressions are both real, but their difference is imaginary.
227 The modulus is -\-x/a^+^^.
228 If the modulus vanishes, a and /3 must vanish.
229 If two imaginary expressions are equal, their moduli are etiual, by (224).
230 The modulus of the product of two imaginary expres- sions is equal to the product of their moduli.
231 Also the modulus of the quotient is equal to the quotient of their moduli.
METHOD OF INDETERMINATE COEFFICIENTS.
232 If A + Re + a«- + . . . = .4' + B'x -\- G'x' + . -_. be an equa=. tion which holds for all values of .^', the coefl&cients .1, B, &c. not involving ;r, then A— A', B = B\ C = G', &c. ; that is, the coefficients of like powers of x must be equal. Proved by putting X = 0, and dividing by x alternately. See (234) for an example.
233 METHOD OF PROOF BY INDUCTION.
Ex. — To prove that
5
Assume 1 + - + •» -\- ... +ir = - -"-.
0
iwirriAL FiLurnoxs.
0
= »(» + !) (2n + l) +6 (n + 1)- ^ (m + 1) {n (2n + l) -l-G r«-H)}
6 ' (5
^ (« + lU» + 2)(2u + 3) _ n(n'+\){'ln+l) G 6 '
where n' is written for «+l ;
o
It is thns proved that i/ the formula he true for n it is also true for n + 1.
But the formula is true when n = 2 or 3, as may be shewn by actual trial ; therefore it is true when >t = 4 ; therefore also when n = 5, and so on ; therefore universally true.
234 Ex. — The same theorem proved by the method of In- determiuate coefficients.
Assume
1^2^ + 3-+.. .+n^ =A + Bn +Cn^ +Du^ +&c.;
.-. 1 + 2- + 3-+. ..+»' + (« + !)- = .-l+5(/^ + l) + (7(« + l)- + D(n + l)'' + &c.; therefore, by subtraction,
«H2n + l = B + C(2n + l) + D{3n' + Sn+l), Avriting no terms in this equation which contain higher powers of n than the highest which occurs on the left-hand side, for the coefficients of such terms may be shewn to be separately equal to zero.
Now equate the coefficients of like powers of n ; thus
1
, and ^ = 0;
3jD= 1,
■•■ ^=i-
2C' + 3D = 2,
•■ ^ = 1'
i + C + D = 1,
0
therefore the sum of tlie Heries is equal to
n TT «=" _ «(» + !) (2n + l) G "^ 2 "^ 3 " 6 ■
PARTIAL FRACTIONS.
In the resolution of a fraction into partial fractions four cases present tlieniselves, \\]\\c\\ arc illustrated in tlie follow- ing examples.
78 ALGEBRA.
235 First. — When there are no repeated factors in the de- nominator of the given fraction.
3a;— 2
Ex. — To resolve :r—, -— — into partial fractions.
{x—l){x—2){x—o)
^''^""^ (a-l)(J"-2)(«-3) " ^:il "^ ^-2 "^ x-3 '
Sx-2 = A(x-2)(!c-S)-\-B(x-S)(x-l) + C{x-l)(x-2).
Since A, B, and C do not contain x, and this equation is true for all values of X, put x = l ; then
3-2 = ^(1-2) (1-3), from whicli A = ^. Similarly, if x be put = 2, we have
6-2 = i? (2-3) (2-1) ; .-. B = -4 ;
and, putting a; = 3,
9-2 = 0(3-1) (3-2); .-. G = \'
H 3a; -2 ^ __1 ^ 7
®°°® (a;-l) (a!-2) (a;-3) 2(a!-l) a;-2 2 (a;-3)'
236 Secondly. — When there is a repeated factor.
Ex. — Eesolve into partial fractions i- — -.,„' j!- ^ (a;— 1)^(33 + 2)
. lx^-\Ox'^^x A . B C ^ D
A^«"^« (-^=i?(.:t^ = (^^» "^ (^::ir "^ ^^ "^ ^rr2-
These forms are necessary and sufficient. Multiplying up, we have
7x'-lOx' + 6x = A ix-\-2) +B (x-1) ix + 2) + C (x-iy(x + 2) +D (x-iy
(I).
Makea; = l; .'. 7-10 + 6 = ^(1 + 2); .-. ^ = 1.
Substitute this valne of A in (1) ; thus
7x'-lOx' + 5x-2 ^ B (x-].){.v i2) + C (x-iy(x + 2)+D (x-iy.
Divide by a; — 1 ; thus
7x'-Sx + 2 = B(x + 2) + C(x-l)(x + 2)+D(x-iy (2).
Make X = 1 again, 7 -3 + 2 = J? (1 + 2) ; .-. B = 2.
Substitute this value of B in (2), and we have
7a;*-5a;-2 = G (x-l) (x + 2) +D (x-iy.
Divide by .T-l, 7a; + 2 = G (x + 2)+D (x-l) (3).
Put a; = 1 a thiwl time, 7 + 2= C (1+2); .-. C = 3.
I'AirriAL FRACTIONS. 79
Ijastl}^ make a; = —2 in (3),
-14 + 2 = X>(-2-l); .-. D = i.
1 2 3 4
Result 7 z—j. + 7 TTT H , H rii"
(a— 1)' (a;-l) «-l a; + 2
237 Thirdly.— When there is a quadratic factor of imaginary roots not repeated.
Ex.— Resolve ,t— tw^2-. — , ix into partial fractions.
Here we must assume
Ax-{-B Cx + D
(a5»+l)(j!» + a! + l) a;^ + l x' + x+l' x-i-l and X- + X + 1 have no real factors, and are therefore retained as denominators. The requisite form of the numerators is seen by adding
too'ether two simple fractions, such as — —- ^ r~,-
° ^ x + b x + d
Multij)l}iiig up, we have the equation
1 = (Ax + B) (x' + x + l) + {Cx + D) (x' + l) (1).
Let a;- + l = 0; z. x^ = —I.
Substitute this value of x- in (1) repeatedly ; thus
1 = (Ax + B) X = Ax' + Bx = -A + Bx ; or Bx-A-l = 0.
Equate coefficients to zero ; .'. 5 = 0,
^ = -1. Again, let ar + .r + l=0;
.-. x-=-x-l. Substitute this value of x^ repeatedly in (1) ; thus
1 = {Cx + D) i-x) = -Cx'-Dx = Cx + C-Dx- or (G-D)x + C-l=0.
Equate coefficients to zero ; thus ^ ' = 1,
1)= 1.
XT 1 _ = ''+1 _. ^
^^'"'^^ (..^ + l)(x^ + * + l) .tHx + 1 a-Hl
238 Fourthly. — When there is a repeated quadratic factor of imaginary roots.
Rv —"Resolve 40.^' — 103 ■ ^ i)artiiil fractions.
80 ALGEBRA.
Assume
40.7; -103 ^ Ax + B _Cx±B _ Ex + F
(x + iy {x'-4x + Sy (.r2_4x + 8y (.'?;--4a; + 8)- a;--'-4^ + 8
4- -^ + -^;
(.r+l)- a; + l'
40.t;-103 = {iAx + B) + {Cx + D)ix-—ix + 8) + iEx + F)(x--4:X + 8y} {x + l)- + {G + H(x + 1)} (x'-4x + Sy (1).
In the first place, to determine A and B, equate rt;-— 4a; + 8 to zero ; thus a;2=4a;-8.
Substitute this value of x- repeatedly in (1), as in the previous example, until the first power of x alone remains. The resulting equation is
40a; -103= (17.4 + 65) a? -48^ -75.
Equating coefficients, we obtain two equations
17^ + 65= 40 ) f .. , A = 2
48^ + 75 = 103)' ^^«--^^«^^ B = l.
Next, to determine 0 and D, substitute these values of A and 5 in (1) ; the equation will then be divisible by a;^— 4a; + 8. Divide, and the resulting equation is
0 = 2x + l3+{Cx + B+(Ex + F)(x'-4x + 8)] (x + iy
+ {G + H(x + l)]{x'-4x + 8y (2).
Equate a;'- — 4a; + 8 again to zero, and proceed exactly as before, when finding A and B.
Next, to determine E and F, substitute the values of (7 and D, last found in equation (2) ; divide, and proceed as before.
Lastly, G and H are determined by equating a' + l to zero successively, as in Example 2.
CONVERGENCY AND DIVERGENCY OF SERIES.
239 Let ai-{-a.^-\-a;i-\-&c. be a scries, and (7„, a„+^ auy two consecutive terras. The foUowino- tests of convergency may be applied. Tlie series will converge, if, after any fixed term — (i.) The terms decrease and are alternately })0sitive and negative.
(ii.) Or if "- is always (j renter than some (piantity
(' n 1-1
greater tlian unity.
SERIES. 81
(iii.) Or if — — i.s never less tluui tlic corrcspoiidiii^ I'atio '''1 + 1 ill a known coiivei\u:ing series.
(iv.) Or if l-^—n) is always tjreafrr than some (juan-
tity greater than unit3^ [% - tl' and iii.
(v.) Or if l^-^—ii — l]\og)i is always i/rrdfcr tlian V^'»j+i ^
some quantity greater than unity.
240 The conditions of divergency are obviously the converse of rules (i.) to (v.).
241 The series ai-^a.,x-\-a.iX^-{-&c. converges, if ^^ always less than some quantity p, and x loss than
1
[By 239 (ii.)
242 To make the sum of the last series less than an assigned
(iiiantitv /s make ,v less than , , I' hvincr the o^reatest co-
efficient.
Grnrral Tltcnron.
243 If •/> ('") be positive for all positive intec^ral values of .r, and continually diminish as <>■ increases, and if )n be any posi- tive integer, then the two series
<^(l) + (^(2) + (^GJ) + ^(l) +
<li{l)-\-m(f>{m)-\-m-4>{m-)-\-m''(t>{m')-Y
arc either both coiivern-ent oi* diverofent.
244 Ajiplication of tliis theorem. To asc<'rtain whotlier the is diverjT^ent or convero-init when p is «^i-eater than unitv.
41
82 ALGEBRA.
Taking m = 2, tte second series in (243) becomes
1.2,4,8,0 ^ 2'^ ^ 4p ^ 8^
a geometrical progression whicli converges ; therefore the
245 I'lie series of which --:- ^,- is the general term is
?i (log ny'
convergent if j9 be greater than unity, and divergent if p be
not greater than unity. [By (243), (244).
246 The series of which the general term is
1
n\{ii)X'{n) V{n){r^'(n)}^'
where \ (n) signifies \ogn,X'^{n) signifies log {log ()i)}, and so on, is convergent if ^ be greater than unity, and divergent if j) be not greater than unity. [By Induction, and by (243).
247 The series ai + cu + SLC. is convergent if
ncu log (n) log2 {n) log''(7i) {log,^i {n)y
is always finite for a value of p greater than unity ; log'' (7;) here signifying log (log iz), and so on.
[See Todhunter's Alr/ehra, or Boole's Finite Bijjcrences.
EXPANSION OF A FRACTION.
42.' — 10a3
248 A fractional expression such as :, — --'. — -- may
\ — bx-\-l\x^ — Q>x,
be expanded in ascending powers of x in three different ways.
First, by dividing the niiraerator by the denominator in tlie ordinary way, or by Synthetic Division, as shewn in (28).
Secondly, l)v the metliod of Indeterminate Coefficients (2:32).
Thirdly, by Partial Fractions and the Binomial Theorem.
SERIES. 83
To expand by tlie method of Indeteriiiiii:ite CoefficiLiits proceed as follows : —
Assume , '^''^ ~\^'^' . , = -1 + ^'•'- + C.>- + J).c' + E.v' + & c.
4x-lUr = .1+ llx+ Cx--\- nx''+ Ex'+ I<\r''+...
— OAx— GBx-— GC'u;*- Gi*./;'— OiiV'-...
+ ll.lj;-4-ll/^a;' + liac*+llA/'+...
- G.-Lc'- 6Bx*- G^.V-...
Et|uate cocUicients of like powers of x, thus
.1 = U,
JJ- 6A = 4, .-. J! = I ;
C- 6B + IIA =-lO, .-. C= li;
D-6C+UB-GA= 0, .-. 1)= 40;
E—6D + IIC— OB = U, .-. E=UO;
F-6E + 11D-6C= 0, .-. i''=;30-i;
The formation of the same coefficients by synthetic division is now exhibited, in order that the connexion between tlio two processes may be clearly seen.
The division of 4a; — lO.r l)y 1— ('..i'-f U.r-G,/' is as follows:— 0 + 4-10 + 6 24 + 84 + 240 + GGO
-11 -44-154-440-1210
+ 6 + 24+ 84+ 240 + GGo
0 + 4+14 + 40 + 110 + 304+
^l n C D E F
If wc> stop :it the term llO.r', then the undivided remainder will lie ;i04.j;''— 'JTO/^ + CtiOi/, and the complete result will be
4.r + 14x- + 40..H110. + ^_^^^,fZ:^-
249 Here the conchidiiig fraction may be regarded as the sum to inlinity after four terms of the series, just as the original expression is considered to be tlie sum to inlinity of the whole series.
250 Tf the general term be reipiired, the method of ex- pansion by partial fractions must be adopted. See (257), wliere tlie Lrcneral term of the foregoing series is oljtained.
84 ALGEBRA.
RECURRING SERIES.
a^^-\-(ii.i'-]- Uod'- -\-ayv'^-\- &c. is a recurring series if the co- efficients are connected by the relation
251 (In = Ih «« - 1 + 7>2 «« - 2 + • • . + Pm (in - m-
The Scale of Relation is
252 1 -PI^V -JhO^ —... —lhn^V''\
The sum of n terms of the series is equal to
253 [The first m terms
—piV (first tn — l terms + the last term) —p^x^ (first m— 2 terms + the last 2 terms) —IhJC^ (first m— 3 terms + the last 3 terms)
-~i>i«-i'^'""^ (first term + the last m — \ terms)
—p,nX"' (the last m terms)] -^ [l—p^.v—pocV^— ... — />,„cr"'].
254 If the series converges, and the sum to infinity is re- quired, omit all " the last terms " from the formula.
255 Example. — Required the Scale of Relation, the general term, and the apparent sum to infinity, of the series
4'c + 14r + 40,v^ + 110,ii'^ + 304^^-8o4/+ ... .
Observe that six arbitrary terms given are sufficient to determine a Scale of Relation of the form l—px — qx' — rx^, involving three constants p, q, r, for, by (251), we can write three equatious to determine these constants ; namely, 110= 40p4- 14(2+ 4r\ The solution gives
304 = llOp + 402 + 14r k p = G, 7 = - 1 1, r = 6. 854 = 304;j+110g + 40rJ Hence the Scale of Relation is 1 — 6.« + ll.r — 6.r^.
The sum of the series without limit will be found from (254), by putting Pi = ^, Pi = — 11» P3 =6, m = 3.
The first th ree terms = 4,c + 1 4.r + 40.i-'' — 6xthe first two terms = —24^-— 84a;* -I- 1 l.r X the first term = + 44a;'
4«-10a:-
RE CURBING SERIES.
^^ 4.r-10x'
1 - G.i; + 1 Ix* -<;.(•»'
tlie meaning i)f which is tliat, if this fmction bo expiunluil in asuentling powers of x, the first six terms will bo those given in the question.
