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Digitized by the Internet Archive 
 
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 http://www.archive.org/details/essayonalgebraicOOjarrrich 
 
AN ESSAY 
 
 ALGEBRAIC DEVELOPMENT, 
 
 CONTAINING 
 
 THE PRINCIPAL EXPANSIONS 
 
 IN COMMON ALGEBRA, IN THE DIFFERENTIAL 
 AND INTEGRAL CALCULUS, 
 
 AND IN 
 
 THE CALCULUS OF FINITE DIFFERENCES; 
 
 THE GENERAL TERM 
 
 BEING IN EACH CASE IMMEDIATELY OBTAINED 
 BY MEANS OF 
 
 A NEW AND COMPREHENSIVE NOTATION. 
 
 By the rev. THOMAS JARRETT, M.A. 
 
 FELLOW OF CATHARINE HALL, AND PROFESSOR OF ARABIC 
 IN THE UNIVERSITY OF CAMBRIDGE. 
 
 CAMBRIDGE : 
 
 PRINTED BY J. SMITH, PRINTER TO THE UNIVERSITY : 
 
 FOR J. & J. J. DEIGHTONS, CAMBRIDGE; 
 
 AND RIVINGTONS, LONDON. 
 
 M.DCCC.XXXl 
 
PREFACE. 
 
 The following pages are intended to illustrate and apply 
 a system of Algebraic Notation submitted to the Cambridge 
 Philosophical Society in the year 1827, and published in the 
 third Volume of their Transactions. In that paper the ap- 
 plications were necessarily few, and the whole was deficient 
 in that development which was indispensable to render the 
 introduction of the system into general use at all probable; 
 but in the present Work it is applied to the demonstration 
 of the most important series in pure Analysis. The methods 
 by which these are demonstrated are partly original, and partly 
 taken from one or other of the works of which a list follows 
 this Preface, but they are in general so much modified that 
 a distinct reference to the inventor of each demonstration 
 appeared useless; so much, however, is due to the admirable 
 works of Schweins, that it would be unjust not to make a 
 distinct acknowledgment of the great use that has been made 
 of his " Analysis."'' The demonstration of the legitimacy of 
 the separation of the symbols of operation and quantity, witli 
 certain limitations, belongs to Servois, and will be found in 
 the "Annales des Mathematiques;" and the proof, that the 
 coefficients of the binomial, (the index being a positive integer,) 
 are integers, is due to Mr. Miller, of St. John's College, and 
 is the only independent proof with which I am acquainted. 
 
IV PREFACE. 
 
 The following apparent innovations in the ordinary notation 
 are not original : 
 
 (1) E^(p{x), for 0(a?+Z>t^?), is partly due to Arbogast, 
 who uses E(p(x) for the same function. 
 
 (2) d"u, for y--, is due to Lacroix, although not used 
 
 by him, being merely pointed out in a single line*; 
 it was suggested to the writer of these pages by the 
 analogous integral notation invented by Professor 
 Airy. 
 
 (3) (w)^^„, for the value assumed by u when x is put 
 equal to a, belongs to Sehweins. 
 
 In order that the work may be as independent as possible, 
 the Reader is supposed to be acquainted only with the first 
 rules of Algebra, and the fundamental theorems of Trigo- 
 nometry; and, for the sake of facility of reference, the whole of 
 the theorems have been arranged in an index at the end of the 
 volume. 
 
 The additions contain a few Theorems of importance that 
 did not suggest themselves till too late to be, inserted in the 
 text, together with a few simplifications of the demonstrations 
 inserted in the body of the Work. 
 
 In conclusion, the Author has to acknowledge the great 
 liberality of the Syndics of the University Press, in defraying 
 a considerable part of the expense of publishing. 
 
 * Calcul Diff. Tome ij. page 527. 
 
LIST OF WORKS 
 
 WHICH HAVE BEEN CONSULTED. 
 
 Annales des Mathematiques. 
 
 Arbogast, Calcul des Derivations. 
 
 Herschel, Examples on Finite Differences. 
 
 Hindenburg, Sammlung combinatorisch-analytischer Abhandlungen. 
 
 Lacroix, Calcul DifFerentiel et Integral. 
 
 Laplace, Theorie Analytique des Probabilites. 
 
 Schweins, Analysis. 
 
 ■ Theorie der DifFerenzen und DifFerentiale. 
 Wronski, Introduction a. la Philosophic des Mathematiques. 
 
INDEX TO THE CHAPTERS. 
 
 PAGE 
 
 CHAPTER 
 
 I, On Series in general 1 
 
 II. On Products and Factorials 12 
 
 III. On Combinations and Arrangements 20 
 
 IV. On Binomials and Exponentials 23 
 
 V. On Finite DifFerences 40 
 
 VI. On Differentiation in general 66 
 
 VII. On Polynomials 78 
 
 VIII. On the Differentiation of Exponential and Circular 
 
 Functions 93 
 
 IX. On the Expansion of Circular Functions 102 
 
 X. On the Integration of certain Definite Functions 123 
 
 XI. On Generating Functions 136. 
 
INDEX TO THE SYMBOLS. 
 
 PAGE 
 
 ART. 
 
 
 
 1 
 
 2 
 
 s,„«,„. 
 
 
 2 
 
 3 
 
 S„,a,„. 
 
 
 5 
 
 15 
 
 S,„S„a..„. 
 
 
 13 
 
 28 
 
 P,„a„, and P,„a„,. 
 
 
 15 
 
 38 
 
 [a , l^a, and [a. 
 
 
 19 
 
 49 
 
 m «i+l n+1 
 
 
 20 
 
 53 
 
 Crtty, and C «,• 
 
 
 21 
 
 59 
 
 ""CrAarA)- 
 
 
 22 
 
 62 
 
 A + mam- 
 
 
 41 
 
 116 
 
 (0 + x//)„M. 
 
 
 
 117 
 
 
 
 45 
 
 125 
 
 (w)^^„, and 0,=„(?*). 
 
 
 
 126 
 
 £.(..). 
 
 
 
 127 
 
 D.{u). 
 
 
 
 129 
 
 A.(^.). 
 
 
 58 
 
 168 
 
 £,,y(M), and Z),,^(7<). 
 
 
 66 
 
 188 
 
 d:u. 
 
 
 67 
 
 195 
 
 £"• 
 
 
 78 
 
 224 
 
 a.^.Cro. 
 
 
 79 
 
 228 
 
 ■3r/0(«), "ar'^Ca), 73-" 
 
 'a", and ■z3-"'«, 
 
 89 
 
 241 
 
 Gam-i- 
 
 
 136 
 
 335 
 
 S,„a,„. 
 
 
 
 336 
 
 Gi.Z*^. 
 
 
ERRATA. 
 
 The Reader is requested to correct the following Errata before he 
 proceeds to the perusal of the Work. 
 
 PAGE 
 
 LINE 
 
 ERRATUM. 
 
 CORRECTION. 
 
 5 
 
 8 
 
 a5 + «2«-l 
 
 «5+-+a2«_i 
 
 6 
 
 last 
 
 s„ 
 
 s„ 
 
 19 
 
 1 
 
 c}...} 
 
 n+1 1 
 
 c 
 
 
 6 
 
 P 
 
 p 
 
 21 
 
 3 
 
 T 
 
 "c 
 
 22 
 
 17 
 
 -(^'"■ 
 
 -c"^"" 
 
 
 last hut one 
 
 +a.a3 
 
 + aia2 0i+a,a3 
 
 
 last 
 
 2«.«,' 
 
 Sa^ar 
 
 24 
 
 19 
 
 A 
 
 m— 1 
 
 A 
 
 47 
 
 last 
 
 ^Ax) 
 
 D,(m) 
 
 48 
 
 3 
 
 ^(w) 
 
 <p(.r) 
 
 
 14 
 
 a.D. 
 
 a.Dx. 
 
 62 
 
 11 
 
 O^x 
 
 a^ 
 
 88 
 
 11 
 
 a;2m+3 
 
 .t2"-3 
 
 89 
 
 10 
 
 |2wi8 
 
 1^ 
 
 98 
 
 6 and last 
 
 (201) 
 
 (200) 
 
 104 
 105 
 
 '1} 
 
 «> 
 
 n 
 
 
 ' last hut one 
 
 s« 
 
 s„. 
 
 126 
 
 8 and 9 
 
 a-" 
 
 x~" 
 
 135 
 
 6 
 
 s„ 
 
 s„. 
 
 146 
 
 last hut one 
 
 a:(«») 
 
 A;r(«,) 
 
ON THE DEVELOPMENT 
 
 ALGEBRAIC FUNCTIONS. 
 
 CHAPTER I. 
 
 ON SERIES IN GENERAL. 
 
 1. In the expansion of Algebraic Functions it has been 
 usual to investigate the first three or four terms, and from 
 these to deduce the remainder of the series by analogy. The 
 unsatisfactory nature of this method in all cases, and the errors 
 into which it may readily lead us in very many instances, must 
 have been obvious to all who have made use of it. In some 
 cases indeed, the connection between the consecutive terms at 
 the commencement of the series is so obscure, that the most 
 patient of analysts have given up the search, and have been 
 compelled to state that " the law of the series is not obvious." 
 In order to avoid this obscurity and embarrassment, we shall 
 adopt a notation by means of which the general term will be 
 obtained in every case, and which will enable us to perform 
 any operation whatever on a series, with the same facility as on 
 a single term. 
 
 2. The m^^ term of a series being usually some function 
 of m, we shall denote it by «,„ ; and, taking the letter S as an 
 
 abridgment of the word Sum, the symbol S,„«„, will be used to 
 A 
 
2 
 
 denote the sum of n terms of which the m"' is a^^ : that is, 
 
 S,„a,„ = Cfi + Oo + «3 + ••• + «n- 
 
 In this notation it will be seen that the index placed over 
 the S denotes the number of terms, and that the index placed 
 under the same letter, is that to which the successive values 
 1, 2, 3, ..., w, must be given, in the function a,„, in order to 
 form the consecutive terms : that is, 
 
 S,„0(w) = 0(1) + 0(2) + 0(3) + ... + (p{n). 
 
 Or, to give examples of a more simple nature, 
 
 S„, m^ = f'' + 2- + 3- + ... + n^. 
 
 n 
 
 8m m' = l'' + 2'" + 3' + ... +7l\ 
 8,„(r-2m,+iy=(r-iy+(r-sy+{r-5y+...+(r-2n+iy. 
 
 3. The symbol §,„ a,„ denotes that the r^^ term must be 
 omitted. 
 
 4. Theorem. If a,„ = 6„„ (ZZl)*, then S„ «„=§„&„. 
 
 For, since a, = 6, 
 a, = h, 
 
 az = h 
 &c. = &c. 
 
 a, + ffo + «3 + &c- +«« = &! + h. + ^3 + &c. + &„, 
 
 or S,„a„, = 8,„i™. 
 
 • By this notation is meant, that this equation is to hold for every integral value of 
 tf>?, from 1 to V. 
 
3 
 
 5. Theorem. S,„ («„. + b,„) = S,„ «,„ + S,„ b,„ . 
 
 For, S„, (a,„ + b„) = a, + b^ + a.> + h^ + . . . + a„ + b„ 
 = «i + rt2 + -+«n + ^ + &2 + ••• + &« 
 
 6. Theorem. If & is independent of m, then S,„ a^fe = 6 . S„ «,„• 
 
 For, S,„a,„ft = Qib + a^b + a^b + ... + a„6 
 
 = 6 (Oi + tta + «3 + ... + «n) 
 
 7- Cor. ^,„b = nb. 
 
 8. Problem. To invert the order of the terms of a given 
 series. 
 
 Now, S,„ «m = o, + oTo + ffg + ... + «„_i + a„ 
 
 = a„ + a„_i + a„_2 + ... + «!, by inverting the series; 
 
 In order therefore to invert the series, we must substitute 
 71 - 7n + 1 for m in the expression for «„, . 
 
 n r n—r 
 
 9 . Theorem. S„, a,„ = S;„ «,„ + S,„ «,+ ,« • 
 
 For, S,„ «,„ = Joi + ao + fla + ... a,j + la^+i + «r+2 + ... + «,i5 
 
 By means of this theorem we can separate from the rest 
 any number of terms, taken either at the beginning or end of a 
 given series. 
 
10. Theorem. 8„,.v"'-'= 
 
 1 - X 
 
 For, a.-"'-^ = .r"-^ 
 
 1 -X 
 
 X X 
 
 \ -X \ - X 
 
 k af'-' = k ~ - k Y^ « (^) and (5) ; 
 
 ,,0 ,-,1 ^^'n „ 1 ^^m ^ 
 
 + 8™-, S„- , (9) 
 
 \ - X \ - X \ - X \ - X 
 
 1 - .r" 
 
 11. Cor. ]. = S„.a" 
 
 12. Con. ^Z. — ^ = k -i""' + ^— 
 \ — X \ - X 
 
 and — — = S,„(-ir-' . ci--' + (-1)" . , 
 
 1 + .f ^ ^ 1 + X 
 
 If X is < 1, the term will diminish as n increases, 
 
 1 -X 
 
 and therefore, by taking n sufficiently great, S,„ a'""' may 
 
 be made to differ from by a quantity less than any 
 
 assignable quantity, although that difference will never vanish 
 
 for anv finite value of //. In this case is said to equal 
 
 \-x ^ 
 
 an ipfinitc series of wliich the m"' term is .r"~'; and this 
 
 relation is denoted bv the eciuation — — = Sm'i''""'- 
 ' 1 - .r 
 
13. If the law which determines the value of a,„ in the 
 series S™ «,„ is such, that a,„ = for every value of m greater 
 
 CO n 
 
 than 72, we may substitute S,„a,„ for S„,«,„; and this substitution 
 will frequently facilitate our investigations. 
 
 14. Theorem. S,„ff,„ = vimfhm-\ + S,„«2mi 
 
 and 8;„«„, = S,„rt2,„_i + S„, «a„,.. 
 
 2ra ,- ■'' 
 
 For, S„, «,„ = ai + ^2 + «3 + «4 + «5-t*-. 4 «;;» ^ 
 
 = («! + ^3 + as + a2„_i) + (^2 + a, + ac . . . + «2„) 
 
 = S,„^„„_l+S,„ff2„,. 
 2n— 1 
 
 and 8„, rt„i = «i + ffo + a,'. + . . . + n->n- 1 
 
 = («! + ^3 + . . . + «o„_ i) + (tto + «4 + • • • + «2n-2) 
 
 « H— 1 
 
 = S,„ «.„_, + 8,„«2;«- 
 COU. S„, «,„ = S,„ tf^m-l + ^m fhm- 
 
 By means of these theorems we can separate the odd and 
 even terms of a given series. 
 
 15. The symbol S,„S,//,„,„ denotes the sum of r terms of 
 which the m"" is 8„ «,«,„: that is, 
 
 )• a s s s ■■! 
 
 S,„ S„a,„,„ = S„ai,„ + Sn(t->,n + Hi«:!,« + + H.«r,n 
 
 = «1,1 + «i,o + «l,3 + + «l,.5 
 
 + «l.M + O^.li + «2.3 + + «2,* 
 
 + «3.1 + rt3,-, + «3,3 + +«3.. 
 
 + &C. + &C. 
 
 + «,-, I + «.-,2 + «>-,3 + + «^r,s- 
 
 It is obvious that the same principle may be extended to any 
 number of svmbols of summation. 
 
6 
 
 16. Theorem. (8,„ a,„) x (8nK) = S„,«„, . kx * • 
 
 For, S,„ «,„ = ffi + a^ + «3 + ... + rt,.- 
 ••• (S.ff,„)x(S„&„) = ai.S„6„ + «,.S„&„ + a3-S„6„+...+ar.S„6„ 
 
 17. Theorem. If r is independent of n, and 5 of tn, 
 
 then S,„ S„ a™, „ = S„ S™ a,„, „ . 
 For S„ «,„, „ = «,„, 1 + o,„, 2 + «m, 3 + . . . + a,„, , . 
 
 r « r »■ »■ )• 
 
 •• S„ S„ a„, „ = S^ «„,, 1 + S,„ ff„,, 2 + S,„ a,„,3 + . . . + S,„ a™, „ (4) and (5) ; 
 
 03 03 03 m 
 
 18. Theorem. S,„S„o,„,„=S,„S„a^_„+i,„. 
 For, S,„S„ff„,„=S„ §,„«,„,„, (17); 
 
 CO CO 03 CO 
 
 =Sma„,i+S,„«,„,2+S,„a,„,3+...+S„«,„,„+&c. 
 
 + «1.2 + «a,2 + «3,2+--- +«,«-!, 2 +&C. 
 
 + ai,3 + «2,3+--- +«m-:?,3 +&C- 
 
 + ai,,l+...+«,„_3,.l +&C. 
 
 + &C.+ &C. +&C. 
 
 +ai,„+&c.4-ff,„_„+i,„ +&C. 
 
 + &C.+ &c. +&C. 
 
 + «!,,„ +&C. 
 
 + &C. 
 
 " That is, the product of S„, «„ multiplied by S„ b„ , is a series consisting of r 
 terms of which the m'^ is fl„, . S„ &„ . 
 
by summing vertically; 
 
 CO m 
 
 CO m » CO 
 
 19. Cor. l. S,„S„a;;,,„=S,„S„a,„+„_i,„. 
 
 X m-1 (» » m 
 
 20. Cor. 2. S„S„a,„,„=S„ai.„+S,„S„a„+i,„, (9); 
 
 » as 
 
 = S„S„a,„+„,„ (19); 
 
 CO r r m 05 r 
 
 21. Theorem. S,„S„«;„,„=S,„S„a,„_„+i,„+S,„S„a^+,_„+i,,. 
 
 05 )• )• r >■ r 
 
 For, S,„S„a,„,„=S„ai,„+S„«2,„+S„a3,„+...+S„a;„,„+&c. 
 
 = «1,1 + «1,2 + «1,3+ •••+«!.«+ •••+«l,r 
 + a2,l + «2,2 + «2,3+---+«2,«+---+«2,r 
 + «3,l + «3,2 + a3,3+---+«3,n+.'-+«3,r 
 
 + &c. + &c. + &c. 
 
 +a,„,i+a,„,2+«m,3+-- +«,«,«+•••+««,,- 
 + &c. + &c. + &c. 
 
 therefore, summing diagonally, 
 
 SmS„a,„,„=ai,i + (%,l + ai,2) + («3,l + «3,2 + «l,3)+--- 
 
 + (a,.i + a,_i,2 + a,-2,3+.--+ai,r) + (ar+l,l + «r,2 + ar-l,3+.--+«2,r) 
 
 + («. + 2,l + «r+l,2 + «r,3+"-+«3,r) + &C. + &C. 
 12 3 >• 
 
 = {S„a2-n,n + S„a3_„,„+S„«4-n,n+...+S„a,-«+l,a| 
 
 + {S„a,._„+2.n+S„a,.-«+3,,.+&c.5 
 
22. Theorem. S,iSra„,, = S„ Sra„+,-i,r 
 
 m m-n+\ 
 
 and =S,j S,«„+r-i, 
 
 m n 1 2 .-J m 
 
 For, S„S,a„,i.= Srai,r+Sra2,r+Sra3,r+ ...+Sra,«,r 
 
 + a2,i + a2,2 
 
 + «3,l + «3,2 + «3,3 
 
 + «4, 1 + «4, 2 + ^^4, 3+ Ct^, 4 
 
 +a5,i+a5,a+a5,3+a5.4+a5,.s 
 + &c. + &c. 
 
 + Cf„,l + a„_g + a„,3+ +Cfm,,n' 
 
 therefore, summing diagonally, 
 
 S„Sra„,r=ai,l + «2,2+a3,3+...+««i,m 
 
 + «2.1 + «3,2 + «4,3+...+«m,m-l 
 + «3,l + «4.2 + a5,3+...+«m,,n-2 
 + &C. + &C. 
 
 +am-i,i + a«i,2 
 
 »n m-1 »n-2 m-n+1 
 
 = Srar,r+Srar+l,r+Srar+2,v+...+Srar+»-l.r+...+am,l 
 m TO— n+1 
 
 and, summing vertically, 
 
 TO « m TO-1 »n-2 m— »i+l 
 
 S„Sra„,r = Sra,-,l + S,ai + r,2+Sra2 + r.3+...+Sra„ + , _!,„+. ..+«„,,,„ 
 TO m—n+l 
 
23. Theorem. S„Sra„,,,=S„ S,(a,„_,,r+«.,:-r+i,r) 
 
 + Sra2m-i + l,r+l + Ki ©r («2 
 
 l + % 
 
 ■+l,r + ji + a)? 
 
 and S„ Sra„,r = S„ S,.(a2n-r.r + «2»-r+l,r) + Sra2m- 
 
 >n— 2 m— n— 1 2)n— 1 
 
 + S„ Sr(a2m-r-l.r + « + l + «2m-.-l.r + „ + 2) + S,.a2m- 
 
 For, S„S,.«„,, = S,a,,r + Sr«2,r + S,a:5.r+ • 
 
 + «2,l + «2.2 
 
 + «3,l + a3,2 + «3,H 
 
 + a4,l + «4,2 + «.l,3+«.J..| 
 
 + &c. + &c. 
 
 + «2m,l + 02m.2+ • • • +«2,n.2m '■> 
 
 + S,.«2» 
 
 therefore, summing diagonally, 
 
 S„Sra„,,= 
 
 («l,l) + («2,l) + («3,l + a2.2) 
 
 + (a4,l + «3.2) + (a5,l + «4.2 + «3,3) + («6,l + «5,2+1.3) 
 + («7, 1 + «6, 2 + «5, 3+ a4, 4) + («8, 1 + «7, 2 + «6. 3+ «.% 4) 
 + &C. + &C. -f&C. 
 
 + ^2m-l,l + ^2m-2,2 + Cf2„_3,3+aj;„_4,4 + a2m-5,5+'"+^wi,m 
 + ^2m, 1 + «2m -1,2 + ^2m - 2, 3 + ^2m - 3, 4 + *2m - 4, 5 + • • • + '^m + 1. '« 
 + «2m,2 + «2m-l,3 + «27«-2,4 + «2m-3,5 + «2m-4,6+---+^m + l.m + l 
 + ^2»n,3 + ^2m-l,4 + «2m-2,5 + «2»n-3,6 + <^2m-4,7+"-+®»n + 2,»i+l 
 + «2m,4 + <^27/i-l,5 + «f2m-2,6 + «2m-3,7 + <*2m-4,8+-"+®m + 2.m + 2 
 + «2m,5 + «2m-1.6 + «2m-2,7 + «2m-3,8 + «2m-4,9+.--+^m + 3,m + 2 
 ^ +^2TO,6 + ^2m-l,7 + %)n-2.8 + *2m-3,9 + ^2m-4,10+ •••+*m + 3,m + 3 
 + &C. 
 
 + V^2m, 2m - 3 + '^Sm - 1, 2»! - 2) + KShm, 2m - 2 + ^2»i - 1, 2m - 1 ) 
 + («2m,2«-l) + (a2m,2m)- 
 
 B 
 
10 
 
 = S„S,(a2»-r,r + a2«-r+l,r)+S,.ao,«-r+l,r+l 
 m— 1 m—n 
 
 2m-l n 2m— 2 n 2m-l 
 
 and S„ Sra„,r=S„ Sra„,r+S,.a2m-i.r, (9); 
 
 m-1 « TO-1 
 
 = S„ Sr(a2n-r,r+«2n-r+l,r)+Sra2m-r-l.r+l 
 
 TO-2 TO-n-1 2)H-1 
 
 + S„ Sr(a27«-r-l.,+« + l + a2m-r-l,r + .> + 2) + Sra2,„-l,r» 
 
 by the former case. 
 
 24. Theorem. If «,„+i = «,„+&,„, C",',_i), then a„=«,+S,„6,„. 
 For, K«„.+i = sl«m+sl6,„, (4) & (5). 
 
 n--2 n-2 n-1 
 
 ••• S,„a„+i + a„=ai + S,„a,„+i + S„&,„, (9)- 
 
 .-. «„=ai+Sm^a? cancelling identical terms. 
 
 25. Theorem. If a,„+i=ca,„+6,„, C:;;_i), 
 
 then a„=c"" ^ . «! +SmC"' " ' . 6„_,„ . 
 For, «„_„ + i=ca„_,„+6„_„,; 
 . • . c"" ' . a„ _ ,„ + 1 = c'" . a„ _ „,+ c'" " ' . />„ _ „, , multiplying by c"' ~ ' ; 
 and S,„c'"-^«„_,+i=S,c"'a„_„+S™c'"-'.6„_„,, (4) & (5); 
 
 n— 2 n— 2 n— 1 
 
 «« + S„, c"" a„_^=SmC'" a„ - ,„ + c" - ' . a , + S,« c™ " ' 6„ _ „, . (9) - 
 
11 
 
 26. Theorem. 
 
 For, (S,„a._i *"'«-') (§«6«-i^^"-^) 
 
 = S„a,„_i ..T^'"-' . S„6„_i.r"-\ (16) ; 
 
 = S,„S„«,„_i.6„_i.a^'"+«-~, (6); 
 
 = S,„S„a^_„.6„_i.a;"'-S (18); 
 
 CO )U 
 
 = S„a^'"->.S„a™_„.6„_i, (6). 
 