256 To obtain more terms of the series, we may use the Scale of Relation ; tlius the 7th term will be
(6 X 854- 1 1 X 30i + 6 X 1 10) a:^ = 2440a;^
257 To find the general term, S must be decomposed into l)artial fractions; thus, by the method of (2'35),
4.B-10a;- _ 1 , 2 8
l-6.c + ll.j;'-(3a;-' l-Sx 1— 2x 1-a; By the Binomial Theorem (128),
, ^, = l+3.c + 3-.r + +S''x'\
1 — Sx
r-=-r = - + 2=.c + 2^r + + 2" * '.c",
l—zx
— =-3-3.c-3.r - -Sx".
Hence the general term involving x" is
(;3'> + 2''»'-3)x''. And by this formula we can write the " last terms" required in (2.")3), and so obtain the sum of any finite number of terms of the given series. Also, by the same formula we can calculate the successive terms at the beginning of the series. In the present case this mode will be more expeditious than that of employing the Scale of Relation.
258 If 5 in decomposing -<S' into partial fractions for the sake of obtaining the general term, a quadratic factor ^vitli ima- ginary roots sliould occur as a denominator, tlie same method must be pursued, for the imaginary quantities will disappear in the final result. In this case, however, it is more con- venient to employ a general formula. Sup[)ose the fraction which gives rise to the imaginary roots to be
L-\-Mx _ L-\-Mx
<i+b,r + x"- ~ {p—r){q-x)'
p and q being the imaginary roots of ri-{-hx-{-x' = 0. Suppose j) = (i-\-i)^,
q = a—ij^y where i = v —1.
86 ALGEBRA.
If, uow, the above fraction be resolved into two partial fractions in the ordinary way, and if these fractions be ex- panded separately by the Binomial Theorem, and that part of tlie general term furnislied by these two expansions written out, still retaining j) and r/, and if the imaginary values of p and q be then substituted, it will be found that the factor will disappear, and that the result may be enunciated as follows.
259 The coeflQcient of a;" ^ in the expansion of
L + iU.r
will be
^g^-^[Ha«-'^-C(«,3)a"-^HC(«,o)a»-^'-...l
^^"+^^ +C(«-l,5)«"-«j8»-...}.
260 With the aid of the known expansion of sin nO in Trigonometry, this formula for the ti^'* term may be reduced to
in wliich 6 = tan — , <^ = tan ^
L + 3/a
If n be not greater than 100, sin (viO — ^) may be obtained from the tables correct to about six places of decimals, and accordingly the «*'' term of the expansion may be found with corresponding accuracy. As an example, the 100*'' term in
the expansion of - — ^ ., is readily found by this method
^ , 41824 „9 to be -^- x^.
To (/cfrrniinc whether a i>;ii'cn Scries i.s rernrri)i;ii- (tr not.
261 If certain tirst terms only of the series be given, a scale of relation may be found which shall produce a recurring
BECUBRINQ Sl^RII'JS. 87
series whose first terms arc those given. The method is exemplified in ('255). The innnber of niikiiowii coefficients j), </, r, &c. to be assumed for the scale of relation must be equal to half the number of the given terms of the series, if that number be even. If tlie nund^er of given terras be odd, it may he made even by })refixing zero for the first term of the series.
262 Since this method may, however, produce zero values for one or more of the last coefiicients in the scale of relation, it may be advisable in practice to deteruiine a scale from the first two terms of the series, and if that scale does not jn-oduce the following terms, we may try a scale determined from the first four terms, and so on until the true scale is arrived at.
If an indefinite nnmber of terms of the series be given, we may find whether it is recurring or not by a rule of Lagrange's.
263 Let the series be
S= A + ]Jx + G.r -h Dx' -f- &c.
Divide unity by S as far as two terms of the quotient, wliicli will be of the form p-\-qx, and write the remainder in the form /S'V, S' being another indefinite series of the same form as S.
Next, divide S by S' as far as two terms of the quotient, and write the remainder in the form S"x-.
Again, divide /S" by S'", and proceed as before, and repeat this process until there is no remainder after one of the divisions. The series will then be proved to be a recurring series, and the order of the series, that is, the degree of tlie scale of relation, will be the same as the number of divisions which have been effected in the process.
KxAMi'LK. — To determine whether the series 1, o, (i, 10, 15, '21, 28, .'U), 45, ... is recurring or not.
Introducing x, we may write
S = l+3.r + 6.r+10.rH15.):' + 21,/' + 2,V' + :](u''+45..:\.. .
Then we .shall have ' = 1 — ;lf4-... ^vith a rcniaimler
6x' + 8.r' + 15.1!* + 24.c'' -f- obx'' + itc. TlH-iir..io S'= 3 + 8.j;+15.7;-4-2kr» + .'3.V + A-c.,
.s" ;; 9
88 ALGEBBA.
with a remainder -^ (.r + 8a;H ().)■' + 10.x-'^ + &c. ...)• Therefore we may take S" = 1 +8.1- + 6.r + 10a* + &c.
Lastly -^„ = 3 — .13 without any remainder.
Consequently the series is a recurring series of the third order. It is, in fact, the expansion of
SUMMATION OF SERIES BY THE METHOD OF DIFFERENCES.
264 Rule. — Form successive series of differences until a series of equal differences is obtained. Let a, b, c, d, &c. be the first terms of the several series ; then the 7^^'' term of the given series is
265 a+i.-m+("-^(r'^e+("-^("-f-^.l+
The sum of n terms
266 =„« + !^M4 + ^iIi=ii§^<- + &c.
Proved by Induction.
Example: a...l+ 5 + 15 + 35 + 70 + 126 +
h ...4 + 10 + 20 + 35 + 5G + ... c ... 6 + 10 + 15 + 21+... (Z...4+ 5+ 6 + ... e ... 1+ 1 + ...
The lOOti' term of tlie first series
■ ^ 1.2 ^ 1.2.3 ^ 1.2.3.4
The sum of 100 terms
,,,.,^100.09 , .100. 99. 98«j^ 100. 99. 98.97 ,, 100.99.98.97.96 == 100+ -^^4+-^-2r3-^+ 1.2.3.4 ^^ 1.2.3.4T.r-
FAiToh'fAl, ,s7';/.7 /•;>'. ^ SO
267 '^'-^ interpolate a term bclwecn two terms of u series by the motliod of ililTtTeiices.
Ex. — Given log 71, log 72, log 73, log 74, it is required to find I<>g 7"2o4. Form tbo scries of diBerences from the given logarithms, as in ('itJG),
log 71 log 72 log 73 log 74
a... l-8ol2".83 l-8573:i2') l-8G:i3229 1-8G02:U7
6 ... -0060742 •00r)9904 -0059088
r ... --0000838 - -0000816
(Z ... —00000-22 coii.sidered to vimish.
Log 72'o4 mnst be regarded as an interpolated term, the number of its place being 2-54.
Therelore put 2-.'')l. for n in formula (265).
Result log 72-54 = 1-8605777.
DIRECT FACTORIAL SERIES.
268 Ex.: 5.7.9 + 7.0.11 -I- <.i.n .1;] + 11 .1:3.15 + .. d = common difference of factors, m= miml)er of factors in each term, n = number of terms, a = first factor of first term —d.
>*♦'• term == (fi-\-n(/) {a + n + i</) (^/ + ;/ + //<- I r/).
269 To find the sum of u terms.
Rule. — Fnnii the last term with the ne.rt hitjlirsf fnrtor take thefi'i\st ti'Dii ir'ilh flie next loiiwst factor, and dlrlde Ixj {m + 1 ) '/.
Proof. — By Induction.
Thus the sum of 4 terms of the above series will be, putting d = 2, vi — 3,
^, 11.13.15.17-3.5.7.9
n = 4, a = 3, h = ^j^^, .
Proved either by Induction, or by the ractliod of Indeterminate Coefficicnt.s.
90 ALGEBRA.
INVERSE FACTORIAL SERIES.
270 Ex.: 5^7^9 + 7^9_ii + 9.11,13 + 11.13.15+---
Defining d, in, oi, a as before, the
1
;}tii term =
(n-]-ud) {a + n-\-l(I) ... {a-^n-\-m — id)
271 To find the sum of n terms. Rule. — From the first term wanting its last factor take the last term wanting its first factor, and divide hy (m — 1) d.
Thus the sum of 4 terms of the above series will be, putting d = 2, m = 3, 1 1
5.7 13.15 ^^ = ^'" = ^^ (3-1)2 •
Proof. — By Induction, or by decomposing the terms, as in the following example.
272 Ex.: To sum the same series by decomposing the terms into partial fractions. Take the general term in the simple form
(r-2)r(r + 2) Resolve this into the three fractions
Substitute 7, 9, 11, &c. successively for r, and the given series has for its equivalent the three series
Ij 1 + 1+ 1 +i + 1 +_1_|
8 1 5 7 9 11 13 2u + 3 )
2^ C _ 2 _ 2 _ 2 _ 2 _ 2 2_ )
8 I '7 9 11 13 2» + 3 2n + 5)
+ 1 j 1 + 1 +1_+ + _i_ + _l_ +^1-1,
8l 9 11 13 2»i + 3 2u + 5 2» + 7)*
and the sum of n terms is seen, Jiy inspoction, to he
1(1_1_ 1 +_l_l=Jil 1_ ]
8(5 7 2« + 5 2n + 7) 4(5.7 (2» + 5) (2« + 7} 3 '
a result ol)taiiicd at once by the rule in (271), taking -— - — for the first •^ ^ '^ 5.7.9
term, and -t- .-- for the »i"' or last term.
(2n + 3)(2n + 5)(2n + 7)
FACTOh'IAL SERIES. 91
273 Analogous series may be reduced to tlie types in (268) and (270), or else tlie terms may be decomposed in the manner
shewn in (272).
Ex.: -J_+_i_+__7__+_10L.+
1.2.3 2.3.4 3.4.5 -i. 5. <;
has for its general term
3»-2 __1 ,_5 4_ , .......
n(n + l)(n-\-2) ,i n + 1 n + 2 '^ '^""^^>''
and we may proceed as in (272) to Hnd the sum of n terms.
The metliod of (272) includes the method known as "Summation by- Subtraction," but it has the advaut;igo uf being more general and easier of application to complex series.
COMPOSITE FACTORIAL SERIES.
274 If the two series
M N-5 i^r _l5.G .,^5.6.7 3^5.6.7.8 ,^
/I ^-3 1,.. ,3.4 .. 3.4.5 3 , 3.4.5.0 t ,
be multiplied together, and the coefficient of .f* in the product be equated to the coefficient of x*" in the expansion of (1 —x)'^^ we obtain as the result the sum of the composite series
5.6.7.8xl.2 + 4.5.C).7x2.3 + :K4.5.6x3.4
4! 2.11!
+ 2. 3. 4. 5X4. 5 + 1. 2. 3. 4X5. G
7! 4!
275 Generally, if the given series be
Aa + Aa.+ --+A,. ,(?„-! (I),
where (^)^ = ,. (r + 1) (/• + 2) ... (r + v- 1),
and r,.= (?i— r) (//-/• + 1) .. {n- ri-p-l) ;
the sum of n — l terms will bo
/>!7! (;,4-;, + r/-l)!
1)2 ALGEBRA.
MISCELLANEOUS SERIES.
276 Sum of the powers of the terms of an Arithmetical Progression.
1+2+;)+...+// =
1 +2»+y +...+«'= 5 ^^^j' =s.
H o.+y+ ... + „. = "(" + 1) (2« + l)(3»-+3» -1) ^ ^, ,
[By the method of Indeterminate Coefficients (234).
A general formula for tlie sum of tlie 7-"' powers of 1.2.3 ... n, obtained iu the same way is
>• + !
wliere Ji, A.,, &g., are determined by i)uttiug j> = 1, 2, 3, &c. successively in the equation
1 2 0^ + 1)!
~(;?+2)!"^r(/>)!'^r(r-l)(i>-l)!^"'"r(r-l)...(r-7>+l)
277 «"' + (^/ + ^/)"' + (^/ + 2^/)"'+... + (^/ + //</)"'
^ (>, + !) ^,'" + ,s^,,,^f'"- V/+.S,(' (m, 2i «'"-->/-
Proof. — By Binomial Theorem and (276).
278 Summation of a scries parthj Arithmetical and pa rill/ Geometrical.
Ex.\m?lt;. — To hud the sum of the series 1 +3,i' + 5.r + to n terms.
Let s = l+;?.r + 5.r' + 7.i:» +... + (2/1-1) x"-',
S.V = .i+3.r + oj;'+ ... + {2n-:i) j""' + (*Ju- 1) x",
.'. by Kiiblractioii,
6- ( 1 -.,•) = 1 + ±v + 2.r^ + 2./.» + . . . + 2.t" - ' - (2« - 1 ) x" 1 — r""'
= l+2a^^- -^--(2H-l).r", 1 — X
l_(2n-l).r" 2x(l-.r''-') • '- 1-x + (I-../--
279 A general formula for tlio sum of n terms of
^- \^' + 0-ry ■
Obtained as in (278).
Rule. — Mi(Itij)Ji/ Inj the ratio and subtract the resulting series.
280 r-'— = l+.r+.»"+.t'+...+cr"-'+-pi
281 Tj-^, = l+2.r+;ja-+lr''+...
, n-i , (n-{-\)r'*~)i.v"^' (l-cf)-
282 (^<-l).r+(/i-2).rH('<-.'i).''^'+... + -V-H'i'""'
= (^^-i);;-"^;+->"". B,(253).
283 i^,,^W//^--])^;/(»-])0/-2)_^^^.^.^^^,„^
Hv making 4^-=^ in (12r»). .etrrat*.
94 ALGEBRA.
284 The series
»-,-j , (n-4)(n-5) _{n-6){n-r,){n-7) , ^~"T""^ ;5! 4! ^*
^ , ( ^y.,0i-r-l)(n-r-2)...(n-2r-[-l) consists of ^^ or ^^~ terms, and the sum is given by
/S' = — if 71 be of the form 6m-\-S,
n
S = 0 if ?i be of the form 6/?i + l,
S = if ?i be of the form 6m,
n
>S^ = — if 7i be of the form 6m±_2. n
Proof.— By (545), putting;) = x^-y,q = xy, and applying (546). 285 The series 7i^-n («-!)'•+ Hd^l^ {n-2y
o !
takes the values 0, n\, ^n{n + l)\ according as r is <n, =n, or =)i + l.
Proof. — By expanding (e^ — 1)", in two ways: first, by tlie Exponential Theorem and Multinomial ; secondly, by the Bin. Th., and each term of the expansion by the Exponential. Equate the coeflicients of a;** in the two results.
Other results are obtained by putting r = n-{-2, n-\-o, &c.
The series (285), when divided by r\, is, in fact, equal to the coefiBicient of aj'" in the expansion of
POLYGONAL NUMBERS.
05
286 By exactly the same process we may detlucc from the I'unctiou {t"*' — f'*}" the result tliat tlie scries
n'-n (N-'2y+"^"~^^ (,,_.l.)'-_&c.
takes the values 0 or 2'*.?^!, according as r is < n or = n; this scries, divided by r ! , being e<j[ual to the coefficient of x'' in the expansion of
f ^.3 ,-,.5 yi
POLYGONAL NUMBERS.
287 Tlie n^^' term of the r'^' order of polygonal numbers is equal to the sum of n terms of an Aritli. Prog. Avhose first term is unity and common difference r — 2 ; that is
= 1 {2+{n-l)(r-2)] = n + ln {n-l){r-'2).
288 I'hc sum of It terms
__uiu-^l) u{u-l)(n-\-l)(r-2) 2 "^ «
By resolving into two scries.
Order.
«*•> term.