 27. Cor. (S,„ «,„_! . J-'™-^^ = S™ ci?™-' . S„ a„-n • ««-!• 
 
CHAPTER II. 
 
 ON PRODUCTS AND FACTORIALS. 
 
 28. The Product of n factors, of which the m* is «,„, will 
 be denoted by P,„ a,„ ; that is, 
 
 P,„o,„ = «! . ttv . «3...a„. 
 
 The symbol P,„ «,„ will denote that the r"' factor is to be 
 omitted. 
 
 29. Theorem. If «,„ = 6,„, Cl,^), then P,„a,„ = P,«6„,. 
 
 For, since Gj = &, 
 a2 = 62 
 
 &c. = &c. 
 a„ = b„ ; 
 therefore, multiplying, 
 
 Oi . a.^ . (Is . . . a„ :^ bi . h.j . bs . . . b„ , 
 
 or P„G„ = P,„6„. 
 
 30. Them-em. 
 
 If 6 is independent of w, then Pm («,«&) = 6" . P„,o,„. 
 
 For, P,„ (o,„ ft) = a, 6 . 02 b . a^ h ... «„ 6 . 
 = a^a.^a-^ ... a„ . b" 
 
 = b".V„M,„. 
 
13 
 
 31. Problem. To invert the order of the factors of a given 
 product. 
 
 Now, P„, «,„ = «! . a.> . a-^... a„ 
 = a„ . «„_, . a„_2 ... «i, by inverting the order of the factors; 
 
 If, therefore, we substitute 7i-m + l form, in the expression 
 for «,„, the order of the factors will be inverted. See Art. 8. 
 
 32. Theorem . P,„ a,„ = P,„ («,„) . 'P,„ (« ,.+ „,) . 
 
 For, P,„a,„ = (fli . a.> . a^... a,) (a^+i • a,^., . ar+3..-««) 
 
 = P,„(a.). ?!(«.+.). 
 
 By means of this theorem we can separate from the rest 
 any number of factors, taken either at the beginning or the 
 end of a given product. See Art. p. 
 
 33. CoR. P„,(«-«)-P«(«v4.J='Pm«™- 
 
 34. Theorem. P„,a,„ = l. 
 
 For, P,„ («,„) . P,„ («,,, ,„) = P'l «,„, {S3) ; 
 or P,„ (a,„) • P,„ «,„ = P,„o,„; 
 
 .". Vrndm = ^ ? by division. 
 
 35. Theorem. P,„ a,„ = = . 
 
 P™«,«-« P."_,„,_, 
 
 For, P,„ a,„_„ = P,„ (a,„_„) . P,„ «„, (32) ; 
 and = 1, (34). 
 
 .'. P«(o.).P,««m-«=i; 
 

 14 
 
 
 
 
 1 
 
 -by 
 
 division ; 
 
 ka^-n 
 
 
 1 
 
 
 
 (31). 
 
 and P,„a,„ 
 
 n—1 
 
 36. Theorem. If a„+j = a,„ . 6„,, then a„ = a^. VmK- 
 
 For, b„, 
 
 «m 
 
 •1 
 
 
 
 ■■■%b,„ 
 
 = p1 
 
 /«,« + 
 
 ^)' 
 
 (29); 
 
 
 «-2 
 «1 
 
 P. 
 
 i) -a 
 
 Om + i 
 
 -, (32); 
 
 — , cancelling the identical factors, 
 a, 
 
 • •• «„ = a, .P,„6„,. 
 37. Theorem. If «„+, = a,„ . 6„, 
 
 then a,„ = a._> . V,„bo,„, and a^,,,, = Uy . 'P„A 
 
 For, ft^^ = 
 
 ••■ p16..= p1(^, (2C)); 
 
 ri—2 
 P™(a2m + 2)-%„ 
 
 „-2 , (32) ; 
 
 «2-P,«ff2«, + 2 
 <^2» 
 
 — , cancelling the identical factors. 
 
 »— I 
 
 ••• «2« = O2 . P,>,„,. 
 
15 
 
 Also b,,„_ 
 
 
 .-. pU.«-. 
 
 Vhm-J 
 
 
 P™(«2«! + l) -^SH-l 
 
 
 "~ n—2 ' 
 «1 • Pm«2m+-1 
 
 
 %n-l 
 
 
 «! 
 
 ••• ao«- 
 
 n-1 
 
 1 = ai-P,«(6„„_,). 
 
 (32); 
 
 38. The symbol [« denotes the product of n factors 
 
 forming an arithmetical progression, of which the first term 
 is a, and the common difference m; if m = - 1, the m may be 
 omitted ; and, if, in the same case, n = a, the n also may be 
 omitted : thus 
 
 jo = a(a + m)(a + 2m)...(a + n - 1 . m), 
 \a = a(a - l)(a - 2)... (a - 7i + 1), and 
 [a = a(a - l)(a - 2). ..2 . 1. 
 39. Theorem. \ab = b".\a . 
 
 For, \ab = ab . (ah + m)(ab + 2ni).,.{ab + n - 1 . m) 
 
 n, m 
 
 = b'' .a 
 
 = 6" . I ffi . See Art. 30. 
 
16 
 
 40. Theorem. I a = 1 a + w - i . m . 
 
 », m n, —m 
 
 For, \a^=a.{a+ m) (a + 2m) ...(a + n -I . m), 
 
 = (a+n-1 .m) (a+n-2 .m) (n+n-3 . m) . . .(a+m)a, 
 by inverting the order of the factors ; 
 
 = \a-h7i-l. m . See Art. 31. 
 
 n, —m 
 
 41. Theorem. \a = \a . \n + rm . 
 
 For, [ff = \a.(a + m) (a + 2m). ..(a + r - 1 . m) j x 
 
 { (a + rm) {a + r + I . m) . . . (a + w - 1 . m) \ 
 
 = \a . \a + rm . See Art. 32. 
 
 r, m n — r, m 
 
 42. Theorem. \a = i. 
 
 0, m 
 
 For, [a. I g + . m = [a , (4.1); 
 
 0, m >i,m U+n,ni 
 
 or \a . \a = \n . 
 
 .-. lo = 1, by division. See Art. 34. 
 
 0,m 
 
 1 1 
 
 43. Theorem. \a = -. 
 
 ' a — n 5 
 
 la 
 
 For, 
 
 |« -7i7n = |a -nm . \a , (41), 
 
 n— jj, m n, m —n, m 
 
 and = 1, (42); 
 • nm . I a = 1. 
 

 
 .-. \a = -. 
 
 1 
 - nm 
 
 by division; 
 
 
 
 
 
 
 
 
 1 
 
 
 (40). 
 
 See Art. 
 
 35. 
 
 
 -[a- 
 
 -m ' 
 
 
 n,-n 
 
 I 
 
 
 44. 
 
 Theorem. -. = 
 
 0. 
 
 
 
 
 For, 
 
 1 
 1 -m 
 
 1 
 
 = \- m + m. 
 
 (43) 
 
 
 
 
 ^ 
 
 = 0. 
 
 45. The theorems in Articles 34, 35, 42, and 43 are ana- 
 logous to the equations a°= 1, and a~"= — ; which last equations 
 indeed may be deduced from them as particular cases. 
 
 Iw In 
 
 46. Theorem. ^^— = 
 
 For, jw . \n- m = \n 
 
 m 
 
 n—m 
 
 [n [w^ 
 
 , Wl fl — TO 
 
 I m \n — m 
 
18 
 
 I w \n \n 
 
 + 1 
 
 47. Theorem. ~— + 
 
 For, 
 
 Im m + 1 I m + 1 
 
 \n In \n 
 
 b- hu hr f n-m\ 
 
 [to [m + 1 Im V m+\) 
 
 \n .{m + 1 +n -m) 
 
 — "» 
 
 |to + i 
 
 (ti + 1) |w 
 
 ly^ + l 
 
 + 1 
 
 \n 
 
 48. Problem. To shew that ^ is a whole number ; n 
 
 and 7» being integers. 
 
 \m 
 
 \r + \ \r \r 
 
 Now, fii = _:2ii_+^, (47); 
 
 |to+ 1 |m+ 1 [TO 
 
 .-. — 2iI_=--2i]^ + S,f-, (24); 
 |to + 1 |to + 1 [to 
 
 Ito 
 
 m 
 
 [to 
 
 If, therefore, ^ were an integer, _2+L_ would be an integer ; 
 
 '— TO+l 
 
 \n 
 
 \n 
 
 but i— is an integer, and therefore j— is an integer. 
 
19 
 
 n 
 
 49. The symbol {a^ + b^ \...\ c ] ...] denotes the re- 
 
 m m+1 n+1 n+1 1 
 
 suit of the combination of the symbols 
 
 {«! +fti {tto + h^ {"-{an + K { c f ... j ; 
 
 1 2 3 n n+1 w+1 1 
 
 the brackets being omitted after the expansion, if they are then 
 without signification. 
 
 n n m—1 m 
 
 50. Theorem. | a,„ + 6,„ 5 . . . {c = S„«,„ . VA + c . VA- 
 
 m m+1 n+1 
 
 n 
 
 For, {arr,'k'h^\...{c={a^+h^{ao+h^\...\ c \...\ 
 
 m m+1 n+1 12 3 n+1 n+I 1 
 
 = {«!+ {61 052+ |&1&2«3+ { ••• 
 12 3 4 
 
 + \hA'-A-i • ««+ \bih..Ac} ... I 
 
 n n+1 n+1 1 
 
 = aj+6ia2+6i&2ff3+... 
 
 + bA"A-lC'n + bA"-bnC 
 
 = S^a^"VA+c-VA- 
 
 51. Theorem. If a„ = 6„ + c„ . a„+„, 
 
 m 4-1 m 
 
 then a„ = S,6„+— i.„ . PtC„+in.« + ««+«,« • PtC«+i=l.« ■ 
 For, a„=56„+c46„+„+e„+„j...{&„+— i.„+c„+,-Zi.4--l«n+'««S-i' 
 
 12 3 s s+1 m+1 m+1 1 
 
 by substitution ; 
 
 m 
 
 = l^n + .—j.a + C„ + ,— i.„ j ... {a« + m«» W ; 
 J s+1 m+1 
 
 = S,6„+-,...PtC„+^ri.a + ««+m«-PtC„+<-ri.«, (50). 
 
 52. CoR. If a„,„ = 6,„,„ + c„,„ . am+«,n+e» then 
 
 >• j-i 
 
 «m,n = Ss&m + iri.«,n + jrT.)3- PtCCm+TTi .«,« + <". /s) 
 
CHAPTER III. 
 
 ON COMBINATIONS AND ARRANGEMENTS. 
 
 53. 1 HE symbol Crffr will be used to denote the sum 
 of every possible combination^ (without repetitions of any one 
 letter in the same combination,) that can be formed by taking 
 m at a time of n quantities of which the /*" is a^; and the 
 
 m, n; s 
 
 symbol C,a, will denote the same thing, with the condition 
 that Oj is to be every where omitted. 
 
 m+l,M+l m+l,n m,n 
 
 54. Theorem. Cr a^ = C,- a,+a,,+■^ . Cr Or- 
 
 m+\,n+l 
 
 For, Cr^r must consist of terms into which a„+i does not 
 enter as a factor, and of others into which it enters as a factor 
 
 once only ; and it is obvious that Cr Or will express the first 
 
 m, n 
 
 set, and that a„4.i.C,«r will express the second. Hence the 
 truth of the theorem is manifest. 
 
 55. Theorem. If 6 is independent of r, then 
 
 "Criarb) = 6'".c"fv 
 
 m, n 
 
 For, Cr («r b) denotes the sum of a certain series, each term 
 of which is the product of m quantities, and into each of which 
 
 quantities b enters as a multiplier ; and Cr a^ denotes the sum 
 of a series, each term of which is the product of the same m 
 quantities, each being deprived of its multiplier h. 
 
21 
 
 56. Theorem. If a is independent of r, 
 
 \n 
 
 m.n L_ 
 
 then C, («) = f— • a'"- 
 m 
 
 ... ^ 
 
 For, the number of terms in C.a, is f— ; and each term 
 
 |m 
 
 consists of m factors. Since, therefore, in C,- («) each of these 
 factors is equal to a, the truth of the theorem is manifest. 
 
 57. Theorem. -^ ='c'X-'. 
 
 Vra, 
 
 For, the numerator of the first member of this equation con- 
 sists of every possible combination of 7i quantities, taken m, at a 
 time ; and, hence, that side of the equation consists of a series of 
 fractions, the numerator of each being unity, and in which the 
 denominators are formed by taking away, in every possible 
 manner, m of n given quantities, and will, therefore, consist 
 of every possible combination of these n quantities taken 7i-m 
 at a time. 
 