1
2
3 4 5 6
n
Inin + l)
n-
iH(3u-l)
(2»-l)»
11 11111 1 2 3 4 5 6 7 13 6 10 15 21 28 1 4 9 16 25 36 49 1 5 12 22 35 51 70 1 6 15 28 45 66 91
r
n+^i^(r-2)
1, r, 3 + 3 0—2), 4 + 6 (r-2), 5 + 10(;— 2), 6 + 15 (r- 2), (fee.
96
ALGEBRA.
la practice — to form, for instance, the 6*'' order of poly- gonal numbers — write the first three terms by the formula, and form the rest by the method of differences.
Ex.: 16 lo 28 45 66 91 120 ...
5 9 13 17 21 25 29 ...
[r-2 = 4] 4 i 4 4 4 4 ...
FIGURATE NUMBERS.
289 The n"' term of any order is the sum of n terms of the preceding order.
The n^^ term of the r*^' order is
njn+l)_^in +r-2) = /j („ ^ ,. _ j ) . [By 98.
(r-1)
290 The sum of n terms is
n{n-\-l)...{n + r-l) _
H{n,r).
Order.
Figurate Numbers.
nth term.
1
1, 1, 1, 1,
1, 1
1
2
1, 2, 3, 4,
5, 6
n
3
1, 3, 6, 10,
15, 21
7?, (w+1) 1.2
4
1, 4, 10, 20,
35, 5G
1.2.3
5
1, 5, 15,_35,
70, 12G
1.2.3.4
G
1, G, 21, 5G,
126, 252
7i(n + l)(7i + 2)(7i + 3)(7z-l-4)
1.2.3.4.5
iiYi'h'iiCKoM irrincAi. si:riks. 97
IIYPEUGEOMETKICAL SERIES.
291 ,+^_^,,. + ^(^+lMim),,.=
l.y 1. -2.7(7+])
a(a+1)(a + 2)/3(/3+1)(^ + 2) ^, ^^, "^ 1.2.;$.y(y+l)(7 + 2)
is convergent if .r is < 1 ,
and divergent if x is > 1 ; (-•5'* ''•)
and if x = 1, the series is
convergent if -y — a — /3 is positive, divergent if y — a — /3 is negative, (239 iv.)
and divergent if 7 — a — /3 is zero. (239 v.)
Let the liypergeometrical series (291) be denoted by F{a, ft, y) ; then, the series being convergent, it is shewn by induction that
292 ria,ft-^\,yjyi} ] concl.ulin- ^vitl.
l-/r, I-/.-,. ,
1-c^c. ... \-k,,z,..
^vlH"^c /.',, I:,, Jr., Sec with .-:,,,, are given l)y tlic foniiiilie
. _(a + r-1)(y+)— 1-y8).r (y + 2>—2j(y + 2>— 1)
_()8 + r)(y+r-a).r (y+2;-l)(y+2r)
f'(a-f-r, /8 + r, yH-2r)
Tlie continued fi-action may be conchuhMl at nny point with k-ir^Ur' When r is infinite, r/o^ = 1 and tlic continurfl fraction is infinite.
o
98 ALGEBRA.
293 Let
"l.y ' 1.2.y(y+l) ' 1 . 2.;{.y (y+1) (7+2)
f{y) ::^ 1 + _!^ + '^' + -^ /"^i 1,/ _1_.>X + '^^•
1.7 1.2.V v+1) 1.2.,{.v(v4-l (-/+2)
the result of substituting — for ic in (291), and making /3 =:= « = X . Tlien, by last, or independently by induction,
/(y + 1) _ 1_^ p^_ P2_ 'Pj!]_
Ay) 1+1+1 + ... + I+&C.
with j),„ =
(y+m — 1) (y+/>«]
294 In this result put y = ^ and -^ for ,r, and we obtain by Exp. Th. (150),
Or the continued fraction may be formed by ordinary division of one series by the other.
295 ('"' is incommensurable, m and n being integers. From the last and (17-1), by putting x = ' .
INTEREST.
If r be the Interest on £1 for 1 year, 11 the inimber of years, /' the I'l-incipal, A the auiouut in n. years. Then
296 At Siiuple Interest A = P{l-^)n').
297 At Compound Interest A = /*(t+r)". % (-^-i)-
i\Ti:i:i:sr AM) .l\.\ ///'//vS. 99
298 But if the payments of ") .„
Interest be made 7 C A = I' h + -j times a year ) ^ ^^
If A be an amount due in 11 years' time, and /' the ])resent worth of -1. Theu
299 At Simple Interest 7* = -j-^ .
By (-200).
300 At Compound Interest /' = ,. By (297).
301 Discount = A- P.
ANNUITIES.
302 The amount of an Annu- 1 ,,(,,_])
ity of £1 in n years, [■ = nA~ .^ >'• ^y (82). at Simple Interest .
303 1 'resent value of same = '^yr — :
,tA-hi{N-\)r n^(09,j).
304 Amount at Compound \ _ (1+r)" — 1 ^ ,^-.
Interest ) "(l+r)-l '
Present worth of same — ~ . ' , '' . 'b' (-J^'*^*)-
(l+r)-l
305 Amount when the pay- ^ ( j 1 _!1 Y"' _ 1
ments of Interest/ _ ___7_ iw (2:is).
are made q times -|)er C / i 1 '* V' 1 ' annum J \ (/ ' ~
1
Present value of same =
100 ALGEBRA.
306 Amount wlicn the pay- )
meuts of the Annuity f _ ( 1 -f >•)"— I are made m times per [ ' i ~
annum J m \{l-{-r)'-—l}
Present value of same
^ l-(l+r)- m{(l-fr)i-l}
307 Amount when the In- terest is paid q times and the Annuity m
times per annum ... J "^ V\ qi'
Present vahie of same
m
(i+v)-
PROBABILITIES.
309 If ^11 tlie ways in which an event can happen be m in number, all being equally likely to occur, and if in n of these m ways the event would happen under certain restrictive conditions ; then the probability of the restricted event hap- pening is equal to n-^m.
Thus, if the letters of the alphabet be chosen at random, any letter being equally likely to be taken, the probability of a vowel being selected is equal to -i^q. The number of un- restricted cases here is 26, and the number of restricted ones 5.
310 Ifj however, all the m events are not equally probable, they may be divided into grou{)s of ccpially probable cases. The probability of the restricted event happening in each group separately must be calculated, anel tlie siun of these probabilities will be the total })robability of the restricted event liappening at all.
I'UOHAIIILiriHS. lol
ExAMPLK. — Tlicro are three bags A, B, and G.
A contains 2 white and .'} black balls. B „ 3 „ t
C „ 4 „ 5
A bapf is taken at random and a hull drawn from it. Required the pro- bability of the ball being white.
Hero the probability of the bag A being chosen = J, and the 8ub.sc([nonfc probability of a white ball being drawn = l-
Therefore the [jrobability of a white ball being drawn from .1
~ 3 5 15-
Similarly the probability of a white ball being drawn from B
- 1' X 3 - l' ~ 3 7 ~ 7
And the probability of a white ball being drawn from G
-1 J* - i ~ 3 ^ 9 ~ 27'
Therefore the total probability of a white ball being drawn
.j2_ 1 4 ^ 401 15 7 27 945'
If a be the number of ways in wliicli an event can liappen, and J) tlie number of ways in wliieli it can fail, then the
311 rrobabilitv of the event lia])penin2r = r.
312 l'rol)al)ihty of the event failing =
Thus Certainty = 1.
If p, p' be the respective probabilities of two iudcpcndcnt events, then
313 rrol)al)ility (^f both liappening = pp'.
314 )} of not /yo//i happening == i—pp'.
315 )) of one happening and one faiHng
316 „ of l)o(h failing = (!—/>) (1 —/>').
102 ALGEBlLi.
If the probability of an event happening in one trial be j>, and the probability of its failing q, then
317 Probability of the event happening r times in n trials
= C{n, r)2fff-\
318 Probability of the event failing r times in n trials
= C {n, r) ^j" "''(/''. [By induction.
319 Probability of the event happening at lea><t r times in n trials = the sum of t\iQ fivHt n—r-[- 1 terms in the expansion of 0; + ry)".
320 Probability of the event failing at least r times in n trials = the sum of the last n — r-[-l terms in the same ex- pansion.
321 The number of trials in which the probability of the same event happening amounts to j/
_iog(i-;/)
log (!-;>)'
From the equation (1 — j^)*" = 1 —])' -
322 Dei'inition. — AVhen a sum of money is to be received if a certain event happens, that sum multiplied into the proba- bility of the event is termed the expectation.
Example. — If three coins be taken at random from a bag containing one sovereign, four half-croAvns, and five shillings, the expectation will be the sum of the expectations founded upon each way of drawing three coins. But this is also equal to the average value of three coins out of the ten ; that is, -i^ths of 35 shillings, or 10s. Qd.
323 The probability that, after r chance selections of the numbers 0, 1, 2, 3 ... 7i, the sum of the numbers drawn will be 6', is equal to the coefficient of .t'* in the expansion of
ri;i)r.M:iLiiii:s. lo:
324 'I'lie probability of the existence of a certain cause of an observed event out of several known causes, one of wliieli vi^ist liave produced the event, is proportional to tlie a jn-iarl probability of the cause existinu: multiplied by the probability of tlie event happening from it if it does exist.
Thus, if the a priori probabilities of the causes be /',, /'. ... Sec. J and the corresponding probabilities of tlie event hap- pening from those causes (^),, (/_, ... itc, then the probal)ility of the ?•"' cause having produced the event is
X{1'Q)
325 If A'> P-i ••• &c. be the a jfrlori probabilities of a second event hap])ening from the same causes respectively, then, after the first event has happened, the probability of the
second happening is t {PQ)
POP'
For this is the sum of such probabilities as \^''//, (i which is
the probability of the r^'' cause existing multiplied by the probability of the second event happening from it.
Ex. 1. — Suppose there are
4 vases containing each 5 wliite and (i l)l:ick ball.s, 2 vases containing each ^ white and 5 black balls, and 1 vaso containing '2 white and 1 black ball.
A white ball has been drawn, and the probability that it came Irani tiie group of 2 vases is required.
Here P, = ^ ]'.. = 'f , P, = !
Therefore, by ('S-l), the pn)l)ability i-c(iuii-fd is
4,:.^ 2.;^ L2 427 7.11 7.8 7.3
Ex. 2. — After the white ball has been drawn and ro])laced. a ball drawn again; required the probability of the ball being lilack.
104 ALGEUltA.
He-o P; = A, p; = |, p. = |
The probability, by (325), will be
4.5.6 2.3.5 1.2.1 7.11.11 7.8.8 7.3.3 58639
4.5 2.3 1.2 112728'
7.11 7.8 773
I£ the probability of the second ball being white is required, QiQ^Qi must be employed instead of P{P'.P'z.
326 The probability of one event at least happening out of a number of events whose respective probabilities are a, h, c,
&G., is P1-P2 + P3-P4+&C.
where P^ is the probability of 1 event happening,
and so on. For, by (316), the probability is
l-(l-a) (l-h) (l-c) ... = y.a-^ah + ^ahc-, ...
327 The probability of tlie occurrence of r assigned events and no more out of 01 events is
where Q„. is the probability of the r assigned events ; Q.,.+i the probability of r + l events including the r assigned events.
For ii a, h, c ... be the probabilities of the r events, and a, 1/, c ... the probabilities of the excluded events, the re- quired probability will be
ahc ... {l-a'){l-h'){l-c') ... = ahc ... {l-1.a-{-'^aU-:ia'l/c'-\-...).
328 Tlu^ probability of ani/ r events hapjiening and no more Note.— If a = h = c = &c., tlien ^Q, = C (n, r) Q,, cl'c.
INEQUALITIES. 105
ii\i<:QLiALrrib;s.
330 ^^'+^^-+- •"^'^" lies between the -Teatest uiul least of />i + '>-.>+ ••• +^'»
the fractions i^, ^, ... -^, the (leiiominators being all of the same sign.
PuoOF. — Let k be the greatest of the fractions, and - any other; then
ar<kbr. Substitute in this way for each a. Similarly if k be the least frapCtion.
331 ^ > V"0.
332 «.+«.+ ...+»„ > y„,7^~^,;
71
or, Arithmetic mean > Geometric mean.
Proof. — Substitute both for the greatest iind least factors their Arith- metic mean. Tiie product is thus increased in value. Repeat tlie process indefinitely. The limiting value of the G. M. is the A. ]\I. of the quantities.
333 q:^' > {^l+!i)'\
excepting when m is a positive pi'opei' IVaet ion.
PlJOOK
„'" + ?,'" =("t'')"'[(l+.r)'" + (l-..)"'},
diere .»■ = " -'. Kinploy Hin. Tli a + b
334 ":'+""'+..+": > ^'.+".+ ■+"..y\
excepting ^vhen /// is a positive proper fraction.
I"
106 Ahdi'JiiUA.
Otherwise. — The ArUhmetic mean of the m"' poivers is greater than the m"' power of the Arithmetic mean, excepting when m is a positive proper fraction.
Pkoof.— Similar to (332). Substitute for the greatest and least on the left side, employing (333).
336 If -'' and m, are positive, and x and mx less than unity ; then (l + ciO-'"> l-mx. (125, 240)
337 K ,1^ m, and n are positive, and n greater than ra ; then, by taking' x small enough, we can make
For X maybe diminished until l^nx is > {l—mx)'^, and this is > (l-\-xy\ by last.
338 If ^ be positive, log {l-\-x) < os. (150)
If X be positive and > 1 , log (l+.r) > <r- ^. (155, 240)
If X be positive and < 1, ^ogj^^, > ■<'• (^''^^^
339 When n becomes infinite in the two expressions
1.3.5... (27Z-1) .^^^^^ o.-^.7...(2y^ + l) 2.-4.(3 ... 2yi ' 2. 4. 6. ..2m
the first vanishes, the second becomes infinite, and their product Ues between -J and 1.
Sliewii by adding 1 to each factor (see 7o), and multi- plying the result by the original fraction.
340 II' '" he > II, and ii > <(,
SrA LES OF NOT. I TloX. 1 ( i7
341 If ", f> 1)0 |)ositiv(> quantities,
„V/' is > ('i+I'f"'.
SinulaHy a'' !,'■.■'> {^' + ',' + ' f"*' ■
These and similar theorems may be proved hy takinf^ lou^a- rithms of each side, and employing the Expon. Th (loH), Sec.
SCALES OF NOTATION.
342 It" iVbe a whole number of h-{- I digits, and /• the radix of tlie scale, ^<' = JJ,,'" +I>u-xr"-^ +p„_,r"-'-+ ... -^j>ir-\-p,,, where j^„, 2>„_i, ...2>„ are the digits.
343 Similarly a radix-fraction will be exju'essed by where 2>i, p.,, cVe. are the digits.
ExAMi'LKS : o-12l! in the scale of 7 == 3 . 7'' + •!■ . 7- + - • 7 + (J ;
•104o in the same scale = - + ^., + -,- + '* . 7 r 7'^ 7'
344 Kx.— To transform :U2G8 from the scale of 5 to the scale of 11.
RuLK. — JJii'ide sucrcssiiu-Jij hi/ tlw iinr radix.
11 3426S 111 1348 -< 111-40-3
1—9 Result \[)'3t, in which t stands fm- Id.
108 ALGEBRA.
345 Ex.— To transform -tOcl from the scale of 12 to that of 7, e standing for 11, and f for 10.
Rule. — Multiply successively hy the neio radix.