 58. 
 
 Theorem. 
 
 0,n 
 
 = 1. 
 
 For, 
 
 0,« 
 
 = 1. 
 
 (57) 
 
 59. The symbol Cr,s(«r-^s) denotes that there are 7i 
 quantities of which the r*"^ is a,., and n others of which the 
 s^^ is 6s, and that every possible combination, (witliout repe- 
 titions of the same quantity in any one combination,) is to 
 be formed of the first series, by taking them m at a time; 
 
22 
 
 and that each combination thus formed is to be multiplied 
 hy n — m quantities of the second series, so taken that in 
 each of the combinations the whole of the natural numbers 
 from 1 to w shall appear as indices : thus, 
 
 + Ozdihibzh^ + a^ar^bih^bi + a^a^hib2b'i. 
 
 m, n—m m, n 
 
 60. CoR. If h, = 6, then Cr, . {a, . b,) = 6""'" . C. «r- 
 
 61. Theorem. 
 
 n— m+l, m n—m, m n—m+1, m—1 
 
 n— m+l,m 
 
 For, Cr,s (^r • &«) will consist of terms into which a„+i enters 
 as a factor, and &„+i does not; and of others into which 6„^., 
 enters and a„+i does not. Also each of these terms must con- 
 sist of n factors, exclusive of the factors a„+i or 6„+i ; and each 
 of them must contain n-m+1 factors of the series a^, a2,...a„+i, 
 
 n-tn+l, n 
 
 and m of the series fcj, 625. .-^n+i- Also C^.sC^r-^i) must con- 
 tain every possible term that can be formed consistently with 
 
 these conditions. Hence «„+i • Cr,s(«r-6s) will contain all the 
 
 w— m+l,m— 1 
 
 terms of the first kind, and 6„+i . C,,s(«r-6s) all those of the 
 second kind. 
 
 62. The symbol A+„am denotes the sum of every pos- 
 sible Arrangement that can be formed of any numljer of 
 quantities of which the wz*'^ is a,„, these arrangements being 
 subject to the condition that the sum of the indices subscript 
 shall in every single arrangement amount to n; repetitions of 
 the same letter being allowed in any arrangement : thus 
 4 
 A+„,o„, = o, + a^a^ + a.^a.^ + ao^i^i + a,ajaiai + a^a^a^ + ffia, 
 
 = ^4 + 20.301 + 02^ + 202 • «i^ + «A 
 
23 
 
 63. Theorem. A + rO. = Smflm- A + r«r 
 n m—\ 
 
 = S„a„_;„+i.A+rar- 
 
 For, A+rfl^r is the sum of all the terms that can be formed 
 of any number of quantities «!, «2> &c. such that the sum of 
 
 n — m 
 
 the indices subscript shall be w; now a^.A+^a,- will include 
 
 n n — m 
 
 every term in which a^ is a factor, and Sma^.A+r^r will 
 include all the admissible values of a„, and therefore every 
 
 term of A+rOr- 
 
 n n n — m 
 
 .-. A + r«r = Smttm- A+r«r 
 M m — 1 
 
 = S«0„_„ + ,.A + rar, (8). 
 
 64). CoR. A+rOr = 1- 
 
 n n 
 
 65. Theorem. If «„ = Sma„_„ . 6„,, then a„ = ao • A+r^, • 
 
 1 
 For, fli = tto • &i = <^o • A + r &,- 
 
 ^2= «l6l + 00^2 
 
 2 
 
 = tto (6l^ + 62) = «o • A + r 6r • 
 
 a^ = «2 ^1 + ^1 ^2 + ^0 ^3 
 
 = «o(&l^ + 6l&2 + 62 6l + 63) 
 3 
 
 = «o. A + r^r- 
 
 ^4 = ag&i + a^br, + ajfeg + 0064 
 
 = tto (6i* + 61^ 62 + 61 62&1 + 61 63 + 6261^ + 62^ + 6361 + 64) 
 
 4 
 
 = Oo-A+rftr- 
 
 n 
 
 Suppose, therefore, a„ = Oq • A+r6r ; 
 
 n+l 
 
 then o„+i = S;„ «„_„+! 6;n 
 
 n+l n— ni+l 
 
 = S;„ao.6„.A+r6,. 
 
 n+l n— m+1 
 
 ao.S.6„.A+.6„ (6); 
 
 n+l 
 
 ao.A+.6r, (63). 
 
24. 
 
 If, therefore, the law were true for n and all inferior integers, 
 it would be true for n+\ ; but it is true for 1, 2, 3 and 4, and 
 therefore for n. 
 
 QQ. Theorem. If a„ = Cn+ S^ a„_m • h^ , 
 
 n TO— 1 n 
 
 then a„= S„.c„_,„+i. A+.fer + «o- A+v6,.. 
 For, proceeding as in the last Article, we shall find that 
 
 4 m-1 4 
 
 «4 = S„C5_,„. A + r^r + Ao- A + r&r- 
 
 Suppose, therefore, a„ = S™ c,_„+i • A+^&r + «o • A+r 6^, 
 
 n+l 
 
 then o„+i=c„+i+S;;,6;„.a„-,«+i 
 
 n+l n— TO+1 i-1 n— m+1 
 
 =c«+i+S^6„SS,c„_^+i_,+i. A+,6,.+ao-A+,64' by substitution; 
 
 n+\ TO— 1 s— 1 «+l n— TO+1 
 
 =c„+,+S;„&„_,„+2-SsC„_,. A+,&r+ao-S„6„. A+r&r, (§) and (6); 
 
 s-\ n m s-l 
 
 n+l 
 
 +ao.A+r&r, (9) and (63); 
 
 n m s~\ n+l 
 
 = C„+i + Sm6„_m+,.S,C„_s + i. A + r^r + Oo- A + ,.6r 
 n n— TO+1 s—\ n+\ 
 
 =c„+i+S„S,6„_„_,+2-c^- A+.&r+«o. A+,6„ (6) and (22); 
 
 n n— TO+1 s-1 >J+1 
 
 = C«+l+S„C„.Ss&„-™-s + 2- A + r^r+flo- A + ,^, (6); 
 
 n «— TO+1 n+l 
 
 =c„+i+S,„o„. A + ,ft,+«n- A+,.6r, (63); 
 
 n m «+l 
 
 = C«+l+SmP„-« + l- A + ,6, + ao- A + ,6,, (8); 
 n+l TO n+l 
 
 = S;„c„_„+2.A + ./>.+an-A+,6„ (9). 
 
 If, therefore, the law were true for n and all inferior integers, 
 it would be true for r/ + l ; but it is true for 1, 2, 3 and 4, and 
 therefore for n. 
 
CHAPTER IV 
 
 ON BINOMIALS AND EXPONENTIALS. 
 
 w+1 n—m+l,m- 
 
 67. Theorem. P,(a, + &,) = S. a,,K.6.)- 
 
 For, by actual multiplication, 
 3 
 P,.(a,, + br) = a^a^a.^ + a^a^h^ + a^a^bo + aza^b^ 
 
 = s™ a,,(«,..&.). 
 
 M+l H— m+1, m—l 
 
 Suppose, therefore, P, {a, + 6,) = S;„ Cr, ^ («, • &»)» 
 
 M+l n+l n-m+l,m-l 
 
 then P,(a, + 6,) = (a„^, + 6„^0 . S. C,.(a.. 6,) 
 
 «+l n-m+l, m-1 n+l n-m+l, tn-1 
 
 = S™a„+i . C,,,(a, . 6,) + S™6„+i . a,,(a, . 6,) (6) : 
 
 = ««+!• C,,,(a, . b,) + S„ a„+i . C,,,K . 6,) 
 
 n—m+l, m—l 
 
 + S.6„+x . a,,(a.. 6.) + 6„+i . a,.K. 6,), (9) 
 
 :C,U«,.6s)+S„{a„+i.d.,.(ar.6,)+&«+i-CU«r.6.)J+C,,,(ar-^), (5); 
 
 a,,K.6.)+s. a,;K.6.)+a,.K.6.), (61); 
 
 n+2 n—m+2, -i 
 
 = s. a,K.6,), (9). 
 
 If therefore, the law were true for n factors, it would be 
 true for n+l ; but it is true for 3, and therefore, it is true for n, 
 D 
 
26 
 
 n M+1 n—m+l, m—1 
 
 68. Cor. 1. Put 6, = a?, then P^(^ + a,) = S,„ C^.^Ca^-r) 
 
 n+l «-m+l, n 
 
 = S„.J?'"-^aa„ (60). 
 
 69. Theorem. If a,, is the r'^ root of the equation 
 
 »+\ n-m+l, n 
 
 = S,„am_] ..2?'""S then shall am-i=Cr(-ar). 
 
 For, S;„ «„_, . .r™-' = P,(a; - «,) 
 
 = S„^™-'.a(-a.), (68) 
 
 .-. «,„-! = (CiV- «r)- 
 
 70. Problem. Given 6r-i = Smffm~' • '^m? C=m)' to find a7(. 
 Multiply both sides of the equation by C,(-a,) 
 then 6._"7(i'(- «.) = L^m . «:-"~U'(-«.)» (6)- 
 ••• kbr-r~K(-a,) = S.a;,„.a<-'".~d'(-«.), (4) and (17); 
 = S™.x-..P,K-a,), (68). 
 
 re, < 
 
 But P,(«m-ffr) = 0, for every value from m=l to ?» = w, 
 except for m = t; 
 
 n n—r, n; t v ; t 
 
 therefore, Sr6,_, . Cs(- a,) = .r, . P,.(at - a,), 
 
 P,(n, - o,) 
 
2-7 
 
 71. Cor. If6,_,=6'-', then a, = ?4^1^lMl°i) 
 
 
 72. Theorem. 
 
 n+l „+i 
 
 .2? - 6 .r - 6 
 
 For, '-6'" ' J'g^^,_,^^,_,_ 
 
 .?? - 6 ^ 
 
 0? - cZ' - o 
 
 n+l H+1 
 
 and r = Ka^-i.S,.x'-\b'"-"\ (4); 
 
 ,v - b X -b ' V / 
 
 = fflo • Sra-''"' . b-' + S,„a,„ . Sr.'c''"' . 6'""% (9) ; 
 
 = S.cr— rS,«„^,_, . 6-\ (6) and (22). 
 
 x - o w -b 
 
 «+i 
 73. Cor. If 6 is a root of the equation 0=SOT«m-i-'2?"'~S 
 the second side of the equation is divisible by x-b. 
 
 For, the remainder after the performance of this division 
 
 n + l 
 
 is Swa^-ift"'"^ ; which =0, since b is a root of the equation. 
 
28 
 
 74. Theorem. 1 0+6 = S™ , "' ' • I « -1 6 ; ^ being any 
 
 positive integer. 
 
 For, |a+& =g+fe=|«+|^ • 
 
 la+6=(a+6+»^)([« + |6) = [a . (a+r+fe) + [6(a+6+r) 
 
 = [a^+[a.[6^+|^. |^a + |6 = |^a + 2. la. [6 + |^. 
 
 2,r \,r l,r \, r \Tr 2, r 27r \,r \~ 2,r 
 
 |a+6 =(a+6+2r)(|a+2.|«.|fe + |6) 
 
 3,r 2, r l,r l.r 2,r 
 
 = |a(a+2r+6)+2 . [a . l6(a+r+6+r) + j6(a+6+2r) 
 
 :[a+ [a^. [6+2 . [a . [6^+2 . [a . |& + |a • [6 +[6_ 
 
 3,r 2,r l.r 2~ \, r l,r 2,r 1 27)^ 3,/- 
 
 2, r l,r },r 2, )• 
 
 A 4— m, »• m- 
 
 Similarly, I a+b = S„, i ""^ • | a . 1 6 
 
 Suppose, therefore, l^/-L/»-<g »'-i U U ; 
 
 ^777 — [to-1 L_m+i,,. brT;7 
 
 then |a+6 = (a+6+wr) . |a+6 
 
 /.+ ] [w_ 
 
 = Sm , '"-' \f'' -ll' .(fl+7l-m + l.)'+h 
 
 + m-l . ;•), (6); 
 
29 
 
 \m — l 7i-m+2,r m-\.r 7i-m+l,r m,r 
 
 n \n » I** 
 
 (9); 
 
 + |a_. 16 
 
 0, r n 
 
 k+TT. 077 \\m \m-l I b;^^-. mT7 o77 k+ITr 
 
 
 = [a_.[6_+S™.:i [^ •l.i+L^-[i_' (^7)' 
 
 i+l,/- 7H,J- 0,r w + l,i 
 
 m-1 lrr;;r+27r ^13177 
 
 If, therefore, the law were true for n, it would be true for 
 n+1 : but it is true for 4, and therefore, for n. 
 