•tOel
7
5-i657
1
6-1931
7
1-0497 7
0-2971 Result -5610
346 Ex. — In what scale docs 2f7 represent the number 475 in the scale of ten ?
Solve the equation 27-- + 10;- + 7 = 475. [178
Result >• = 13.
347 The sum of the digits of any number divided by ?' — 1 leaves the same remainder as the number itself divided by r— 1 ; r being the radix of the scale. (401)
348 The difference between the sums of the digits in the even and odd places divided by r-\-l leaves the same re- mainder as the number itself when divided by r + 1.
THEORY OF NUMBERS.
349 If ft is prime to b, , is in its lowest terms. b
Proof. — Let = — i, a fiaction in lower terms. b 1)^
Divide a by «,, remaimler n., (juotieiit 5,,
h by 6,, remainder h., quotient 7, ;
and so on, as in liiidinii- the H. C. F. of a and a„ and of b and i, (see 30).
Let «„ and b„ be the highest eoninion factors thus determined.
THEORY OF NUMBERS. 100
Tlu'ii, hocaiise — = ' , .-. = ■'•'=', (70;
6 ^, /' '' — Q\''\ "i
and so on ; thus - = ' = • = txc ~ 7 •
T hero ft )!■(.' u and b arc C(|uimnlti|)k's of a„ and b„ ; that is, a is not prime to b if any fraction exists in lower terms.
(t (I
h ('(jiiiimiltiples of a and />.
350 It" " is prime to A, and -^= '.-; then n' and //arc
Pkoof. — Let - reduced to its lowest terms be ^ . Then ^- = — , and, b 1 1 '>
since p is now prime to 7, and a prime to /', it folhiws, by ;M-ft, that ^ is neither greater nor less than ; that is, it is (.Hjual to it. Therefore, &c.
351 If '^^ is divisible by c, and a is not ; then h must be.
II T , (lb a 0
Pi;nOl'.— Let =7; •■• ' = ■
c c l>
Hut '( is prime to c ; therefore, by last, b is a multiple of c.
352 If <^f Jind h be each of them prime to r, ah is i)rime to r. [By (351).
353 If abed... is divisible by a prime, one at least of the factors a, h, c, &c. must also be divisible by it.
Or, if p be prime to all but one of the factors, that factor is divisible by j>. (:ir.l)
354 Therefore, if a" is divi.sihle by ^^ i> cannot be jirime to '/ ; and if j) be a prime it must divide a.
355 If " is prime to h, any power of <i is prime to any } tower of It.
Also, if a, 0, c, &c. are prime to each otlier, the product of any of their powers is prime to any other protluct of their powers.
110 ALGEBRA.
356 No expression with integral coefficients, siicli as A + B,e + Cx' -h . . . , can represent primes only.
Proof. — For it is divisible by x if ^1 = 0 ; and if not, it is divisible by A, when x=: A.
357 '^^^G nnmber of primes is infinite.
Pkoof. — Suppose if possible p to be the greatest prime. Then the pro- duct of all primes up to p, plus unity, is either a prime, in which case it would be a gieater prime than p, or it must be divisible by a prime ; but no prime up to p divides it, because there is a remainder 1 in each case. Therefore, if divisible at all, it must be by a prime greater than p. In either case, then, a prime greater thanp exists.
358 If (I be prime to h, and the quantities a, 2a, oa, ... (b—i) a be divided by h, the remainders will be different.
Proof. — Assume ma — nb = ma — )ib, iii and n being less than b,
"" - "~"' Then by (350).
b m — in
359 A number can be resolved into prime factors in one way only. [By (353).
360 To resolve 6040 into its prime factors. Rule. — Divide hij the prime numbers snccessiveli/.
2x51 5040
2 1 504
21 252
2 1 126
7 1 63 3L9_
3 Thus 5040 = 2'. 3-.. 5. 7.
361 Required the least multiplier of 4704 wliicli will make the product a perfect fourth power.
By (19G), 4704 = 2'. 3. 7-.
Then 2'. 3'. 7- x 2'. 3». 7- = 2». 8'. 7' = 84',
the indices 8, 4, 4 being the least multipli's i)f I which :ire not less than 5, 1,2 I'espcctively.
Tlius 2". 3'. 7- = 3584 is the multiplier reciuired.
TllEUliY UF MJMUl'JIiS. Ill
362 All mmilHTS :uv ..f one of tin- forms 2// <.|- 2//-f 1
•2n(n'2n — \
„ ,, ',\n ov:\n±\
,, ,, 1// or l//il oi- I// + 2
,, ,, ly/ or ly/±l or 4/<— 2
,, ,, r)/M)i- ."iz/il or r>//i-
aiul so oil.
363 AH square numbers are of tlie form .w< or .u/il-
l'i;00K. — By squarino; tlii" lonns o/t, .">» ± 1, I'tuzt-. wliii-h ciniiitrelieiHl all numbers whatever.
364 All cube numbers are of the form "Jn or 7^'dzl- And similarly for other powers.
365 The highest power of a prime jh which is contained in tlu> i)roduct III ! , is the sum of the integral parts of
m m m p
1> F V
For there are ' factors in //^ ! which p will divides '., which
it will divide a second time ; and so on. The successive divisions are eejuivalent to dividing b}'
ExAMi'LK. — The hitrliesfc power of 8 which will divide 29!. Heio the
factors 3, <3, l», 12, 15, 18, 21, 24, 27 can be divided by 'A. Their ntinilier is
'" I
"■ = !> (the inte,t,M-al pai-t).
Till' factors it, 18, 27 can be divided a second time. Their nmnlx'r is
"'_ = ;5 (the integral part).
One factor, 27, is divisil)le a third time. "7;^ = 1 (intei^'ral i):irt).
9-f 3+ 1 = 13; that is, 3" is the highest power of 3 which will divide 29!.
366 'I'hc [)roduct of :mv /• consecutive integers is divisiljlc byr!.
PkooK: «("-!) ••• (»-'•+!) is necessarily an inteu'cr, by ('.'<;).
112 ALCIEBEA.
367 If ^^ be a prime, every coefficient in the expansion of
{a-\-hy\ except tlie first and last, is divisible by n. By last.
368 If " l)t> a prime, the coefficient of every term in the ex- pansion of {(i-\-h-\-c ...)", except a", &", &c., is divisible by n.
Proof.— By (367). Put /3 for (6 + c+ ...).
369 Frrmaf.s Theorem.— U p be a prime, and N prime to p ; then iV^"^ — 1 is divisible by p.
Proof : W={l + \ + ...y = N^Mp. By (368).
370 If V be any number, and if 1, r/, ^>, c, ... (p — 1) be all the numbers less than, and prime to p ; and if n be their number, and x any one of them ; then ,if — 1 is divisible by p.
Proof. — If x, ax, hx ... (p — l)x be divided by p, the remainders will be all different and prime to^ [as in (358)] ; therefore the remainders will be 1, a, b, c ... (p — l) ; therefore the product
x"ahc... ip—l) = ahc ... {p — \)+Mp.
371 Wilson's Theorem.— If p be a prime, and only then, l + (^— 1) ! is divisible by p.
Put j) — 1 for r and n in (285), and apply Fermat's Theorem to each term.
372 If i' be a prime = 2y« + l,then {vlf + {-iy is divisible hy p.
Proof. — By multiplying? together equi-di.'^tant factors of (j» — 1) ! in Wilson's Theorem, and putting 2n + l for /).
373 \,Qt N =(('' !>''(''' ■'• in prime factors ; the number of in- tegers, including 1 , which are less than u and prime to it, is
Proof. — The number of intogcrs piimo to N contained in ri" is n"- Similiirly in //", /•'•, &c. Take the ])r()duct of those.
TUEORT OF NUMBERS.
113
Also tlio miinhcr of intcfjfors less tliaii mikI ])i-imo to (Xx ^fxScc.) is the ])roduct of the coiTcspoiuliiig miiuhcrs for X, ^[, &c. separately.
374 The number of divisors of N, incliidiiif^ 1 and ^V itself, is = (y + l) (v 4-1) (/' + !) ...• For it is equal to the number of terms in the product
(l+./ + ...+r7'')(l+/.-h...+^")(l+r+...+'")---<-'tc.
375 The number of ways of resolving N into two factors is half the number of its divisors (374). If the number be a S(juare the two equal factors must, in this case, be reckoned as two divisors.
376 If the factors of each pair are to be prime to each other, put j;, 7, r, &c. each equal to one.
377 The sum of the divisors of ^V is
a^^-'-l h'^^'-^ c^-^'-l a-l ' h-1 ' c-1 '"
Proof. — Bj the product in (-374), and by (85).
378 If 7^ be a prime, then the j? — 1*** power of any number is of the form mj^ or )fij)-\-l. By Fermat's Theorem (3(30).
Ex. — The 12"* power of any number is of the form 13fM. or 12m-\-l.
379 To find all the divisors of a number ; for instance, of 50-1.
I.
II.
1
504
2
2
252
2
-i
126
2
8
63
3
3
6
12
2J,
21
3
9
18
3G
72
7
7
7
14
28
56 21 42
84
168
63
126 252 504
ExprANATiON. — Kesolvc 504 into its priino factors, placing thi-m in column 11.
114 ALGEBRA.
The divisors of 504 are now formed from tlie numbers iu column II., and placed to the right of that column in the following manner : —
Place the divisor 1 to the right of column II., and follow this rule — Multiply in order all the divisors which are written down by the next number in column II., which has not already been used as a multiplier : place the first netv divisor so obtained and all the folloiving products in order to the right of column II.
380 >^r tlie sum of the r^^' powers of tlio first n natural numbers is divisible by 2}i-\-l.
Proof : x {x"- V) (x" - 2") . . . (»' - »0
constitutes 2» + ] factors divisible by 2w+l, by (36G). Multiply out, re- jecting X, which is to be less than 2ji-|-l. Thus, using (372),
x'"-S,x"'-' + S2X-"-'- ... S,,.,x-' + (-iy([ny = M(2n + l).
Put 1, 2, 3 ... (« — 1) in succession for x, and the solution of the (n— 1)
equations is of the form Sr = M{2n + l).
THEORY OF EQUATIONS.
FACTOKS OF AN EQUATION. iicncral form of a rational integral equation of the w^'' degree.
The left side will be designated /(,/•) in the following summary.
401 If /('■) bt-' divided by « — rt, tbe remainder will be /(")• By assuming /(,(;) = P {x — a)-\-lL
402 If « be a root of the equation /(.i;) = 0, tlien/(rt) = 0.
403 To compute /(rt) numerically; divide f{.r) by x — a, and the remainder will bef{a). [101
404 Ex^^MPLE.— To find the value of 4x''-2x' + l2x'-x'' + 10 when x = 2. 4-3 + 12 +0 -1 +0 +10
8 + 10 + 4-i + 88 + 174 + 34.8
4 + o + 22 + -lrl + 87 + 174 + 358 Thus /(2) = 358.
If a,h,c...h be the roots of the eciuation f {(r) = 0 ; then, by (401) and ( 1U2),
405 /Or) =7>o (.t-^/) G^-'>) O'-O ... ('^-/O.
By multiplying out the last cijuation, and equating coefficients with cquatiou (100), cousidcriiig j'u = 1, the following results lu-o obtained :—
116 THEORY OF EQUATIONS.
406 —ih = the sum of all tlie roots of /(a?).
_ ( the sum of the products of the roots taken ^'" ~ \ two at a time.
_ ( the sum of the products of the roots taken ~~i^ ~ \ three at a time.
X 1 y _ ( the sum of the products of the roots taken ^ /ir — "^ rata time.
( — l)"j?,j = product of all the roots.
407 The number of roots of /(.^') is equal to the degree of the equation.
408 Imaginary roots must occur in pairs of the form
a + /3v/^, a-i3\/^.
The quadratic factor corresponding to these roots will then have real coefficients ; for it will be
■x'-2ax + a-+ii\ [405, 226
409 If /(■'«) be of an odd degree, it has at least one real root
of the opposite sign to p,j.
Thus a;^ — 1 = 0 lias at least one positive root.
410 If /('O ^Q of an even degi^oe, and jhi negative, there is at least one positive and one negative root.
Thus a;'*— 1 has +1 and —I for roots.
411 If several terms at the beginning of the equation are of one sign, and all the rest of another, there is one, and only one, positive root.
Thus x^ + 2x*-\-Sx^ + x'^ — 5x—4< = 0 has only one positive root.
412 If all the terms are positive there is no positive root.
413 If all the terms of an even order are of one sign, and all the rest are of another sign, there is no negative root,
414 Thus **— a;" + «'—;« + 1 = 0 has no negative root.
DISCRIMINATION OF ROOTS. 1 1 7
415 If :>11 tlio indices are even, aiul all tlie terms of the same sijjfii, there is no real root; and if all the indices are odd, and all the terms of the same sign, there is no real root but zero.
Thus x* + x^ + l = 0 has no real root, and x'^ + x^ + x = 0 has no real rout but zero. In this last equation there is no absolute term, because such a terra would involve the zero power o( x, which is even, and by hypothesis is wanting.
DESCARTES' RULE OF SIGNS.
416 In the following theorems every two adjacent terms in /(.r), which have the same signs, count as one " continuation of sign"; and every two adjacent terms, with different signs,
count as one chanw of siefu.
417 /(<'■)' multiplied by (■/.■ — a), has an odd number of changes of sign thereby introduced, and one at least.
418 ./* (c) cannot have more positive roots than changes of sign, or more negative roots than continuations of sign.
419 Wlien all the roots of f{.r) are real, the number of positive roots is equal to the number of changes of sign in f{.r) ; and the number of negative roots is equal to the number of changes of sign in/(— .r).
420 Thus, it being known that the roots of the equation
a;*-10.t'' + 3o.c2-50.c + 24 = 0 are all real ; the number of positive roots will be equal to the number of changes of sign, which is four. Also f(—x) =x*+lOx'^ + 3bx- + 50x + 2i! = 0, and since there is no change of sign, there is consequently, by the rule, no negative root.
421 If the degree of /(./•) exceeds the number of changes of sign in f{x) and /(—a;) together, by /t, there are at least /j. imaginary roots.
422 If, between two terms in /(,/■) of the same sign, there be an odd number of consecutive terms wanting, then there must be at least one more than that number of imaginary roots ; and if the missing terms lie between terms of different
118 TEEORY OF EQUATIONS.
sign, there is at least one less than the same number of imaginary roots.
Thus, in the cubic equation x^ + 4x — 7 = 0, there must be two imaginary roots.
And in the equation x°—l =■ 0 there arc, for certain, four imaginary roots.
423 If an even number of consecutive terms be wanting in f{x), there is at least the same number of imaginary roots.
Thus the equation x^ + 1 — 0 has four terms absent ; and therefore four imaginary roots at least.
THE DERIVED FUNCTIONS OF f{.v).
Rule for forming the derived functions.
424 Multiply each term hy the index of x, and reduce the index by one; that ^s, differentiate the function with respect to X.
Example. — Take
/ («) = x^+ a;*+ a;'- x'-x-l
f (x) = 5x'+ 4x^ + dx"" - 2a; - 1
f(x) = 20x' + 12x' + 6x-2
f(x) = 60a;2 + 24« +6
f(x) = 120x+24.
f (x) = 120 /' (*)j /' ('''')y <^c. are called the first, second, &c. derived functions of/ (a;).
425 To form the equation whose roots differ from those of f{x) by a quantity a.
Put x = y-\-a infix), and expand each term hy the Binomial TJieorem, arranging the results in vertical columns in the foU lowing manner : —
/(a + 2/) = (« + 2/)'+(a + 2/)H(a + 7/)»-(a + y)=-(a + 7/)-l
a"
+ a* +
a" -
a? -
- a -1
+ ( 5a*
+ 4a'' +
3a^ -
2a -
• l)i/
+ (10a»
+ Go? +
3a -
1)^^
+ (10a»
+ 4a +
1)/
+ ( 5a
+ 1)/
+ f
TBANSFOJIMATION OF AN EQUATION.
119
('om paring this result with that seen in (421), it is seen that
426 /•("+.'/) =/(")+/'(")//
\A lA LI L±
80 tliat tlie coefficient generally of ?/'' in tlie transformed
equation is ' , ^ ^
r
427 To form tlie equation most expeditiously when a has a miinerical value, diride f{,i') continuously hi/ x — a, and the succcssli-e remainders ivlU furnish the coefficients.
ExAiirLE. — To expand f(y + 2) when, as in (425), f(x) = Z' + x' + x'-x'-x-l. Divide repeatedly by x — 2, as follows : —
1 + 1 + 1 - 1 - 1-1 + 2 + 0 + 14 + 2G +50
1 + 3 -f 7 + -f 2 +10 +
13 +
31 +
1 + 5 +17 + 47 + 2 +14 + 02
+ 119
+ 49=/(2)
= rC2)
1 + 7+31 + 2 +18
+ 109 =
(i)
11
1 + 9 I +49 =
+ 2 I
14
_/'(2)
|3
+ 11
1 - f(^\ |5
That these remainders are the required eoeflicients is seen by inspecting the form of the equation (420) ; for if that equation bo di- vided by x — a = II repeat- edly, these remainders aro obviously produced when a = 2.
Thus the equation, whoso roots are each less by 2 than the i-oots of the proposed equation, is ?/+ll?/' + 49(/"'+109//'+ 119// + 49 = 0.
428 To make any assigned term vanish in the transformed equation, a must be so determined that the coeffieient of tliat term shall vanish.
Example. — In order that there may be no term involving if in equation (420), we musi have /*(a) = U.
Find /'(a) as in (424); thus 120« + 24 = 0; .-. a = -\.
The equation in (424) must now be divided repeatedly by a; + | after the manner of (427), and the resulting equation will be minus its seeoud term.
120 THEORY OF EQUATIONS.
429 Note, tliat to remove the second term of tlie equation
y(-t') = 0, tlic requisite value of a is = — ^ ^ ; tliat is, the
coefficient of the second term, with the sign changed, divided hy the coefficient of the first term, and hy the numher expressing the degree of the equation.
430 To transform /'(''') "^^o an equation in y so that // -■ <p {.<■), a given function of x, 2^ut x = (j)~'^{y), the inverse function of y.
Example. — To obtain an equation whose roots are respectively three times the roots of the equation x^ — Gx + 1 = 0. Here 2/ = 3a; ; therefore x = —, and the equation becomes ^ — -^ + 1 = 0, or ?/— 54^ + 27 = 0.
431 To transform /(,v) = 0 into an equation in which the coefficient of the first term shall be unity, and the other coefficients the least possible integers.
Example. — Take the equation
288x^ + 24>0x''-176x- 21 = 0.
Divide by the coeflacient of the first term, and reduce the fractions ; the
5 11 7
equation becomes x^ + —• x^— — x — KB — ^'
Substitute -^ for x, and multiply by h^ ; we get
Next resolve the denominators into their prime factors,
3 57^ , life' _ yic" ^Q
'^ "^2.3^ 2.3'^^ 2^3 The smallest value must now be assigned to h, which will suffice to make each coefficient an integer. This is easily seen by inspection to be 2'-. 3 = 12, and the resulting equation is i/ + 10if — 88y — l26 = 0,
the roots of which are connected with the roots of the original equation by the relation y = I2x.
EQUAL ROOTS OF AN EQUATIOli^.
By ox})audiiin^ 7X''^ + ^') i^^ powers of ;■• by (i05), and also by (1'2()), and c(iiiating the coefficients of z in the two ex-
EQUAL nOOTS. 121
pansions, it is provctl tliat
tVoiii wliicli result it appears tliat, if tlie roots a, b, r, &c. arc all uTUMiual, /'(.*') and /'(,»■) can liave no common measure in- volving ,*•. If, however, there are r roots each equal t<^ d, s roots e(|ual to h, t roots equal to r, &c., so that
f{^v) = iK{,'-ay {.v-hy {.v-i'Y ... then
433 /(..■) = ^ + ;^;! + (-ia^ + *..;
and the greatest common measure <'f/((') and /'(,<■) will bo
444 {.v-(iy-' {.v-hy-' {.v-cY-\..
When .v, = a, /(..), /'{,'), .•■/'-'Oi') all vanish. Similarly when .v = b, &c.
Prdctiral mctlnnl offfiH/in^' the (-([Udl nKits.
445 Lft / (.0 = A', X: a1 X\ Xl . . . X;::, where
A', = product of all the ftictors like {x — <^^, A1= „ „ {x-a)\
Xl= „ „ (.-«)».
Find the greatest comraon measure of /(j-) and /'(f) = I'\ (.') say,
I'\(x)aud'F;Cv) = F,(x), F^(x) And F:(x) = F3(x),
Lastly, the greatest common measure of F,„.i(x) and /''„,-i(*) = I'\,0') = 1- Next perform the divisions
f(x) -i- F,(.r) = <p,(.r) .say, F,(x)-^F,(') ='P,{.r),
And, iinully, 0, (.,') -4- <Pi{x) = A',,
i''«.-.C'-)=V'..C'-) = A',,.. [T. 82.
1:
122 THEORY OF EQUATIONS.
The solution of the equations Xj = 0, X, = 0, &c. will furnish all the roots of/ (a;) ; those which occur twice being found from X.^; those which occur three times each, from X^ ; and so on.
446 If /('^) lias all its coefficients commensurable, X^^X^^X^, &c. have likewise their coefficients commensurable.
Hence, if only one root be repeated r times, that root must be commensurable.
447 III all the following theorems, unless othermse stated, /(-/•) is understood to have unity for the coefficient of its first term.
LIMITS OF THE KOOTS.
448 If the greatest negative coefficients in /(,/') and /( — <^') be j) and q respectively, thenp + 1 and —(^ + 1) are limits of the roots.
449 If x''-'' and x''-' are the highest negative terms in /(<i') and /( — aO respectively, {l + \/p) and —{l-\-^q) are limits of the roots.
450 If /-^ be a superior hmit to the positive roots of /( — j » then — will be an inferior limit to the positive roots of /(-f)-
451 If each negative coefficient be divided by the sum of all
the preceding positive coefficients, the greatest of the fractions so formed + unity will be a superior Hmit to the positive roots.
452 Neivtons tnethod.—Viit x=h-\-i/ in /(./•) ; then, by (420),
Take // so that /•(/')' /(^O^ fW yf'W ^^^ ^11 positive; then // is a superior limit to the positive roots.
453 According as /{a) andf{b) have the same or different signs, the number of roots intermediate between a and b is even or odd.
INTEGRAL BOOTS. 123
454 Rollrs Thcorrm.—Ono real root of the eriuation f (r) lios iH'lwoeii every two adjacent real roots of /(./')•
455 ('OH. I.— /(•'•) cannot have more than one root gi-eatcr than the greatest root in /(./■); or more than one less than llic least root in/'(,r).
456 CoR. 2.— If f(.r) has m real roots, /'•(') has at least III — /■ real roots.
457 CoH. 3.— If f(u') lias /t imaginary roots, f{.r) has also /i at least.
458 CoK. -1.— If a, /3, y ... K be the roots of /(.<) ; then the 11 umber of changes of si^-n in the series of terms
f{^), /W, /(«, /(7).-/(-^) is equal to tlie number of roots oif{.c).
NEWTON'S METHOD OF DIVISORS.
459 To discover the integral roots of an equation.
ExAMTLE. — To ascertain if 5 be a i-oot of 5 ) 105
«*-6x» + 80x*-i;6.i- + 10-. = 0. 21
— 1/b
If 5 be a root it will divide 105. Add the quotient to the ^ ) —\55
next coefficient. Result, —155. —'Si
If 5 bo a root it will divide -155. Add the quotient to _86
the next coefficient ; and so on. 5 ) 55
U — 6
If the number tried be a root, tlic divisions will be effectiblo to the end, and the last quotient will bo -1, or —}>o, if/u he >- n _ e
not unity. 2_Z_.
-1
460 In employing this method, limits of the roots may first be found, and divisors chosen between those limits.
461 Also, to lessen the number of trial divisors, take any integer m ; then any divisor a of the last term can be rejected if a — m does not divide /(///).
In practice take /// = -f-1 and — 1 .
To find wliether any of the roots determiiK'd as above are repeated, divide f{x) by the factors correspijndiTig to them, and then applv tlie method of divisors to the resulting ecpiation.
124 THEORY OF EQUATIONS.
Example. — Take the equation
Putting X = 1, we find/(l) = —24. The divisors of 144 are
1, 2, 3, 4, G, 8, 9, 12, IG, 24, &c. The values of n — m (since rii = l) are therefore
0, 1, 2, 3, 5, 7, 8, 11, 15, 23, &c.
Of these last numbers only 1, 2, 3, and 8 will divide 24. Hence 2, 3, 4, and
9 are the only divisors of 144 wliich it is of use to try. The only integral roots of the equation will be found to be ±2 and ± 3.
462 If /('^') and F{X) have common roots, they are con- tained in the greatest common measure of /(a:) and F{X).
463 If /(■'') l^as for its roots a, (p {a), h, (jt {b) amongst others ; then the equations /(.*) = 0 and/[(^(A')j = 0 have the common roots a and h.
464 But, if all the roots occur in pairs in this ^vay, these equations coincide.
For example, suppose that each pair of roots, a and b, satisfies the equation a + h = 2r. We may then assume a — b = 2z. Therefore/ (2 + r) = 0. This equation involves only even powers of z, and may be solved for z'.
465 Otherwise: Let a& = z ; then /(;«) is divisible by {x — a){x — l) = x^ — 2rx + z. Perform the division until a remainder is obtained of the form Px + Q, where P and Q only involve z.
The equations P = 0, Q = 0 determine z, by (462) ; and a and I arc found from a + b = 2r, ab = z.
EECIPROCAL EQUATIONS.
466 A reciprocal equation has its roots in pairs of the form a, — ; also the relation between the coefficients is
Pr =Vu-r^ OY else p, = —Pn-r-
467 A reciprocal equation of an even degree, with its last term jwsit ire, may be made to depend upon the solution of an equation of half the same degree.
BINOMIAL EQUATIONS. 125
468 KxAMiM.K : •l-,/''^-24,r''-f57r'-7o./-''-f :)7./--2-1../ + t = 0 is a rocii)rtic;il ('(lualioii of :iii cncii dcLi-i'd', with its last term positive.
Any reciprocal equation w liicli is 7iot of this form may bo reduced to it hi/ diridiiuj hi/ ,/ -f I //" the la.sf term l>e posit Ire ; (UkI, 1/ till' lust term he neijafirc, l>i/ diriduuj hi/ ,r— 1 or r' — ], so us to liriiiij thi' t'tjiiiition tn mi < rm. tlrr/ree. Then proceed in tlie foHowinn' manner : -
469 First brin<^ toovther e(|nidistant ti'rins, and diviih' the equation by ,/'"*; tlius
By })utting ,v -\ = //, ;ind by making- repeated use of tlie
relation ,/'- -| =. i,r -\ ) "~ -> ^^'<^' eipration is reduced to
a cubic in ?/, the degree being one-half that of the original equation.
]"*ut ;) for .r -| , and p„, for x„^-\ .
470 Tlie relation between the successive factors of the form j)„, may be exi)ressed by the e(juatioii
471 'I'he equation for ji,,,, in terms of ^>, is P,u = p'-nip'" -+ \ ,, p'" '- ...
_L f_i V ^" f»/ — >•— 1) ... (in — '2r-\-\) „,_..^,
I>y (^')i")), putting 7 = 1.
BINOMIAL EQUATIONS.
472 If a be a root of .r" — 1 = 0, then a"* is likewise a root where m is any positive or negative integer.
473 If " be a root of ./" + 1 = 0, then a-'"^' is likewise a root.
126 THEORY OF EQUATIONS.
474 If ^''' ai^d ^2- be prime to each other, x'"' — l and x^—1 have no common root but unity.
Take iim — qn = 1 for an indirect proof.
475 If n be a prime number, and if a be a root of ct;" — 1 = 0, the other roots are a, a^, a^ ... a'\
These are all roots, by (472). Prove, by (474), that no two can be equal.
476 If ^i be not a prime number, other roots besides these may exist. The successive powers, however, of some root will furnish all the rest.
477 If r*/'— 1 = 0 has the index n = m2)q; m, ]}, q being prime factors ; then the roots are the terms of the product
(l+a + a^+ ... +a-^)(H-/3 + /3'--[- ... +/3''-^)
X(l + 7 + 7'+ - +7'"')> where a is a root of «'"— 1, /3 „ x^-l,
7 » »''-!'
but neither a, /3, nor 7 = 1. Proof as in (475).
478 If n = m^ and
a be a root of x""—! = 0,
(i „ x^>^-a = 0,
7 » r.--|3=0;
then the roots of x''—l = 0 will be the terms of the product
(l+« + a^+ ... +„-^)(l+/3 + /3-^+...+r-^) X(l + 7 + 7^"+... +7""')-
479 a^" + 1 = 0 may be treated as a reciprocal equation, and depressed in degree after the manner of (468).
480 The complete solution of the equation
.1 - -1 = 0
is obtained by De Moi\Te's Theorem. (757)
The 71 different roots are given by the formula
0? = cos ± V — 1 sill
71 n
in which r must have the successive values 0, 1, 2, 3, &c., concluding with ^ , if n be even ; and with -~ , if // be odd.
CUBIC EQ UA TIONS. 1 2 7
481 Similarly the n roots of the ofiuatiuu
.r" + 1 = 0 are given by the formula
n u
r taking the successive values 0, 1, 2, 3, &c., up to ' ~^, if n be even ; and up to ' , if )i be odd.
482 'I'he number of different values of the product
is equal to the least common multiple of m and n, when m and 7/ are integers.
CUBIC EQUATIONS.
483 To solve the general cubic equation
a;^ + jj.r + qx -f /• = 0. Remove the term j^at^ by the method of (429). Let the trans- formed equation be .v'^-\-q,r-\-r = 0.
484 Cardan s mcfhoiL — The complete theoretical solution of this equation by Cardan's method is as follows : —
Put x = i/-\-:i (i.)
yH,v^ + (3v.v + 7)(y + ,v) + r = 0. Put Si/:: + q = 0; .'. ^ = - 3^
Substitute this value of //, and solve the resulting quadratic in //^. The roots are equal to 1/ and .r* respectively ; and we have, by (i.),
485 r
{-iWf+j^r+i-^-vj+f;}'
128 THEOBY OF EQUATIONS.
Tbe cubic must have one real root at least, lij'- (400).
Let »i be one of the three values of j ^ "^ \/ TT "*" '^ ( ^' ^°^ " "°® I
of the three values of j ^ \/ X "^ 9" [ '
486 Let 1, n, a- be the three cube roots of unity, so that
a=-l +1- y^, and ci' = - 1- - L yZs. [472
487 Then, since Viu^ = my I, the roots of the cubic will be
m + ii, am-ta'-n, u'in-\-n)i. Now, if in the expansion of
I 2 ^V 4 ^ 273
by the Binomial Theorem, we put
fx = the sum of the odd terms, and
V = the sum of the even terms ;
then we shall have m = /u. + y, and « = ^ — v;
or else m = /u + v/ — 1, and n = fji — y v— 1 ;
according ^^ \/ 'T '^ §^ i^ ^^^^ ^^ imaginaiy.