 75. CoR. 1. \a-b = S,n'r^^^^- \a • | -& 
 
 n, »• W— 1 M-m+1, r m-1, »■ 
 
 But 1-6 =(-1)™-^ [6 , (39). 
 
 m-l,r m-l,-r 
 
 i— 1 »_m + I,»- TO-1, 
 
 76. CoR. 2. Since lw=U . U-w+l , .-. |w = - — 
 
 U .16 
 
 
 andiiL:-«=S„(±ir -7 r 
 
 n —m+l,r m —^^r 
 
30 
 
 77- Theorem. 
 
 \ |m-l ^ ^ \n-l ' 
 
 a+6. A'" 
 
 For, S."-i=i^^ S„^^^if^: 
 
 ^ \m-\ I V \n-\ ' 
 
 \ a .\b_ 
 
 = S.^"'-' •§ "'-- "-^'- , (26); 
 \m—n. \n — \ 
 
 = S.a''«-'.pi:^, (76). 
 
 78. Cor. 1. (s.^^^f^^ ) = S 
 
 n being any positive integer 
 lo-fe.<r'" 
 
 h . ,v" 
 
 79. coK.o. (s. -;- ^ ) (a^^^i^^— -) =s. 
 
 ja-6. 0?*""^ 
 
 
 80. CoR. 3. S, 
 
 a 
 n 
 
 ^^.-1 
 
 |m-l 
 
 m-l.r 
 
 
 \ » eo ~,m - 1 
 
 
31 
 
 1 1 
 
 81. Cor. 4. (S4a -^) = S™|a_. ,-^) 1 
 
 I t a?™-' 
 
 \n \m,-l 
 
 m-\,r 
 
 82. Cor. 5. Put S,„ k '.t^^ =/(a)5 
 
 then ;/(«)?-"= 
 
 /(O) 
 
 
 =/(0-wa), (79); 
 
 
 83. Cor. 6. S/(a) J -'=/(-«) 
 
 TO-l.r 
 
 84. morm. (SJa .^Wsj6 .^1 
 
 = Sm 1 0'-\rb . h terms in xK 
 
 For, (S.^.J^)(S46 .^) 
 
 = S;„ S^ ,"-^"- ;-^''- . w'^+"-% (16) and (6) ; 
 
32 
 
 a . \b 
 
 = S^ S„ ■."'""•'" — p^i^ . x'" - ' + terms in w\ (l 8) ; 
 m-w. \n-l 
 
 a . b 
 
 85. 
 
 Sm x'"~ ' • Sn "'-"■'• "-I''- + terms in ,r*, (6) ; 
 , \a+b.w"'-\ 
 
 = Sot —-^-^^ 1- terms in a;\ (76). 
 
 COR.I. [Sm^ )=S„.^ 
 
 ^ \m—l I \m-\ 
 
 Theorem, (a + bY = Sm , "^' .«""'"+ ^ ft*" "' ; 7i being 
 m-l 
 
 + terms in ^r*. 
 
 any positive integer. 
 
 n n+1 m-l,n 
 
 For, P, (a+b,)= Sm «"'"■'' • C. (6.), (68) and (8). 
 Put 6,= 6 then ia+by=S,na"-'"+\c"(b) 
 
 = S^-r^i-.a''-"'^'.b"'-\ (56). 
 
 This theorem may also be proved as follows : 
 
 Put r=0, then la+6 = (a+6)", \a =a" 
 
 n,r n—m+\,r 
 
 and [6 =6'"-^ 
 
 
33 
 
 87- Cor. 1. If we invert the series we shall get 
 
 («+6)"=S,„ """^' . «'«-' .6"-"'+', (8); 
 
 n-m+1 
 
 but 
 
 n-m+l \m-l 
 
 4, (46); 
 
 therefore the coefficients are the same when taken in an inverted 
 order. 
 
 88. Cor. 2. Since \n=\n .\ n-m+l, 
 \n ~ '" "" \n-m+l Un-l 
 
 89. Theorem. 
 
 \m-l 
 
 according as n is even or odd. 
 
 n+l I 
 
 For, (a+6)«=S„,7-^:i^:i-.a'— " + '.6"-', (86). 
 \m—l 
 
 (1) Let w be even ; then 
 
 \n \n 
 
 2' 
 
 
 + S,T-=^.a-'.6— ', (9), (8) and (46). 
 \m—l 
 
34 
 
 m-1 \^n 
 
 (2) Let »z be odd, then 
 
 'm— 1 m— 1 
 
 (8) and (46) ; 
 
 = S, T-2::i- . (aft)'"-' (a''-='^-+-^+&"-^'«+^), (5). 
 
 m-1 
 
 90. Cor. If n is even, 
 
 Im — 1 
 
 in 
 
 H-'^y".r^.{aby"; 
 
 and if w is odd. 
 
 (a-fevJS™ (-])'"-'. •;^:^^^. (0.6)'"-'. («"--'" -^'-6"--"'+0- 
 m-1 
 
 91. Theorem. + .vf '■ = S,u 
 
 n W" ~ ' 
 ± - . -, ; n and r 
 
 r \m-l 
 
 being any positive integers, and x being less than unity. 
 
 For, (S„ 
 
 r \m-1 
 
 = Sm\n_ • T—— , (78) and. (82) ; 
 
 = S,.j^.,v"-\ (13) 
 m-l ^ ^ 
 
 = (l+.vy, (86). 
 
35 
 
 r ' \m-l 
 
 \n_ 
 
 92. Cor. l. (l +•«■)" =SmT^"-^ • -^""S where n is any 
 \m—l 
 
 rational quantity ; x being less than unity : (86), (l3) and (91). 
 93. Cor. 2. (l + xy=Sn 
 
 = S.f^- - , (39). 
 
 94. Cor.. 3. (l+.v)»=Sj-. 
 
 11 x""-' 
 \n' Im—l 
 
 = i+sj-.f^, (9); 
 
 But 
 
 l-=|i — 
 
 (39); 
 
 l-n.~^, (41) 
 
 = \n-l. 
 
 :^ (-l)'"-^n'" 
 
 m 
 
 \n—i 
 
 .-. (.+.)^=.+s.(-.)-.^.0'. 
 
1 A>"'- 
 
 n ' \m- 
 
 95. Cor. 4. {\+x) "=S„ 
 
 =i(-ir-.f^. (-)"■", (39). 
 
 \m-l \n) 
 
 96. Theorem. 
 
 V |m-l / V |w-l / ^m-1 
 
 m-l /V w-1 
 
 S„-^ — l=S„a?"'-'.S„, i -, (26); 
 
 -n\n-l 
 
 n 
 
 97- CoR.1. P,.(s /-, "' -Us.- ^^-7 '"^ 
 
 \ \m-i J m-l 
 
 98. Cor. 2. Put a,=a, 
 
 then S„-n — =S/ ^ 
 
 [m-l 
 
 m-l 
 
 99 
 
 fo fay-' cv"'-' Y " a-^-^a?"-' 
 
 . Cor. 3. {S,„ - ., 7}=S.-^ — ; 
 
 ( \nl I m-l J I m-l 
 
 ■ i^- I m-l / -^""UJ -f^^r 
 
 100. Cor. 4. 
 
 
 ,m - 1 „m - 
 
 [m-l ' 
 
37 
 
 \ \m-l I \ ■ \n-\ J 
 
 (a -ft)'"-' .x'"- 
 
 101. Cor. 5. 
 
 g (-a)'"-^c^;'"-^ /g g"-'.^"-' -. - (^z^O;!llif!l! =i 
 
 "" 1 7/2.-1 J \ " jw^l / "" im-1 
 
 .'. s, 
 
 =s„ 
 
 102. Cor. 6. 
 
 [ lw-1 j [ m-l J 
 
 and 
 
 =S.-^-^^f^ ^ , (101); 
 
 \m-l ^ ' 
 
 2^ ar-\x"'-\ " fp^ (^pay'^.w'"-^ 
 
 Om 1 I = ^m j ; 
 
 \m-l / V m-.l 
 
 (±£:.a)'"-i.,T" 
 Im-l 
 
 (.?.«)-. 
 
 103. 
 
 /CO a'"-' \ * " O 
 
 Cor. 7. Put .t=1, then S™ , =8;, — f- 
 
 V \m-lj \m-l 
 
 104. Cor. 8. Put a=V±l, then 
 
 
 8„ 
 
 m-l j jm-1 
 
 where .r is any rational number. 
 
The series S„ 
 
 38 
 
 1 
 
 j occurs very frequently in algebraical 
 
 investigations, and therefore we shall use the symbol e to re- 
 
 present it; while e"^-' will be used to denote Sm — ; • 
 
 ^ \m-l 
 
 Hence the above equation may be written 
 
 ■^-^=s„ 
 
 m-l 
 
 105. Cor. g. Let a? be irrational ; and suppose y and % 
 are two rational numbers very nearly equal, such that x > y, 
 and x<%. 
 
 Then e", e^ and e^ are in order of magnitude ; that is, 
 
 S„ 7- ? €'% and S,„T^ , are in order of magnitude. 
 
 \m-l m-l 
 
 But 8,ni . S^r 7' ^rn^ -, are also in order of 
 
 |m-l jm-l \m-l 
 
 magnitude however near the values of z, and y are taken to 
 
 that of x; 
 
 106. CoR. 10. Hence, whatever the value of x may be, 
 
 CO .,,m-l 
 
 we shall have €"^=8^ 
 
 lw-1 
 
 107. Theorem. „'=S,/-^J^ 
 
 m-l 
 
 For, a = 6""^'-"*; 
 
 By log( a is denoted tlic logarithm of a in the system whose base is 
 
39 
 
 » (a-.log^.a)" 
 
 (106). 
 
 108. Theorem. {a^hf=ar^2ah^-h-. 
 This will appear from actual multiplication. 
 
 109. Theorem. (S^ «,„)'= S,„a„,'+ 2. S„fl„.S, «,„+,. 
 
 n— r+l n-r 
 
 For, S;„ar+«-i=a.+S„ar+m- (9); 
 
 n-r+l n-r n-r 
 
 .-. (S..a,.+.„_0'=«.'+2a,S,„«,.+,„+(S,„a,^,„r, (108) ; 
 
 n n— r+l n n n—r n n—r 
 
 and S,(S„ar+™-,)'=S,.a,^+2S.a,.S™a,^,„+S,.(S,„a,^,„)^ (4), 
 (5), and (6) ; 
 
 • •• (S,„aJ^+K(Slar+,«)'=S.«r+2aa,..sl«,.+ . 
 
 n— 1 n—r 
 
 +a(S.„a,.,»)-^+(S,„a,„„)^ (9); 
 
 n n n n—ni 
 
 ••• (S™«™)'=S,„a^'+2S^a;„.S,a,„+,. 
 
CHAPTER V. 
 
 ON FINITE DIFFERENCES. 
 
 110. If (p(u) is any function of u, then (p(p{u) will 
 denote the same function of (p{u). This last is expressed by 
 0^ (m) ; and, the same notation being extended, we get the 
 equations 
 
 (p(p"'{u)=(jf+^{u), and 0"'0"(7/)=0"'+"(w). 
 
 111. CoR. 1. <^°.0»(m)=<^"+"(m) = 0"(w). 
 
 .-. (If(u) = u. 
 
 112. Cor. 2. 0-'. 0"(w)=0-*+"(w) 
 
 113. Definition. If 0(w) is such a function of u that 
 ^(u+v)=(l){u)+(p(v), then 0(w) is called a distributive func- 
 tion of «. 
 
 114. Definition. If 0(m), and >//(w) are such functions 
 of u that ^^(u) = \ly(p(u), then the functions <^(m) and \//(w) 
 are said to be commritative with each other. 
 
 115. Instead of 0(w) + \|/(w), it is frequently convenient 
 to write (0 + \|/)?^; in which case the latter expression must be 
 carefully distinguished from the product (0+>//)xw, and must 
 be considered merely as an abridgment of the full form 
 0(m) + x/,(m). 
 
41 
 
 116. We shall express {(0+>|/)(0+^//)}w, by (0+x//)2W, 
 and \((p+^L){(f)+^l^)((j)+^l,)^u, hy (cj)+ylf%u: 
 
 that is, (^+\//) w=^(w)+\//(w), 
 
 and, similarly, (0+'v|/)„?^=0(0+\//)„_iW+-v//(0+>|/)„-iW. 
 
 117. The symbol |(0r+^r)lw, will be equivalent to the 
 
 r r+1 
 
 expression 
 
 l((pi+^^M2+i^d-'((pn+i^n)\u. (See Art. 49). 
 