By substituting these expressions for in and n in (487), it appears that —
488 (i-) If V" + ^ ^® positive, the roots of the cubic will be
2/^, —/i + >'%/— 3, —fi — yv—o.
r- cj^ (ii.) If -r "^ 97 ^^ negative, the roots will be
2/1, —fx + y^S, — /J — vn/S.
,2 3
(iii.) If + t^ = 0, the roots are 4 27
2ot, —:?/!, -m; since m is now equal to fi.
489 '/^/'^' Trigonometrical method. — The equation
.1'^ + r/.r + r = 0 may be solved in tlie following manner, by Trigonometry,
when -p + 77= is negative. 4 27
Assume <6' = ?; cos a. Divide the equation by n^; thus
cos'' a + -2- cos a -\ ^ = ^^
But cos« a - -? cos a - -^ = 0. By ((357)
4 4
TilQVADRATW EQUATIONS. 129
Equato coefficients in tlic two (M|n;itions ; the result is
n must now be found ^villl the aid of tlie Trigonometrical tabh's.
490 The roots of the cubic will l)e
n cos a, n cos (jTr+a), n cos (^tt— a).
.2 3
491 Observe that, according as -- + ^- is positive or nega- tive. Cardan's method or the Trigonometrical wall be practi- cable. In the former case, there will be one real ami two hnaginary roots ; in the latter case, three real roots.
BIQUADRATIC EQUATIONS.
492 Descartes' Solution. — To solve the equation
.v' + qA' + r.v -\- s = 0 (i.)
the term in .r' having been removed by the method of ('t29).
Assume (.t.-+(u'+/) {.i--i\v-\-^-) = 0 (ii.)
Multiply out, and equate coefficients witli (i.) ; and t1ie fol- lowing equations for determining /", g, and e are obtained
ir+/='/ + ^'% .ir- /=A Kf=''*' i'''')
493 ^.«4.2rye'+(r/-4v)e2-j- = 0 (iv.)
494 The cubic in t? is reducible by Cardans method, when the biquadratic hits two real and two imaginary roots. For proof, take « ± //5 and — a ± y as the roots of (i.), since their sum mu.st bo zero. Form the sum of eucb pair for tlio values of e [see (ii.)]. and "PP'Y t^° ^^^^^^ ^" ('i88) to the cubic in e*.
1/ the biquadratic has all its roots real, or all imaginary, the cubic will liave all its roots real. Take n ± //3 and —a ± iy for four imaginary roots of (i.), and form the values of e as before.
495 If' «'> f^f y' ^e the roots of tfie cubic in e', tlie roots of tlie biquadratic will he _i(„+/5 + y), l(a+/3-y), 4(/9 + y-n), ^ (y + a-/3).
130 THEORY OF EQUATIONS.
For proof, take w, x, y, z for the roots of the biquadratic; then, by (ii.), the sum of each pair must give a value of e. Heuce, we have only to solve the symmetrical equations
1/ + 2 = CI, w + ,7;=— a,
Z -I- iC = /3, ?o + ;/ = — /j,
a- + ?/ = y, w-\-z = —y-
496 Ferraris solution. — To the left member of the equation x^+lKv^-\-qx^-\-ra:-\-s = 0,
add the quantity ax^ + bx + —, and assume the result
= (,.+|,+,„y.
497 Expanding and equating coefficients, the following cubic equation for determining m is obtained
8m^—4iqm^-\-{2j)r—8s)m-\-4!qs—2^^s — r = i).
Then x is given by the two quadratics
2 , » , , 2(Lv-{-b
^ + 2 '*' + '*' = ± -TvTT
498 The cubic in m is reducible by Cardan'' s method ivhen the biquadratic has two real and tioo imaginary roots. Assume a, /3, y, B for the roots of the biquadratic ; then aft and yB are the respective products of roots of the two quadratics above. From this find m in terms of aftyS.
499 Elders solution. — Remove the term in x^; then we have .V'*' + q.v"^ -\-r.v-\-s = i).
500 Assume x = i/-\-z-\-u, and it may be shewn that i/, z^^ and u^ are the roots of the equation
fA.±f-4-'^"'~^'t-^=0 ^ 2 ^ ^ 10 ^ (>i
501 The six values of y, z^ and //, thence obtained, are
restricted by the relation yzii. =
Thus X = //-!-;.■ + /< will take four different values.
COMMENSURABLE ROOTS. 131
COMMENSURABLE ROOTS.
502 '1'" t'""l tlio commonsuraUlo roots of an ofiiiation. First transform it by pultinjr •'' = 'ir "^^^ ^^^^ equation of
tlic form .r"+/>i.r"-'+;>.>.r"--+ ... +/>„ = 0,
liaving j>o = 1» iiii^l tlio remaining coefHcients integers. (l-Gl)
503 This ecpiation cannot have a rational fractional root, and tlie inte<]:ral roots may be found Ijy Newton's method of Divisors (451)).
These roots, divided each by h, will furnish the commen- surable roots of the original equation.
504 Example. — To find the comrnensurablo roots of tlie equation
8 1 /' - 20 7x' - 9x» + 89.r + 2,r - 8 = 0. Dividing bj 81, and proceeding as in (431), \vc find the requisite substitu- tion to bo a; = -^.
The transformed equation is
,/_23/-V + 801)/- + lG2//-5832 = 0. The roots all lie between 24 and —34, .by (451). The method of divisors gives the integral roots G, —4. and 3. Therefore, dividing each by 9, we find the commensurable roots of the original equation to be 3, — f , and |.
505 To obtain the remaining roots ; diminish the transformed equation by the roots G, —4, and 3, in the following manner (see 427) : —
1_23- 9 + 801 + 1G2- 5832 6— 102— GGG + 810 + 5832
6 -4
1-17-111 + 135 + 972 - 4+ 84 + 108-972
-21- 27 + 243 3_ 54-243
1_18- 81
The depressed equation is therefore
if - 18(/ - 81 = 0. The roots of which arc 0 (1+ \/2) and 9 (1— v/2) ; and, consequently, the incommensuiable roots of the proposed equation are 1+ V'l and 1— v/2.
132 THEORY OF EQUATIONS.
INCOMMENSURABLE ROOTS.
506 Sturm's Theorem. — lff(x), freed from equal roots, bo divided by /('>')> and the last divisor by the last remainder, changing the sign of each remainder before dividing by it, until a remainder independent of x is obtained, or else a re- mainder which cannot change its sign; then /(a^), /'('<-')' ^^^ the successive remainders constitute Sturm's functions, and
are denoted by f{a-) , / (<r) , f^ {.v) , &c /« Gv) •
The operation may be exhibited as follows : —
M^r) = q,f,{.v)-f,{.v),
507 Note. — Any constant factor of a remainder may.be rejected, and the quotient may be set down for the corres- ponding function.
508 An inspection of the foregoing equations shews —
(1) That /„, (ft') cannot be zero; for, if it were, /'('^O ^^^ /i (x) would have a common factor, and therefore /(<^') would have equal roots, by (432).
(2) Two consecutive functions, after the first, cannot vanish together ; for this would make/*„ (x) zero.
(;3) AVhen any function, after the first, vanishes, the two adjacent ones have contrary signs.
509 If, as X increases, f(x) passes through the value zero, StnrrrC s Junctions lose one change of sign.
For, before ./'(-O tf^kes the value zero, /(a-) and/, {x) have contrary signs, and afterwards tliey have the same sign; as may be shewn by making h small, and changing its sign in the expansion off{x + li), by (420).
510 If d-nij other of Sturm's functions vanishes, there is neither loss nor gain in the number of changes of sign.
This will appear on inspecting the equations.
511 Ri:sui;r. — The nnmhvr of roots off(,v) hrtu'cen a and h is equal to the difference in the number of cluoiges of sign in Sturm's functions, when x = a and when x = b.
INCOMMEN suit ABLE ROOTS.
133
512 <'"";• — Tlio total iiiiin])ci- ui" roots of /■(,*■) will he foiiiid by taking a = -\- cc mid h = — cc ; tho aU^n of each fuiictioii will then bo the same as that of its first term.
\VTien tho number of functions exceeds the degree of f{x) by unity, the two following theorems hold : —
513 If thr jir^t trrins hi all flii'fdnrtionii, aj'trr t lie first ^ are /losifirc ; all flu' roots off{j) arc real.
514 If the first terms are not all positive ; then, for every r/iiiiK/c of si(j)i, there will be a pair of imaginart/ roots.
For the proof put .r = + co and — oo, and examino the number of changes of sign in each case, applying Descartes' rule. (-tl6).
515 If ^ (•'') ^^^ no factor in common with /(<>'), and if <p (x) and _/"(,/') take the same sign when /(,<■) = 0; then the rest of Sturm's functions may be found from f{.f) and <p (</;), instead of /'(./■). For the reasoning in (509) and (510) will ajjply to the new functions.
516 If Sturm's functions be formed without first removing equal roots from /(,/'), the theorem aWII still give the number of distinct roots, without repetitions, between assigned limits.
For if/(.r) and /, (x) be divided by their highest common factor (see 444), and if the quotients be used instead of /(./■) and/, (.r) to form Sturm's func- tions ; then, by (olo), the theorem will ajjply to tho new set of functions, which will dilfer only from those formed froin/(,c) and/, (./.) by the absence of the same factor in every term of the series.
517 Example. — To find the position of the roots of the equation
x'-4x^ + x- + Gx + 2 = 0.
Sturai's functions, formed according to f(x)=x*—4a^+ x^+ Gx+ 2 the rule given above, are here calculated.
The first terms of the functions are all positive ; therefore there is no imaginary root.
The changes of sign in the func- tions, as X passes through integral values, are exhi- bited in the adjoin- ing table. There are two changes of sign lost while x passes from — 1 to 0, and two more lost while X passes from 2 to 3. There
A(^') =
2x»— G.c*+ x+ 3
A(^) =
5x--l(Jx- 7
Ai-^) =
X- 1
Ai^) =
12
x =
-2
-1
0
1
2
3
4
f(x) =
/,(.«) =
A(^) = A(^) =
1
+
+
+ + +
+ +
+
+ +
+
+ +
4- + + + +
+
+
+
No. of changes ) of sign )
4
4
2
2
2
0
0
134 THEOEY OF EQUATIONS.
are therefore two roots lying between 0 and — \ ; and two roots also between 2 and 3.
These roots are all incommensurable, by (503).
518 Fourier's Theorem. — Fourier's functions are the fol- lowing quantities f{x), f'{.v), f'i^v) /"(cv).
519 Properties of Fourier's functions. — As x increases, Fourier's functions lose one change of sign for each root of the equation /(.f) = 0, through which x passes, and r changes of sign for r repeated roots.
520 If any of the other functions vanish, an even number of changes of sign is lost.
521 Results. — The 7iumher of real roots of f{x) hetween a and j3 cannot he more than the difference hetween the numher of changes of sign in Fourier's functions when x = a, and the numher of changes ivhen x = j3.
522 When that difference is odd, the number of intermediate roots is odd, and therefore one at least.
523 When the same difference is even, the number of inter- mediate roots is either even or zero.
524 Descartes' rule of signs follows from the above for the signs of Fourier's functions, when x = 0 are the signs of the terms in /(«); and when a3 = oo, Fourier's functions are all positive.
525 Lagrange's method of approA'wiating to the incom-
mensiirahle roots of an equation.
Let a be tlie greatest integer less than an incommen- surable root oif{x). Diminish the roots of /(,r) by a. Take the reciprocal of the resulting equation. Let h be the greatest integer less than a positive root of this equation. Diminish the roots of this equation by h, and proceed as before.
526 Let a, h, e, &c. be the quantities thus determined ; then, an approximation to the incommensurable root oif{x) will be
the continued fraction x = a + -— -,-
h-\- f-l-
INCOMMENSURABLE ROOTS.
]3i
527 AV/c/o/<'.v mcfluK/ <tf(ii>pr(uinnfti<ni.- If r, hv a ((uaiitity a little less than one o£ the roots of the eciuatioii fU) = 0, so that /(r, + //) = 0; tlieu c^ is a first approximation to the value of tlie root. Also because
/(,-, + /,) =/(,.,) + /,/■{,-,) +^'/"(.-,) + &c (KG),
and h is but sniall, a second approximation to the root will be
In tlie same way a third ap]:»roximation may be obtained from Co, and so on.
528 Fourier's limitation of Newton s method. — To ensure that Ti, ^2, Cg, &c. shall successively increase up to the value Ti + h without passing beyond it, it is necessary for all values of X between c^ and Ci-\-h.
(i.) Thatf{;ii) andf'{,v) should have contrary signs. (ii.) Thatflx) and f" {x) should have the same sign.
Fia. L
Fia. 2.
A proof may be obtained from the figure. Draw the curve y=f(^x). Let OX be a root of the equation, and ON = Ci; draw the successive ordinates and tangents NPy PQ, QBy &c. Then OQ = c^, OS = Cg, and so on.
Fig. (2) represents Tg > OX, and the subsequent ai)proxi- mations decreasinof towards the root.
530 Newton s Rule for Limits of the Roots.— hot the co- ethcients of /(./) be respectively 'divided by the Binomial coefficients, and let ^o, (T,, a, ... o„ be the quotients, so that
f{x) = ao-'j" + uai,/;"-
+ ''t^"''
+ ... + » ",.-!»;+«
13G TUEOUY OF EQUATIONS.
Lot ^Ij, ^lo, ^Ig . . . An be formed by tlic law A^ = a^— ri^_i(X^+i. Write the first series of quantities over the second, in the fol- loA\dng manner : —
«0, rti, ('2, ((3 (^n-l, ««,
ylo, Ai, ^2, ylg .-l,,_i, A,,.
"Whenever two adjacent terms in the first series have the same sign, and the two corresponding terms below them in the second series also the same sign; let this be called a double 2Jermanence. AVhen two adjacent terms above have different signs, and the two below the same sign, let this be known as a variation-permanence.
531 Rule. — The number of double 'permanences in the asso- ciated series is a superior limit to the number of negative roots
The number of variation-permanences is a superior limit to the number of positive roots.
The number of imaginary roots cannot be less than the number of variations of sign in the second series.-
532 Sylvesters Theorem. — Let /(f + X) be expanded by (426) in powers of x, and let the two series be formed as in Newton's Rule (530).
Let P (X) denote the number of double permanences. Then P (X) ~ P [fi) is either equal to the number of roots of /(ft'), or surpasses that number by an even integer.
Note. — The first series may be multiplied by [n_, and will then stand thus,
/«(X), /-^X), [2/«-2(X), [l/"-^(^)-h/W-
The second series may be reduced to
On{^), r/„_,(x), r/„_,(x)...rr(x),
where G, (X) = {/'• (X)}^ - '^y^^ f-' (A) f^' (X).
533 Horner's Method. — To find the numerical values of the roots of an equation. Take, for example, the ecpiation
x'-4t^ + x^-\-6x-\-2 = 0,
and find limits of the roots by Sturm's Method or otherwise.
INCOMMENSUnAIiLi: HOOTS.
137
It has been shewn
in (ol7) that this eqnation has two
incominensiirablo roots
between 2 and
:?. Th(«
|)rocess of
calciil;
itiiig the least of these roots is here
exhibited.