 118. Theorem. If (px{u)^ (p%{u)-> (psiu), &c. and -vZ/^iCw), 
 \//o(?«), yJTiiu), &c. are all distributive functions, and commuta- 
 tive with each other, then shall 
 
 )• r+1 
 
 For, (0i+>//i)w=0i(m)+\//i(w), (115). 
 
 2 
 
 K0r+^^,)!w = ((^2 + ^/'2)!0l(w) + ^|/,(w)i, (H?) ; 
 
 )■ r+l 
 
 =02S0i(?^)+^i(?^)S+f2{0i(«O+V'i(«*)i' (115); 
 =</)20i(w)+<^2'^i(w) + >//o0i(w)+\//2x|/i(w), (113); 
 
 = 0102(w)+01^2(w) + 02^l(w)+>|/l>/'2(«*)» (114). 
 
 ! 
 
 ;(0,+\/.,.) 5 M = ((^3+^/^3) 1</)i<^2(w)+0ix|/2(m)+02v//i(w) 
 
 +v|,ix|,,(?OS, (117); 
 
 = 03 5 <^1<^2(w) + <^,>//2(w) + <^2^1 (W) + ^^1^/'2(W) 5 
 
 + ^3S</)102(W) + 01^|'2(W) + 02^|'1(W) + X/^,>|/O(W)}, (115) 
 
 F 
 
42 
 
 = (010203 + 0102'^3+ 0103^2 + </)203^1 + 01^2^3+ 02'^l'^3 + 03'^l'^2 
 
 + ^ii^2i^3)u, (115); 
 
 4 4— m, m— 1 
 
 n+l n— m+1, m- 
 
 Suppose, therefore, j (0r+^r) {^=8;;, | Cr,s(0r-^s) | w ; 
 
 n+l n+l n-ni+1, m-1 
 
 then {((/),+>/.,) fw=(</>„+,+v//„+OS.{ C,.((^.4.)W. (117); 
 
 r r+1 
 
 ji+1 n— nj+1, m— 1 n+l n— m+l, m— I 
 
 =0„+i[S4 C,X0r4^)}w]+>^«+i[S4 CU^.-^/'.)}^]. (115); 
 
 =s.0„^:[{ c„(<^.4s)!^]+s.v/.„^i[5 c,,(<pr4.)!^]' (113); 
 
 M+l n—m+l, m—l n+l n— m+1, m— 1 
 
 =S4(^„+,CU0..x|/,)Sw+S4>/.„+:C,,X0..>/^.)iw, (115); 
 
 n+l n— 7n+l, m—V n+l n—m+l, m- 
 
 = ss.(3!)„^ia,x0r4O+s.>/.„^xC,,.(0,4O J -w, (115) 
 
 = S0n+l.C,X0r4^) + S™0„,,.C,X0.4.) + S.x/.„,,a,((^r4O 
 
 oTn 
 
 + ^/'« + l•C„(0r.^|'.)Sw, (9); 
 
 n+l, n n—m,m n—m+l, m—l 
 
 sc,x</)r->i'«)+s„[0„,.,.a^(<^.4,)+^^n+i.a,((/)..>|..)] 
 
 oT^i 
 + C,.,(0r->^.)Sw, (5) and (114); 
 
 n+l, n n-m+l,m 0,n+l 
 
 =!a,(0r4o+s. a„(0r4o+a,((/).4.oSw. (^o 
 
 n+2 n-m+2,m-l 
 
 ss„ a,(<^,40Sw' (9) 
 
 n+2 n-m+2, m- 
 
 =s.| a,(0.>^,)>, (115). 
 
43 
 
 If, therefore, the law were true for w, it would also be true 
 for w+1 ; but it is true for 3, and hence it is true for w. 
 
 119. CoR. 1. Let the functions (piiu), 0a(w), &c. be all 
 similar to each other, and to <p{u) ; also let \//i(w), ^,>{u), &c. 
 be all similar to >|/(?^) ; then 
 
 {i<pr+'^r){u becomes (0+>|/)»m, and 
 
 -I »— m+l,m- 
 
 S,„ 1 C,.X(pr.^s)\ u becomes SJj^^.(/)"-'"-^'.x//'"-7m, 
 (56) and (60) ; 
 
 120. Cor. 2. But if cf), and \// denoted quantities instead 
 
 n 
 
 of functions, then would (0 + vl,)« = Smr^^^^^- 0''"" + '^^"'"' ; and 
 
 m-1 ^ 
 
 hence we may express the preceding result by the equation 
 
 (0 + \//)„W=(0 + \//)".7f. 
 
 This must by no means be considered as an identical equa- 
 tion ; for the first side is merely an abridged expression of 
 certain functional operations to be performed, while the second 
 is a compendious method of denoting the expanded result of 
 these operations. In fact these expressions will not generally 
 be equivalent unless (p(u) and -^{u) are both distributive and 
 commutative with each other. 
 
 121. CoR. 3. If v//O^) = S„,«,„_,.0,„_iOO + X«('^)' where 
 ^„(?«)=0 for some value of 71 and for all succeeding values; 
 then we may put 
 
44 
 
 and, if 0i(w), ^2(w)5---0«(w) are distributive functions, and 
 commutative both with each other, and with any constant factor, 
 we shall have 
 
 >/'Hw) = (S„a„_i.0„-irw, (120). 
 
 122. CoR. 4. If, in the same case, a„,_i = a™~\ and 
 0m_i=0'""S then 
 
 r/,"(w) = (S„a'«-'(^'«-'rw 
 
 = (rZ^)"^^' (12) and (13); 
 = (l-a.(p)-\u. 
 
 123. It will be readily seen that the preceding theorems 
 of this chapter will hold not only when (p and \|/ are symbols 
 denoting functions of which the successive orders are deduced 
 by a series of substitutions, but also when they denote functions 
 of which the successive orders are deduced by performing a 
 series of operations all of which are subject to any given law. 
 An exception, however, must be made with respect to the theorem 
 
 ^-".(p"(u) = u: 
 for, if (p^ denotes an operation such that 
 0,,(z^ + o)=<^,(?*), 
 where a is independent of w ; then 
 
 0r"^-0.r(w)=W + Ci, 
 
 0."'-<^/(w)=0.(w)+C„ 
 
 (p.T ' ■' ■ (px' (tl) = W + 0.r " ' (Cl) + C,, 
 
 and 0,r"-0/(M)=M+S,„0.r*"-'"*.c,„; 
 
 where c„ is some quantity independent of ,v, and is to be de- 
 termined l^y the conditions of the problem. 
 
45 
 
 124. If M is a function of any number of quantities, two 
 of which are a; and y; then (p,r(u) may be used to denote the 
 result of an operation in which x only undergoes a change, 
 and (by{u) a result similarly obtained on the supposition that 
 y is the only variable; while (p^-fpyiu) will denote 0^. \(p,j. (u) | . 
 
 125. The symbols (w)^^^, and <p^^^{u) denote respectively 
 the values of u, and 0r(w), when x is put equal to a; this 
 substitution, in the latter case, not being made till after the 
 operation indicated by (p^. has been performed. 
 
 126. Definition. If in w, any function of x^ we sub- 
 stitute x + h for X, u will, in general, assume a new value, 
 which is called the New State {Etat) of u taken with respect 
 to X, and is denoted by the symbol E^(u). 
 
 127. Definition. The excess of the new value of u above 
 its original value is called the Difference of u taken with re- 
 spect to X, and is denoted by the symbol D^.(u). 
 
 128. CoR. 1. D,,{u)=E^{u)-u, E,{u)=u+D,^{u), and 
 u=E,(u)-D^(u). 
 
 129. CoR. 2. If w=0(.r), and x + h is substituted for x, 
 we shall get EXu)=(p{x+h), and DXu)=(p(v+h)-(p{x) ; or, 
 since h is the difference between the two values of x, 
 
 EXu)=(f){x+Dx), and DXu)=<p{x+Dx)-cj)(x). 
 
 The case that most commonly occurs being that in which 
 Dx=l, we shall denote the difference of u with respect to x, 
 on this supposition, by the symbol A,r(w) ; that is 
 
 A,,{u)=(p(x+l)-<p(x). 
 
 130. CoR. 3. Z);«.Z)/(w)=Z>/+"(w), andZ)/(w)=?*; 
 
 also £;"£;'(?o =£;""*"" (w), 
 
 Ef(u) = u, and E_,-".E,"{u) = u : (llO), (111), and (112). 
 
46 
 
 131. Theorem. £/.<j^(ct?)=0(a?+wi>a?). 
 For, E^c^(ai)=(p{x+Dx). 
 
 Ef(l)(w)=E„.E,,.(p{.v), (110) 
 =^E,.(j)ix+Dw) 
 
 &c. = &c. 
 E^'^{x)=(p{x+nDx). 
 
 132. Theorem. E,-'(p(.v)=(j)(cv-Dcv). 
 For, put E^-^(p(iv)=(pyl^(x); then 
 
 cj)ia!)=E,.E,-'(j){a;), (130); 
 =£,.0x|,G^) 
 =0\//(cr+Z).t'). 
 .-. x=\l/{,v+Da;), and 
 A'-Z).p=\|/(.r) ; 
 ■•• £,-^(/)Or) = 0(,r-Z).x). 
 
 133. CoK. E^-"(p(x)=(j){x-7iDa,). 
 
 134. Theorem. E/\u+v)=E,'^'(u)+E,:'\v). 
 For, put u=(h{w), and u=\|/^(<;i') ; 
 
 then JE;/^(w+u)=£/'{0Ot)+x|.(A')S 
 
 =0(a?±Z>a?) +\//(c't?±Z>.v) 
 
 =£/>CtO+£/'x/'(.t') 
 =j5;/'(70+-E/U«)- 
 
47 
 
 135. Theorem. J5/' (a if) = a. iJ,** (m) ; a being indepen- 
 dent of X. 
 
 For, put u=(p(x) ; 
 
 then EJ^' {an) = Ef' .{a. 0(.r) } 
 
 = a.Ef'(u). 
 
 136. Theorem. D,{u+v)=D,(ii)-^DXv)- 
 For, put 7i=(p{.r), and i' = \//(cr) ; 
 
 then D,(7<+u)=Z).,{0(cT)+\//(<r)} 
 
 =^(a'+Z>.T?) -!->//( r+Z>tr) -0(r') -\|/(.t) 
 =(^Or+i)cr) -<^0r) +x//(,r+Z).r) ->/,(.v) 
 
 137. Theorem, i)^ (a ?/) =a. 2)^(2^) ; a being independent 
 of X. 
 
 For, put ti=<p{x) ; 
 
 then Z),,(«w)=Z>4a-0(i)i 
 
 = a.0(cr+Z).r)-a.0(cr) 
 
 =a|0Cr+Z).t7)-0(c77)i 
 
 = a.D^.(p{x) 
 
 =a.D^(.T). 
 
48 
 
 138. Theorem. D,-M>/(M)=M+S,„c,„.Z).r*"-"''(l)- 
 
 For, let a be any quantity independent of cc, and put 
 u=(p{u) ; 
 
 then D,;(u-\-a)= \(p{x+Dcv)+a\ - {0(.i7)+a| 
 = (j){a;+Dx)-(l)(x) 
 
 .: Z),-».A"(w)=w+S,.Z),r*"-'"*-(c„), (123); 
 =u+huC,n-D.r-^''-"'Hi), (137). 
 
 139. Cor. 1. D,-'(u+v)=D,'' {D,.D,-'{u)+D,.D,-'iv)], 
 (128); 
 
 =D,-\D4D,-'(zi)+D,-\v)], (136); 
 
 =D,,-'(u)+D,-'(v), (138). 
 
 The arbitrary constant must be added after the performance of 
 the operations indicated in the second member of the equation. 
 
 140. CoR.2. D,-'{a2i)=D,-'{a.D.D,-'(u)\, (130); 
 
 =D,-\D4a.D,-'(u)\, (137); 
 = a.D,-^{7i), (138). 
 
 141. Theorem. E,.D,{u)=D,.EXu). 
 For, D,{u) = E,{u)-u. 
 
 .: E,D,{ti) = E,E,{u)-E,(u), (l34) 
 =D,.EAu). 
 
142. Cor. E,.D,-'{u)=D,-' E,{u), 
 
 and E,,-' D,-'{u)=D,-\E,-\u). 
 
 143. It follows from the last nine Articles that the func- 
 tions denoted by the symbols E^'\ -O/"? are distributive, and 
 commutative with each other and with any factor independent 
 of w. 
 
 \n_ 
 
 144. Theorem. Z)/(w) = S„(_l)— i.-Jii;^.jE/-"'+H«)- 
 
 For, A(w)=^,(w)-w 
 
 = {E-\).u, (115). 
 .-. Z)/ (z«) = (£,,-!)„ w, (116); 
 
 =hr,{-\T-'--r^^.Er'"^'{u), (119) &(143). 
 \m—\ 
 
 1 45. Theorem. E,^ (w) = S™ r^^^^ -J^^' iu). 
 \m-\ 
 
 For, EXu)=u+D^(u) 
 
 = (l+2>,)w, (115); 
 
 ••■ Ef(u)=il+D,)„u, (116); 
 
 \n 
 
 =S^.^lJ_.2>;"-i('^), (119) and (143). 
 \m—l 
 
50 
 146. Theorem. i>/.A'''=l w./i" ; where /< = Z).i'. 
 
 For, D^ .a,'"=(ct'+//)"-a"=w..i''''^A+inferior positive powers of r^- 
 Z>/ . <r" = I w.. 2?" ~-./r+ inferior positive powers of ,r. 
 
 •2 
 
 Z)/"..x'"=lw..r''~"'./i"'+ inferior positive powers of ir. 
 
 147- When the quantity of which either the difference or 
 new state is to be taken is a power of the independent variable, 
 the index subscript of the letters 2), A, or E may be omitted ; 
 Mnd hence the above theorem will be expressed thus : 
 
 D" .x"=\n.h". 
 
 148. Theorem. ^".x'''=^,(-\Y-\~-^.(x^-n-r^\y' 
 
 \r-\ 
 
 For, i)"..r"' = S,.(-])'-'.|^;^^. £/-'"+'. .x'"', (144). 
 .-. A"..r''' = S,(-l)'-'.-r^:^i— .(.i'+7i-r+l)'", (131). 
 
 149. CoR. 1. S,(-l)'-^~-.(.^■+7^-r+l)"=A^.^■", (HS); 
 
 [r-l 
 
 = [n, (147). 
 
 150. CoR. 2. A".0'«=S,(-l)'-'.r^.(w-r + T)"'. 
 
 The numbers comprehended under the symbol A".0"' are of 
 great utility in the expansion of various functions. The following- 
 values may be readily calculated bv the theorem of this article : 
 
'51 
 
 < 
 
 
 
 
 
 
 
 
 
 
 8 
 
 CO 
 CO 
 
 i 
 
 < 
 
 
 
 
 
 
 
 
 
 CO 
 CO 
 
 o 
 
 •o 
 
 < 
 
 
 
 
 
 
 
 
 o 
 
 CO 
 
 1 
 
 i 
 
 '<[ 
 
 
 
 
 
 
 
 »o 
 
 o 
 
 o 
 
 00 
 
 00 
 
 CO 
 
 
 < 
 
 
 
 
 
 
 o 
 
 1 
 
 o 
 
 o 
 
 i-O 
 
 o 
 
 3 
 
 CO 
 
 < 
 
 
 
 
 
 1 
 
 o 
 o 
 CO 
 
 1 
 
 1 
 
 o 
 00 
 
 CO 
 
 o 
 
 <I 
 
 
 
 
 0> 
 
 o 
 
 o 
 
 00 
 
 CO 
 
 o 
 
 CO 
 
 o 
 
 00 
 
 00 
 
 < 
 
 
 
 ^ 
 
 CO 
 
 o 
 
 
 1 
 
 
 o 
 00 
 
 o 
 
 00 
 Ci 
 
 <l 
 
 
 0{ 
 
 o 
 
 ^ 
 
 o 
 
 CO 
 
 0! 
 
 CO 
 
 0( 
 
 c 
 
 i-O 
 
 0) 
 
 o 
 
 <1 
 
 - 
 
 - 
 
 - 
 
 - 
 
 - 
 
 - 
 
 - 
 
 - 
 
 - 
 
 - 
 
 
 o 
 
 o 
 
 o 
 
 o 
 
 b 
 
 b 
 
 b 
 
 b 
 
 c 
 
 c 
 
52 
 
 151. Theorem. D/.«^=a^(a''-l)"; 
 For, Z>,.a^=a^+*-a^ 
 
 D/.a''=(a''-l).D,.a% (l37) ; 
 
 and, similarly, Z>/.a^= a''. («''-!)". 
 
 152. Theorem. D^.^a+bx=bnh.\a+b.(a^+h). 
 
 For, I>^. [a+&a? = [a+6.(A' +A)-[a+63? 
 
 w, 6A n, bk n, bh 
 
 = \a-\-h{x\nh)-{a^hoe)\ \a-vh.{x-\-h), (41) ; 
 
 «-i, bh 
 
 = hnh. \a+h .{x^h). 
 
 n-l, bh 
 
 153. A.,.[.'r=w.[^. 
 For, ^^.[^=[0^+1-1^ 
 
 n 71 n 
 
 = {a!^l)\x-\x_.{x-n-^\), (41); 
 
 n-l n-l 
 
53 
 
 For, D. 
 
 7^. i),--L— = 
 
 -hnh 
 
 \a+bw 
 
 n,hh 
 
 n+\, bh 
 
 1 
 
 1 
 
 a+hx \a+h.{x+h) \a+hx 
 
 a+btv a+b.{x+nh) 
 
 \a+bx 
 
 n+TTbTT 
 
 bull 
 
 \a-\-bx ' 
 
 n+\, bh 
 
 \a+b. 
 
 155. 
 
 1 -n 
 
 
 n n+l 
 
 For, 
 
 1 1 1 
 
 ■''\x \x+l [a? 
 
 
 w-n-ir\ w+\ 
 
 
 \x+\ \x+\ 
 
 n+l n+l 
 
 
 -n 
 
 
 ~ \ai+l ' 
 
 
 n+l 
 
 156. Theorem. 
 
 D,.Vr^{x+{r-\).h\ = \(l){x-^nh)-(p{x)\ rPr(p(x+rh) 
 For, D,.'Pr<p{^v+(r-l).h\='Pr(li(x+rh)-Vr(p{.v+(r-l).h\ 
 
 = \(j){x+nh)-(p(^v)\ ,P,^(,r+rA), (32). 
 
157- Theorem. D,.\y ,(p\a-v{r-\).h\Y' 
 
 = - \(p{a'-r7ih)-(l){ai) \ [P,0 ja'^- (r- 1) .li\\ 
 For, D,,.[Vr(p\x-^{r-\)Ji\y' 
 
 = 5P,(i)Gi-+rA)i-'-[P,.0Sa-+(r-l). //(]-' 
 = -{(p{x+nh)-(pi,x)\.(Vr(p{<JC^{r-\).h\y\ 
 
 158. Theorem. A.P.(w,) = S,'cl't.(?^,. A-";)- 
 For, £, . P,, (« ,, ) = P, (7<,. + D,u.) 
 
 m+l m-r+l.r-l 
 
 =P,0O+!S. C,.t(«.-^..«,), (9). 
 
 »« m 7)1— r, r 
 
 .^ Z>„.P,(w,) = S,. C.K.A^O. (128). 
 
 159. Theorem. D,{7iv) = u.D,,v+D,{u).E,,v. 
 
 For, jE,,(?<u) = (M+2>,7/)(u + D,.r) 
 
 = // 75 + Z>,, (?< ) . y + ?^ . D, V + D, {ii) . D, V 
 ^iiv + u. D, V + 1),,. (?<) . E,. V ; 
 . . D,{nr)^?(.I),v + D,{H).E,,v. 
 

 55 
 
 
 160. Theorem 
 
 
 v-u.D,v 
 
 .E,,v 
 
 For, D, f- 1 == 
 
 
 
 
 HV +I>X'f^i) .v-uv-u .D^v 
 
 
 v.E,v 
 
 
 
 D,:(u).v-u.D,,.v 
 
 
 V . E,^v 
 
 161. Theorem. If (pin), 0i(w), \//(u), and v//i(y) are 
 distributive functions, commutative with each other and a con- 
 stant factor, and if (p{u).\p{v)+(p^(u).\l^^(v) is denoted by 
 ((p^+(pi^i)uv, then shall 
 
 ... l!L 
 
 For, by proceeding precisely as in Art. 118, we shall find 
 that 
 
 (0x//+(^,x//,)3w^'=s.7^.0*-"'0r-'Oo-^/''""'V'i'"~'(^)- 
 
 Suppose, therefore, 
 
 lm-1 
 then 
 
 (0x//+0i>//O«wt'=S,„-^.0''-'"+'0r-'(w).>p-'''-^'>/'r-'O') 
 
 
56 
 
 S™]S"-^^-f^""''"'^'""'^''^^'^-t^"''""^^'''''("^^ 
 
 +0,[0'-'«+^(^r-'(w)].fi[>^"-"^^>^r-H^)]^ (113): 
 
 n 
 
 + (^"-'« + >.0rO<). >/,«-'« + !. x/,,'«(v)}, (114); 
 
 = 0«+'(w).x/,''+H^)+s,„j^.0''-"'^>r(?^).>l'"-™^^->/'r(^) 
 
 + S.j^.0"-™^^^inw)->/'"-'''"'-KH")+^i''"^(w)-^/'i"-^'(^), 
 
 (9); 
 
 1^+1 
 
 = 0-^(«).x|,-^(t;)+s.~^.^"-"-^<^roo-^"-'''^'-^r(«) 
 
 +0,«+>(w).>/./'+'(i)), (5) and (47); 
 
 \n + l 
 
 ^">T^-r'"'^'-i'^""'('')-^""""''-^^"'-'(^^h (9)- 
 
 If, therefore, the law Avere true for n it would be true for 
 n + 1 ; but it is true for 3, and therefore it is true for n. 
 
 162. Cor. l. The equation just found may be written 
 thus, 
 
 or ((^>//+0i>|/i)„?/t' = (^x|/+0i>|/i)'''wv. See Art. 120. 
 
 It must be carefully observed that, in the expansion in- 
 dicated by this last expression, the symbols (f) and (h^ are to be 
 prefixed to w, while yj^ and \//, are to be prefixed to v. 
 
57 
 
 1G3. Cor. 2. It' <i,, ^.,,,..0,, and \|/i, v/zsi...^,? denote 
 distributive functions, and commutative both with each other, 
 and with a constant factor ; then shall 
 
 164. Cor. 3. With the same limitations, 
 
 m m 
 
 where the symbols '0,, -0,., . . .'0,., are to be prefixed to 7/1, 7^3,. . .Wj, 
 respectively. 
 
 165. Theorem. 
 
 For, Z), . {u V) = u.D,v+ D, (w) .E,{v), ( 1 59) ; 
 
 = (Z>,,+ 'Z>,..£j,)mu, (161) ; where 'Z)^ only belongs to u. 
 
 =S.-^ii^i-.z>;"-'(w)-A''-"'"'£;"-'(^), (161)- 
 
 \m—\ 
 
 166. Theorem. 
 
 Im-l 
 For, Z), (^< d) = E^ {u) . E,, ( y ) -uv 
 
 = (E,.E,-l)uv, (161); 
 .-. D/0<«) = (£,.£,-l)„.M« 
 
 =S.(-i)'"-^T-::^^.^/-™^'(w).£/-'«-^K»), (161). 
 
 m-l 
 H 
 
58 
 
 1 67- C o R . 1 . D," (u y ?<o) = ) ( 1 + ' i> , ) ( 1 + 'Z>,, ) - 1 ( " ?/ , n, ; an d 
 
 D/.V,{il,)={{l+'Dr)(l+W,)...(l+'"D,)-l]"7l,7t,.,.U„. 
 
 in r, m 
 
 = }S,.aCA)S" Uxu,-..ic,n. (68) and (9). 
 
 168. Theorem. E,.E,/u) = E,,.E,,{7i). 
 For, Tput n=(p (a, y) ; then 
 
 E,(7i) = (p(.v,y+Dy) 
 
 E,,.E„(u) = (l)iw+D.v,y+Dy) 
 
 = E,,(t)(.v + D.v.y) 
 
 = Ey.E,,.(p{w,y) 
 
 = E,.EAu)- 
 Hence, we may express either E^.Ey(n) or E,,.EXn), by 
 Er,y{u) ; while D,,,^{u) will denote E,,^,^{n)-i(. 
 
 1 69. Theorem. E.E^ {u) = u+ D, (u) + D„ (w) + D , D, (u). 
 For, E^(u) = u + D,^(7c), (128). 
 
 E, . E„ (u) = E, (71) + E, . D, (?/) , (134); 
 
 = 7i+D,(n) + D,iu)+D,D,,(7i), (128). 
 
 170. Cor. 1. Since u+D,(u) + Dy{u)+D,D^{7i) = E,Ey{u) 
 
 = EyEXu), (168); 
 = 7i+D,^{u)^D,{u)^D,,D,,{ii). 
 .-. D,D„{u) = D.,D,{ii). 
 
 171. Cor. 2. £;,,,(^0 = (l+A + i>.,+A/>,,)7/, (n.^); 
 
 = {l+D,+D,^+D,D,,)".'n, (121) and (143). 
 