-4
+ 1
+ 6
+ 2(2-414213
2
-4
—6
0
_2
-\i
0
A
20000
2
0
-6
-19584
0
-3
i?, — 60U0
A,
4160000
2
4 6'/ 100
1104 -4896
^U
-2955839
o
1204161UUU0
2
176 276
1872
^4
-11437245184
P, 40
B^ — 30240U0
604364816
4
192
68161
-566003348
44
468
-2955839
A
38361468
4
208
68723
-28285470
48
C'j 676UO
n, -2887116000
A
10075998
4
561
27804704
- 8485368
52
6bl(;i
-2859311296
A
1590630
4
562
27895072
7)., oGO
68723
B, -2831416224
1
563
139948
282843)1590630(562372
6G1
Cg 6i>2H600
— 283U01674
1414215
1
22576
139970
28284)176415
562
6951176 22592
B^ -282861704 700
2828)"
169706
1
6709
5G3
6973768
- 28285470
5657
1
22608
700
282) 1052
D3 5040
C, 0996376
B, -28284770
848
4
11
21
28) 204
5044
69974
-2828456
197
4
11
21
2) 7
5648
69985
B, -2828435
5
4
11
0
5652
C, 69996
4
CL 7
P, 5,656
Root =
: 2-414213562372.
METnOD.— Diniinisli tlie roots by 2 in tlic manner of (427). The resulting coefficients are indicated by .4,, J?„ C\, 7>,.
By Newton's rule (527), - {^r'A 5 ^^^^^ ^^' ~ V.' ** "" approximation to the romaininfT pait of llic root. Tliis gives '3 for the next figure ; '4 will bo found to be the eorrect one. The highest figure must bo taken which will not change the sign of ^1.
Diminish the roots by -4. This is accomplished most easily by affixing ciphers to J„ i^„ C'„ I\, in the manner shewn, and then employing 4 instead of -4.
Having obtained ^1.,, and ob.scrving that its sign is +, retrace the steps,
T
138 THEORY OF EQUATIONS.
ti'ying 5 instead of 4. This gives A., with a minus sign, thereby pi-oving the existence of a root between 24 and 2-5. The new coetficieuts are A.^, B.^, C.,, D.^.
— Y? gives 1 for the next figure of the root.
Affix ciphers as before, and diminish the roots by 1, distinguishing the new coefficients as A^, B^, C^, D^.
Note that at every stage of the work A and B must preserve their signs unchanged. If a change of sign takes place it shews that too hirge a figure has been tried.
To abridge the calculation pi'ocecd thus : — After a certain number of figures of the root have been obtained (in this example four), instead of adding ciphers cut ott" one digit from B^, two from C'^, and three from D^. This amounts to the same thing as adding the ciphers, and then dividing each number by 10000.
Continue the work with the numbers so reduced, and cut off digits in like manner at each stage until the D and C columns have disappeared.
Aj and Bj now alone remain, and six additional figures of the root are determined correctly by the division of A^ by By.
To find the other root which lies between 2 and 3, we proceed as follows : — After diminishing the roots by 2, try G for the next figure. This gives Jj negative; 7 does the same, but 8 makes J., positive. That is to say, f{2'7) is negative, and/ (2'8) positive. Therefore a root exists between 2*7 and 2-8, and its value may be approximated to, in the manner shewn.
Throughout this last calculation A will preserve the negative sign.
Observe also that the trial number for the next figure of the root given at
f(c) each stagfe of the process by the formula — , {, will in this case be always
^ I y ^ /(c)' ^
too great, as in the former case it was always too small.
SYMMETEICAL FUNCTIONS OF THE HOOTS OF AN EQUATION.
Notation. — Let a, h, c ... bo the roots of tlie equation
/(••'•) = 0.
Let .s,„ denote a'" + //" + . . . , tlic sum of tlie 7/^^'' powers of the roots.
Let .s,„,p denote a"'h^' + lr(iP-\-a'"<'''-{- ... througli all the permutations of the roots, two at a time.
Similarly let .9,„,p ,, denote (rh'\'''-\-a'"h^\V'-\- ... , taking all the permutations of the roots three at a time ; and so ou.
SYM}n:rRICAL FUNCTTONS of TIIF h'OOTS. 13!)
534 SUM>^ OF nil': /vmi7;a'.s' of tuf norrrs.
wliere m is less tliau n, the degree oiJ'{.i).
Obtained by expanding by division each term in the vahio of/'(.i) given at (432), arranging tlie whole in powers of .r, and equating coeiricieiits in llie result and in the value ofy^^r), found by differentiation as in (1-21).
535 If "i ^0 greater than //, tlie forimihi will be
Obtained by multii)lying /(./•) = 0 by .'""", substituting for .i; the roots a, h, c, &c. in succession, and adding the results.
By these formula? .<„ s.,, .«„ &c. may be calculated successively.
536 To find the sum of the negative powers of the roots, put m equal to n—1, )i — 2, ?^ — 3, &c. successively in (535), in order to obtain s_i, s_o, s_3, &c.
537 To calculate .'?,. independently.
Rule : s,. = — r X rorjjicieut of x~'' in the e:q)amlo)i of
^'^"J n i'i '^''•^''''ii'^''>t'J I^OlV^^iS of .V.
Proved by taking f(x) = {x~ a) (x-h)(x — c) ... , dividing by x", and expanding the logarithm of the right side of the equation by (loO).
538 SYMMETniCAL FUNCTIONS WTUGU AJ^E
NOT POWERS OF THE BOOTS.
These are expressed in terms of the sums of ])o\vers of the roots as under, and thence, l)y (531), in terms of the routs explicitly,
539 'V„,,^,.,, = .V,,,*,,*,/ — .V„, + /, A-,, — *•„, + ,,*,, — A', ,+ ,.?,„ +2.V„, + ^ + , ,.
The last equation may be ]iroved by mnlti})lying .<„,.,, by ."f, ; and expansions of other symmetiical functions may be obtained in a similar way.
540 If </> ('•) be a rational integi'al function of ,r, then the symmetrical function of the roots of ./'('), denoted by
140 TEE OUT OF EQUATIONS.
^ (a)-{-(j> (h) -{-<!> ((■) + &c., is equal to tlie coefficient of x''''^ in the remainder obtained by dividing <{> 0*') /'('') ^7 /('*') •
Proved by multiplying the equation (432) by yji, and by tlieorera (401).
541 To find tlie equation wliose roots are tlie squares of tlie differences of tlie roots of a given equation.
Let F{^) be the given equation, and S,. the sum of the r*'' powers of its roots. Let/(/c) and s^ have the same meaning with regard to the required equation.
The coefficients of the required equation can be calculated from those of the given one as follows : —
The coefficients of each equation may he connected tvith the sums of the ])oivers of its roots hy (534) ; and the sums of the poivers of the roots of the tivo equations are connected hy the formula
542 2^, = nS,,,-2rSA._,-\- ^''('^''-^) S,S,,_,- ... +«S,,
Rule. — 2s,. is equal to the formal expansion of {S—Sf'' hy the Binomial Theorem, with the first and last terms each mul- tiplied hy n, and the indices all changed to siffixes. As the equi-distant terms are equal we can divide by 2, and take half the series.
Demonstration. — Let a, h, c ... be tbe roots of i^(a;).
Let <l>(x) = (x-ay-' + (x-hy+ (i-)
Expand each term on the right by the Bin. Theor., and add, substituting Si, S2, &c. In the result change x into a, b, c ... successively, and add the n equations to obtain the formula, observing tliat, by (i.), 0 (a) +0 (&) + ... =2.v
If r? be tlic degree of 7^ (,*•), then 'k)i(ii — l) is the degree of /(.'). % (UG).
543 The last term of the equation 7" ('') = 0 is equal to
n^T{a)FmF{y)... where a, ft, y, ... are the roots of F' {•>'). Proved by shewing that F'{a)F'{h) ... = n"F{a)F(ft) ...
544 If -^(''O ^^^s negative or imaginary roots, f(.i) must have imaginary roots.
SYMMETRICAL FUNCTIONS OF THE ROOTS. 141
545 The sum of tlic v«"' powers of tlie roots of the (|u:i(l- ratic ecjuation .*-— />.r+7 = 0.
,„ ., , in {ni — .*») ,„_t - .v„, = ;>'" — ni]>"' -(/ + -j p (/-...
By (io") expaudiug the logarithm by (15G).
546 'I'he sum of tlie m^^' powers of the roots of </y* — 1 = 0 is n if ni be a multiple of )i, and zero if it be not. By (537) ; expanding the logarithm by (15G).
547 If <!>{.,•) = a,-\-a,x-\-n,x- + &c (i.):
then the sum of the selected terms
^-illbe .v= — ;a"-"'<^(a.r)+y8"-"'<^()8./0+y"""'*(r'O + ^^'C.} where a, /3, y, &c. are the n^^' roots of unity.
For proof, multiply (i.) by a""'", and change x into a.v; so with /?, y, &c. and add the resulting equations.
548 To approximate to the root of an equation by means of the sums of the powers of the roots.
By taking m large enough, the fraction tuill ^vill approx- imate to the value of the numerically greatest root, unless there be a modulus of imaginary roots greater than any real root, in which case the fraction has no Umiting value.
549 Similarly tlie fraction •^''"•^'"^•-~'^'"":.' api)roximates, as m
increases, to the grrnfr.^f product of any pair of roots, real or imnginary ; excepting in the case in which the product of tho pair of imaginary roots, though less tlian the product of tho two real roots, is greater than the scpiare of the least of tliem, for then the fraction has no limiting value.
142 THEORY OF EQUATIONS.
550 Similarly tlie fraction ' '" '"'^^ '"'^y '""^^ approximates,
^'m'^'m + 2 ^m + 1
as m increases, to tlie sum of tlie two numerically greatest roots, or to the sum of the two imaginary roots with the greatest modulus.
EXPANSION OF AN IMPLICIT FUNCTION OF .r.
Let r{A,v''+)+!f{By-]-)^...-\-fiS.r^+) = 0 (1)
be an equation arranged in descending powers of y, the co- efficients being functions of x, the highest powers only of x in each coefficient being written.
It is required to obtain y in a series of descending powers of X.
First form the fractions
a — b a — c a — d n — s ,c)\
a — y8' a — y' a — S a — a
Let — — - t be the greatest of these algebraicalh^, or
a — n
if several are equal and greater than the rest, let it be the
last of such. Then, with tlie letters corresponding to these
equal and greatest fractions, form the equation
Au''-^ -^Ku'^i) (3).
Each value of ?/ in this equation corresponds to a value of v/, commencing with ux^.
Next select the greatest of the fractions
/.• — / Ix—m I'—s /jx
r» — v^V •
K — A K — /x K — cr
/. _ ^,, Let — -— ' := t' be the last of the greatest ones. Form
K — V
the corresponding equation Kh"-\- ...-\-Nh'' = (I (5).
Tlien each value of u in this equation gives a cori'cspondiiig value of y, commencing with »./.
EXPANSION OF AN IMI'LICIT Fi'M'TIDN. 143
Proceed in tliis way until llie last tVactiun of tlio series (2) is reached.
To obtain the second term in the exjjansion of //, put
,/ = .,.'(//+//,) Ill (1) 03),
cmplojnng tlie dilTerent vahies of n, and again of /' and //, /" and ti, &c. in succession; and in each case this substitution will produce an equation in // and x similar to the original equation in //.
Eepeat the foregoing process ^vith the new eqnation in y, observing the following additional rule : —
Wien all the values of t, t\ t", ^c. have heeu ohtained, the negative ones only must be employed informing the equations in n. (7).
552 To obtain y in a series of ascending powers of ■/■. Arrange equation (1) so that a, /3, y, &c. may be in as-
ccndliKj order of magnitude, and a, h, c, &c. the lowest powers of .t' in the respective coefficients.
Select f, the greatest of the flections in (2), and proceed exactly as before, with the one exception of substituting the vford jmsitive for negative in (7).
553 Example. — Take the equation
{x^ + x') + ( 3.^^ - hx^) 2/ + ( - 4.7; + 7x- + a;") >f - >/ = 0. It is required to expand y in ascending powers of x.
The fractions (2) are - -'^. "qZo' ~ ^E ^ ' ^^' ^' ^' ^"^^ ^' The first two being equal and gi-catest, we have t = 1.
The fractions (4) reduce to — - — ^ = i = ' •
Eqnation (3) is \+3n- 4ir = 0,
which gives ?/ = 1 and — ^, with / = 1.
Equation (5) is —4ir—u'' = 0,
and from this u = 0 and —4*, with t' = ^.
"NVe have now to sub.stitute for ij, according to (G), cither ^(1 + y,), a:(-i + J/,), .tV, or a;*(-4^ + i/,).
Tut y = x (1 + 7/,), the tir.st of tlieso values, in the originul equation, and arrange in ascending powers of y, thus
-4x* + (-rjx'-{-)y,+ (-4.c' + )yl - 10x'y\ - bx'y\ - .c'y] = 0. The lowest power only of x in each cocflicient is here written.
144
THEORY OF EQUATIONS.
4-5 0-4'
5'
The fractions (2) now become
4-3 4-3 _4-5 0-1' 0-2' 0-3'
1, h -h -h -h From tlieso /= 1, and equation (3) becomes
— 4— 5?t = 0; .•. « = — 4.. Hence one of the values of y■^ is, as in (G), yi = x (— f + 2/2)- Therefore y = x {1 + «; (-f + 2/2)} = aJ-f'»'+ ...
Thus the first two terms of one of the expansions have been obtained.
DETERMINANTS.
554 Definitions. — The determinant
b, hi
is equivalent
to a^h-i—dtbi-, ^Tifl is called a determinant of the second order. A determinant of the third order is
a.^ a^ = rti {Ihc^ — 63^2) + a. %c^ — b^c^) + a.^ {b,c., - b.,c^) . b, b^ 63
Ci C.2 C;
Another notation is 2 ± (lyh.^c^, or simply {a^h.yC-^.
The letters are named constituents, and the terms are called elements. The determinant is composed of all the elements obtained by permutations of the suffixes 1, 2, 3.
The coefficients of the constituents are determinants of the next lower order, and are termed minors of the original determinant. Thus, the first determinant above is the minor of ^3 in the second determinant. It is denoted by C^. So the minor of % is denoted by vli, and so on.
555 A determinant of the 7^*^ order may be wintten in either of the forms below
b, b.
,. «,. ... «„ .. 6, ... b.
... «„, ... a.,.
In the latter, or double suffix notation, the first suffix indicates the row, aud the second the column. The former notation will be adopted in these pages.
DETERMTXAXTS. ^^■
ft I a., «! h, h., b.
A Compoiiift' ilotrrminant is one in which the Tuimber of cohimns exceeds the innnber of rows, and it is wntten as in the annexed example. Its vahie is the sum of all the determinants obtained \)\ takiiif^ a number of rows in every possible way.
A Simple ih'tormlnnnt lias single terms for its constituents.
A Compound drtrrmuinut has more than one term in somo or all of its constituents. See (')70) for an exam[)le.
For th(^ definitions of Si/mmetrical, Rcripronil, Parfial, and Complcmciitari/ determinants; see (574), (575), and (576).
General Theory.
556 The number of constituents is n^.
Tlie number of elements in the complete determinant is [?^^.
557 The first or leading element is aih.,<\ ... /„. Any element may be derived from the first by permutation of the suffixes.
The sign of an element is + or — according as it has been obtained from the diagonal element by an even or odd number of permutations of the suffixes.
Hence the following rule for determining the sign of an element.
RuT.E. — Take the suffixes in order, and put them hach to their places in the first element. Let m he the whole number of places passed over ; then ( — 1)"* "'^7/ give the sign required.
Ex. — To find the sign of the element n^h^CrJ\c.^ of tlio detei-minanfc
«4 h ^s 'h '^i
Move the .suffix 1, tlircc pliiccs 1 4 3 ''> 2
2, three places 1 2 4 3 ')
„ „ 3, one phwje 1 2 3 4 5
In all, seven places; therefore (—1)' = —1 gives the sign required.
558 n two suffixes in any element be transi)o.^ed, the sign of the element is changed.
Half of the elements are plus, and half are minus.
559 The elements are not altered ])y changing the rows into columns.
If two rows or columns are transposed, the sign of tho
U
146 TUEOUY OF EQUATIONS.
determinant is clianged. Because each element changes its sign.
If two rows or columns are identical, the determinant vanishes.
560 If all the constituents but one in a row or column vanish, the determinant becomes the product of that con- stituent and a determinant of the next lower order.
561 A cyclical interchange is effected by n — \ successive transpositions of adjacent rows or columns, until the top row has been brought to the bottom, or the left column to the right side. Hence
A cyclical interchange changes the sign of a determinant of an even order only.
The r*^ row may be brought to the top by r— 1 cyclical interchanges.
562 If each constituent in a row or column be multiplied by the same factor, the determinant becomes multiphed by it.
If each constituent of a row or column is the sum of m terms, the compound determinant becomes the sum of m simple determinants of the same order.
Also, if every constituent of the determinant consists of m terms, the compound determinant is resolvable into the sum of m^ simple determinants.
563 To express the minor of the r"' row and ¥^ column as a determinant of the n — V^ order.
I*ut all the constituents in the r^^ row and /.-^^ column equal to 0, and then make r— 1 cyclical interchanges in the rows and L — 1 in the columns, and multiply by (_iy'-+^)(«-i).
r._. _ f_Y\(r-\*k-\){n-l)^
564 To express a determinant as a deter- minant of a higher order.
Continue the diagonal with constituents of " ones," and fill up with zeros on one side, and with any quantities whatever (o, /3, y, &c.) on the other.
1 0 0 0 0
a 1 0 0 {)
fi e a h <>'
V C h h f
h r) - /• c
vi:TEnMTy.iNTS.
U7
565 Tlio sura of tbe products of each constituent of a column by tlio corresponding minor in another given cohimn is zero. And the same is true if we read ' row' instead of * cohimn.' Thus, referring to the determinant in (555),
Taking tlie /)''' and r/^' cohimns, Taking tho a and r rows, For in each case we have a dotcnuinant with two columns identical.
566 I" i^ny row^ or column tlie sum of the pnxhicts of each constituent by its minor is the determinant itself. That is,
Taking the j)^^ cohimn, Or taking the c row.
567 The hist equation may be expressed by Also, if i'lpCq) express the determinant
' ^- r = A. p p
a„ a
then
'2{apr,^) will represent the sum of all tlu^ determinants of the second order wdiich can be formed by taking any two columns out of the a and r rows. The minor of {dp, (\) may be wi'itten (Ap, Cy, and signifies the determinant obtained by suppress- ing the two rows and two columns of Op and c,. Thus A = S {dp, Cg) {Ap, Cq). And a similar notation when three or more rows and columns are selected. ^
568 Analyji'is of a determinant.
Rule. — To resolve into its elements a determinant n*^ order. Express it as the sum of n determinants (n—iy'' order by (5G0), and repeat the process with e the new determinants.
EXAMl'LE : «l «J «3 "i
c, q r, c^ «/, (/j d^ ll^
Again,
of thn of the ach oj
= a, b, b, b,
-a.
63 b^ 6,
+ ^.
b, b, 6,
-«4
b, b, b.
<'i <^i ''a
c, c, c,
r^ c, c,
d, d, d.
d, d, d,
dt (/, d.
./, c/, d.
= b, I c, r, I + b,
+ b.
i d, d, I
^ b, b, . .
c, C, C, I l/j t/j I 1 c/j ./,
di rfj d^
and so on. In the first scries the determinants have alternately ji minus signs, by the rule for cyclical interchanges (501), tlie order bei
lus and ngeven.
148 TUEOUY OF EQUATIONS.
569 Si/nthesh of a determinant.
The process is facilitated by making use of two evident rules. Those constituents which belong to the row and column of a given constituent a, will be designated *' a's con- stituents." Also, two pairs of constituents such as a^, c, and dg, c^, forming the corners of a rectangle, will be said to be *' conjugate" to each other.
Rule I. — No constituent loill he found in the same term with one of its oion constituents.
Rule II. — The conjugates of any two constituents a and b will he cuinnion to a's and Us constituents.
Ex. — To write the following terms in the form of a determinant : uhcd + Ifyl +fh^ + ledf+ cghp + 1 ahr + elpr —fhpr—ahlr—ach^ — Ifhg — bdf^ — efhl — cedp.
The determinant will be of the fourth order ; and since every term must contain four constituents, the constituent 1 is supplied to make up the number in some of the terms. Select any term, as ahcd, for the leading diagonal.
Kow apply Rule I., a ia not found with e,f,f, g,p,0...(l)- c is not found with /,/, I, r, 1, 0... (^). 1) is not found with e, h, h,p,l,0 ..(2). dis not found with g, h,h,l,r,0...(-i).
Each constituent has 2 (»— 1), that is, 6 constituents belonging to it, since n =?'4. Assuming, therefore, that the above letters are the constituents of a, b, c, and d, and that there are no more, we supply a sixth zero constituent in each case.
Now apply Rule II. — The constituents common
to a and b are e, p ; to a and c — -/, /; to b and c — 1, 0 ;
to a and d — g, 0 ; to h and d — h, li, 0 ; to c and d-
The determinant may now be formed. Tlie diagonal being abed,; place e, p, the eoiijugatus of a and b, either as in the diagram or transposed.
Then /and/, the conjugates of a and c, may be written.
1 and 0, the conjugates of b and c, must be placed as indicated, becau.so 1 is one of y/s constituents, sini-e it is nt)b found in any term with /i, and must therefore be in the second row.
Similarly tlie places of y and 0, and of Z and r, are assigned.
In the c-ase of b and d we have h, h, 0 from wliieh to clioose the two conjugates, but, we see that 0 is not one of them because that would assigu two zero constituents to b, whereas b has but one, which is already placed.
By similar reasoning the ambiguity in selecting the conjugates I, r is removed.
The foregoing method is rigid in the case of a complete determinant
/, /•, 0.
a e
f
7
l> b
1
h
/' U
(■
r
U It
/
d
DETEKMJXAXTS.
1 to
having different constituents. It becomes uncertain when the zero con- Ktituonts increase in number, and when several coustitueuts are identical. Ihit even then, in tlie majority of cases, it will soon afford a cluo to the required arrangement.
570 I'RODUCr OF TWO DETERMINANTS OF Tin: „"• (lUDEU.
C)
</, (I., ... a, h, h, ... h,
L /..
{Q)
a, a> ... a„
X, X,
.U A, ... J„
/;, li ii,,
L, L, ... L,
. Ai = ay ai-\-(L,a.,-\- .
Ly r= aiXi4-^^X.+ ... +(?,.X,
Tlie values of A^, By ... L, in tUo first column of S arc an- nexed. For the second column write //s in the place of as. For the tliird columu write f's, and so on.
For proof substitute the values of Ay, By, &c. in the determinant S, and then resolve ,b' into the sum of a number of determinants by (oC>2), and note the determinants which vanish through having identical columns.
Rule. — To form the determinant S, which is the product of two determinants P and Q. First connect hij plus siijus the constituents in the ro2VS of both the determinants F and Q.
Nou) place the first row of F upon each row of Q in turn, and let each two constituents as thnj touch become jn-oducts. Tliis is the first column of S.
Perform the same operation upon Q with the second rou- if P to obtain the second column of S ; and again with the third row of F to obtain the third column of S, and so on.
571 If the number of columns, both in F and Q., be n, and the number of rows r, and if n be > r, then the determinant S, found in tlie snme way from F and Q, is equal to the sum of the C{ii, r) products of prirs of determinants obtained by taking any r columns out of F, and the corresponding r columns out of Q.
But if n be < r the determinant S vanishes.
For in that case, in every one of th be two columns idcutical.
component determinants, there will
150 THEORY OF EQUATIONS.
572 The product of tlie determinants P and Q may be formed in four ways by changing the rows into columns in either or both P and Q.
573 Let the following system of n equations in XicV,.^ ... x^ be transformed by substituting the accompanying values of the variables,
The ehminant of the resulting equations in ^^ ^., • • • ^» is the determinant S in (570), and is therefore equal to the product of the determinants P and Q. The determinant Q is then termed the modulus of transformation.
574 A Symmetrical determinant is symmetrical about the leading diagonal. If the E's form the r^'' row, and the K's the k^^ row ; then B;, = K^ throughout a symmetrical deter- minant.
The square of a determinant is a symmetrical determinant.
575 ^ Reciprocal determinant has for its constituents the fii'st minors of the original determinant, and is equal to its ?i — l'^ power; that is,
A, ... A,
h ... 4
Proof. — ^Multiply both sides of the equation by the original determinant (o55j. The con- stituents on the left side all vanish excepting the diagonal of A's.
576 Partial and Complementary determinants.
If r rows and the same number of columns be selected fi'om a determinant, and if the rows be brouglit to the top, and the columns to the left side, without changing their order, then the elements common to the selected rows and columns form a Partial determinant of the order r, and the elements 7wt found in any of those rows and columns form the Com- plementary determinant, its order being n — r.
DETERMINANTS.
151
Ex.— Let the selected rows from the dctcrmiiiaTit («,''3r,(/^^J be tlio Becond, third, and fifth ; and the selected columns bo the third, fourth, aud fifth. The orijjinal and the transformed determinants will be
(J,
«3
"8
"4
^
h.
K
^
C^
C-l
fs
^4
d,
d.
d.
d.
C\
Ct
Cs
^4
d.
The partial determinant of the third order is (/'. mentary of the second order is {'i^d.^.
The complete altered determinant is plus or minus, according as the permutations of the rows and columns are of the same or of ditierent class. Jn the example they are of the same class, for there have been four trans- positions of rows, and six of columns. Thus ( — 1)'" = + ! gives the sign of the altered determinant.
K
h K . K K
^8
Ci c, r, c.
^8
e^ Cs e, e.
flj
a^ Oj : a, «3
^^8
d, d, d, d.
del
is 0'3''i''i), find its comple-
577 TuEOREii. — A partial reciprocal determinant of the r^^ order is equal to the product of the r— 1"' power of the original determinant, and the complementary of its corres- ponding partial determinant.
Take the last determinant for an example. Here n=5, r=3 ; and by tho theorem,
^8 B, B„
= A=
«i
a,
where B, C, E are the
G, C, C,
d.
d.
respective minors.
Es E, E,
Proof. — Raise the Partial Reciprocal to the original order five without altering its value, by (5G4) ; and multiply it by A, with the rows aud columns changed to correspond as in Ex. (57G) ; thus, by (570), we have
J?3 B, B, B, B, c, C, C, C, C, E^ E^ E^ ..^>....^»
0 0 0 ;""i 0"
0 0 0 0 1
h K h
C, C4 O5
«. e, e.
A 0 0 ti 60 0 A 0 c, c, 0 0 A e, c, 0 0 0a, Oj 0 0 0 </, (7,
= A» a, a, d,d.
(h «4 05 d, d, d.
a, a, ./, d.
578 The product of the differences between every pair of 11 quantities fli, a., ... "„, {a, — (i.;){a,—(t,){a,-a,) ... (r/,-^/„) X {(li — ^l■,^)(^(., — <l^) ... ((i.. — (i„)
X(an-X-(in)
Proof. — The determinant vanishes when any two of the quantities are
1
1
1 ...
1
(t^
(I.,
a-, ...
(1,
<
(tl
< '"
(I-
152
THEORY OF EQUATIONS.
equal. Therefore it is divisible by eacli of the factors on the left ; therefore by their product. And the quotient is seen to be unity, for both sides of the equation are of the same degree ; viz., ^n {n — 1). ' '
579 The product of the squares of the| _ differences of the same n quantities j ~
... S,;
Proof. — Square the determinant in (578), and write Sf for the sum of the r"' powers of the roots.
580 With the same meaning for s-^,s.2..., the same deter- minant taken of an order r, less than n, is equal to the sum of the products of the squares of the differences of r of the n quantities taken in every possible way ; that is, in C {n, r) ways.
Ex.:
^1
§1 S-2
^0
^1
■'2
•''l
*2
-''3
^'2
•''3
*4
(«! - fl2)H (a, - a^y + &c. = 2 (flfi - «2)^
The next determinant in order
= S (a^-a^y (ffli— «3)^ (% — «i)^ («a-«s)^ ((^■i—ci'iy (as — a^y.
And so on until the equation (579) is reached.
Proved by substituting the values of s^, s.2 ... &c., and resolving the deter- minant into its partial determinants by (571).
581 The quotient of
is given by the formula
qo^i"'-'' + f/i.r*"-'^-' + ... + </,.i^'"-"-'' + ...
where
/>o 0 0 ... «o />2 fh K ••• ffi
h, h,
b,a.
Proved by Induction.
ELIMINATION. 153
ELIMINATION.
582 Solution of 11 li)i('(ir rquntious in u rnriahlrs.
The equations and the values of the varialjles are arranged below :
«,.r, + f/,.r,+ ... + r/„.r„ = ^„ .r,A = J^^, + /;, f,+ ... + /., ^„
where A is the determinant annexed, and J^, B^, «, ... a &c. are its first minors.
/i ... /.
To find the value of one of the unknowns x^.
HuLE. — Multiply the equations respectively by the minors of the r''' column, and add the results, x^ will he equal to the fraction whose numerator is the determinant A, with its ■?•'* column replaced by Hj, ^2 ••• s„, and whose denominator is A itself.
583 If SI, ^2 ••• ^« ^^''tl A all vanish, then x^, Xo ... x^ are in the ratios of the minors of any row of the determinant A. For example, in the ratios C^: Cj : C^ : ... : €„.
The eliminant of the given equations is now A = 0.
584 Orthoi^onal Transformation.
If the two sets of variables in the n equations (5y2) be connected by the relation
.»•, + 4 + ... +.<•;= ?; + f: + ... + f;, (D.
then the changing from one set of variables to the other, by substituting the values of the Ts in terms of the xs, in any function of the former, or rice versa, is called orthogonal transformation.
When equation (1) is satisfied, two results follow.
I. The determinant A = ± 1.
X
154
THEORY OF EQUATIONS.
II. Each of the constituents of A is equal to the corre- sponding minorj or else to minus that minor according as A is or
Proof. — Substitute the values of 4i, l^ ... 4„ in terras of x^, x^ ... a;„ in equation (1), and equate coefficients of the squares and products of the new variables. We get the n^ equations
a\^-V + = 1 fliCj + h^h^ + = 0
a.,ai + hjb^ + = 0 a\ + hl + = 1
a- + hl + = 1
a-,an + hA),, -f
Also A
Form the square of the determinant A by the rule (570), and these equations show that the product is a determinant in which the only con- stituents that do not vanish constitute a diagonal of ' ones.' Therefore
A'^ = 1 and A = ±l.
Again, solvi