59 
 
 172. CoK. 3. D,,y{u) = {D,+Dy+D,D^)u, (lbs) 
 
 = {D^+Dy+D,Dyy.7(, (121) and (l43) ; 
 
 173. Theorem. i);;,^(;,) = S„X-l)"'-'- j^-^^T^'^^O- 
 For, D,^y{u) = E,r,y(u)-y, (16"8); 
 
 = {E,,,j-lYn, (121) and (143); 
 
 [^ 
 
 .s„,(-ir-'.^.^:-;-oo. 
 ... t 
 
 174. Theorem. E;^^{y) = S„.-~-^ -D^l^iu). 
 
 For, £,,,,X?0=^^ + A.;y(w)' (168); 
 = (l+A,.v)W' 
 
 = (1+Z> ,,,,)". 7/, (121) and (143): 
 
 m-l 
 
60 
 
 175. Theorem. 
 
 \m . \n 
 
 'r— s \s—l 
 
 \m 
 
 For, E:'(u) = Sr^,^^^^^-D/-'(u), (145) and (13). 
 r—l 
 
 Ej".E;{zi) = SrT^-D.r''E;'{u), (168), (134), (135) & (141); 
 
 = Srr^-Dr'Ar^-D,rHn), (145) and (13) 
 
 r-1 «-l 
 
 \m ^ \n 
 
 = S.-r^.S.-r^.Dr''^.D;-^(u), (136) and (is?): 
 
 r-1 \s-l 
 
 \m . \n 
 
 -kkr'^ /-' .D/-^.D,r^{u). (6) and (is). 
 
 \r-s \s-l 
 
 176. Theorem. 
 
 D,-'(uv) = n.D,-Uc-D,,'' \D,{ii).D,-'E,v] . 
 
 For, D,{ti.D,-'v) = 2iv + D{u).D,\E,(v), (159) and (142). 
 .-. u.D,-^v = Dr-^(uv)+D,-' \D,(ti).D,-'E,(v)\, (139); 
 and D, '(7iv):^u.Dr^v-D,^' \DM).D,'' .E,.v\, 
 
Gl 
 177- Theorem. 
 
 + (-!)'•. A-' \D;(n).D,-'E;v\ . 
 
 For, />,-' |2)/-H«)-2>,r<'«-^*£;"-4'J =D:--\u).D,r''E;'-'{:v) 
 
 -D,-' \D,:"{u).Dr"-E,:'v\ , (176) ; 
 = (-iy"-KDj"''(u).D,-'" .E,'"-' V 
 
 + ( - 1 )"' . i>„ - ^ 5 Z)/ (7^ . Z), - '" • e;" V \ 
 
 r. ^,„(-iy"-' .d,,-'\d,:''-'(u).d,.-^"'-'\e;''-'v\ 
 
 +S,„(-i)'"Z>,r' )Z);«(w). A"" £;"!;;, (5); 
 D.- ' (n v) + s!„ ( - 1 )'« . z>,r M ^^"' («') • ^.^^ '" EJ" V ] 
 
 = S^^ ( _ 1 )'« - 1 . Z);« - 1 (w) . i>, " '" EJ" - ' V 
 
 + sl ( - 1 )"' . 2>,r ' 5 z>,/' (w) . />,,-'" z.';" tj 5 
 +(_i)'.i>,,-^ jz)/(m)-A.~'.z:/";-, (9); 
 
 ^{-\y.D,-^\Di{u).D-''E:v\. 
 
 178. CoK. If, for some value of r, Dju=0, then 
 
 2)^ - • (w v) = S.« ( - 1 )'" - V Z>,;" -'u. D,-'" E;' -'v, (13); 
 
 = \D,-'{\+'D,.D,r'E,)-'\tiv, 
 Avhcre 'Z), only belongs to 7/, (12) ; 
 
62 
 
 .-. D,r'{uv)=\Dr-".{\+'D,.D,-KE,;)-"\uv, (l63) ; 
 
 \m-l ' 
 
 (92) and (39) 
 
 : s,„ ( - 1 )'" - ' . fi-ii . z>,,"' - ' (w) . i>r '" +"'-!'. £;« - ' y . 
 
 m-l V / . 
 
 179. Theorem. D.,-'{a'')= j^ +consi. 
 a' — l 
 
 For, a^(a"-l) = Z>,,.o.% (151), 
 
 and D^-^a"=-7-— +const. (l38). 
 a - 1 
 
 180. CoK. D,r-".a-' = a'(a"-l)-"+S,„D,-^"-"'>.c„„ (138) 
 
 181. Theorem. 
 
 Z),-' (a,,. 70 = «'-S,„(-l)"'-^a*"'-"'. («"-!)-"'. A"'"^^' 
 
 +(-l)^A.-MA^(w)•«'^^'•"•(«"-l)"'■^ 
 
 For, A.-'-(«''.?/) = S,„(-l)""'-A'"-nw).A-"'-^;"~'-fl'" 
 
 +(-!)'•. Z>,rM^''('0-A-'.£/-«1, (177); 
 
 = S„,(-i)'"-'. 
 
 («"-!)" 
 
 Z);«-^7*+(-i)' 
 
 
 (180); 
 
 =«^8„,(-l)"'-^r/<"'-"^(r/'-])-"'.Z);"-'7f 
 
 + (-])'./).' lA'('/).r/'^'".(^"-l)-'(, (fi). 
 
G3 
 
 182. D,r\\a+hx 
 
 — — . I a^b{x-h). 
 
 Vov, l)]i{n+\).ya+hx = D,.YaU).{,v-h), (152): 
 and D^~^\a + bx = .\a + h.(oo-li). 
 
 183. A, 
 
 t 
 
 w+l 
 
 For, (w + i).|^= A^.[.'p , (l.^,'^). 
 
 |a'= a, .-ILL, (i.-jy); and 
 
 
 184. D,rK 
 
 \a-\rhx hh{n-\) .\a+bx 
 
 For, I \ - = 2>,..| J-, (1.54). 
 
 \a+b£C '"I ?>/i(w-l).[ 
 
 I, 
 
 w-l).|a+6ci'j 
 
 (1,']7): 
 
 and D, 
 
 \^a+bx b/i(n-J).]^a+b.i' 
 
 ■iTbfi n-l,bh 
 
G4 
 
 185. A,-^ 
 
 1^ (n-l)\iv-l 
 
 -Cw-l) 1 ^ , 
 
 For, -^i ^=A..,— -, (155). 
 
 .r? [ (w-l)cx>-lj 
 
 L^ 
 
 and A^~ 
 
 [*■ (n-\)[x-l 
 
 186. A-'..i'"=S,« A"'-'0".f-; .i- being any positive integ 
 
 integer, 
 
 For, (j,+.,;)"=S,.7^.A-"-'?/'. ('«)• 
 
 and A-'..t^"=sl ^"' "^ -A.-^h-g , (139), and (140); 
 
 = S.A'"-'.0''.f-, (183). 
 
 187. Theorem. S,„ «,„= ( A„ ' ' - AjJ,,) «« f i 
 
 For, A„.S„,ff,«=S,„«,„-S;„a„ 
 
65 
 
 n 
 
 •■• Sm«,n=^»~'-«H+i+ constant. 
 
 
 
 and S^a,„=A~=o«^„+i+ constant. 
 or, as it may be conveniently expressed, 
 
CHAPTER VL 
 
 ON DIFFERENTIATION IN GENERAL. 
 
 188. Definition. The quantity \{Dx)'" Dj'u] j)^^^) is 
 called the n^^ differential coefficient of u, taken ivith respect 
 to X, and is denoted by the symbol d/z<. 
 
 189. Theorem. d/'d,!'u=d,'"+"u. 
 For, put Dw=h; then 
 
 and dj\d:u= {h-^.D,:^{h-\D,:^u),^,]„^, 
 
 ^{h-^.D:\{h-\D:u)},^, 
 
 = J/i- ('«+«)./);«+««* J ^^^, (137) and (130) ; 
 
 190. Cor. dj>u=u. 
 
 191- Theorem. d^.(u+v) = d^u+d^v. 
 
 For, 4 . (w+tj) = { A- ' . D,, (w+r?) } ;,=„ 
 
 = (h-' .D,u+h-' .D,v)/^^,„ (136); 
 
 = (/i- ' . 2>.r?^)/,=0+ (^i" ' ■ A ^^)/,=f. 
 
 =d-u+d^v. 
 
67 
 
 192. Theorem. d,{u+a) = d^u; where a is independent 
 
 of w. 
 
 For, d^.(M+a)= j/i"M>,(w+a)5,,^o 
 
 = (h-'.D,u),^,. (138); 
 
 193. Cor. l. d,.a=0, (l91). 
 
 194. Cor. 2. rf,-".d/ii=w+S,„4'*"""'*-c,„, where c,„ is 
 independent of a', (123). 
 
 195. The symbol f^u is equivalent to d^~"u, and is read 
 the n**^ integral ofu, taken with respect to x. Hence the equa- 
 tion just found may be written thus : 
 
 196. Theorem. d^(au) = ad,,u. 
 For, dx.{au)=\h~\DXau)\^^f, 
 
 = \h-'a.D,u]^^„ (137); 
 = a{h-'.D^u],^^^ 
 = ad^u. 
 
 1 97 ' Theo rem. 4 . .r = l . 
 For, d^.x=(h'\Dx)/^^^^ 
 
 = (1)^=0 
 
 = 1. 
 
68 
 
 198. Theorem. cb{a!+h) = Sn.-r^ .d,'"-' .d){w). 
 
 ' W2 — 1 
 
 For, T^ni Doo=k, axvA h=nk ; then 
 
 ... \!L 
 
 whatever the value of k may be; 
 
 ■■S,„, Ak-^"'-'\D,"'-\(p(wy 
 
 
 199. CoK. 1. E,u=SmT^ .d;"-'w, 
 
 m-1 ' 
 
 and B, u = Sm i — • rf,'" w. 
 
 200. CoK. 2. D,u = {e'"'^-\)ii, (lO()) and (l 15). 
 .•, Dj'7i={e.'"'^ -i)"n, (121). 
 
69 
 
 201. Cor. 3. r/;".d)(.i) is the coefficient of j— in the 
 expansion of (p{<v+h). 
 
 202. Theorem. If w is a function of ,v, then 
 
 d^.(p{u) = d,f.(l){u).dj.u. 
 
 For, D,..(l){iiy=sJ"'''f^''^ ■iDric)'% (199); 
 
 -S. 1^^ .(Bn^-dfic)'% 199 ; 
 
 
 and f/.,.0(?O=>/'''-^-0OOi/.=o 
 = d„.<p{u).d^u. 
 
 203. Theorem. d,,.{uv)~vd^u+tid^v. 
 For, i>.r (w u) = (/* + D^ i() (y + D^v) -uv 
 
 = u . D^ u + u. D^v + D^ {u) . D,, V . 
 .: h-'.D^(uv) = v.h-\D,ti + u.h''.D,v + h-\D^{u).D,v; 
 and \h-KD,(uv)],^,=tJ. \h-KD,,:u\,^, + u\h-\D,v\,^, 
 
 + [\h-\D,u\,^,.(D.v),^o or (Dr.u)„^,.\h''-DM,=.] 
 .-. d^.(uii)=vd,u + ud_,v+ \d,u.O or 0.d,^v\ 
 
 =vd^7t + ud,rii- 
 
 d,.(uv) d,.u dAi 
 204. Cor. -^ A_^ = _!_ + ^ 
 
70 
 
 205. Theorem. =S;«-^ 
 
 For, 1+1 » ' ^ ^^^ r , _^ . „,+ l ^ ^^^^^ 
 
 V,.Ur P,W, '''"+' 
 
 -1. = +S,n , (24); 
 
 P,w, 
 
 = S.— , (9). 
 
 206. Coil. 4. P,?^,= S™c/,w„, . P,«,. 
 
 207- Theorem. d^.u"=nu''-^ d,:U, for every rational 
 value of *^. 
 
 ^ d,.P,.w,. " d^w, 
 
 For, \ =S,. ^-^, (20.5). 
 
 P,.w, 
 
 Tut Ur=u.i then =1:5,. =w. . 
 
 u^ u u 
 
 n being a positive integer. 
 
 Also, since 1=?/". ?^"", 
 
 .-. 0=7r".nii"~^d_^u + 7c".d^.u~'', (193) and (203); 
 and d,.u-"=-nu~'''^d_^7i. 
 
71 
 
 Again, «*'" = (?< ")"; in and n being any positive integers. 
 .-. ^mu~"'~^d^u = n.(u "y~Kd,.(u ") 
 
 ±!!i m ± !^ _ 1 
 and dj.. (u ") = i — . r« " . d^u. 
 n 
 
 Hence, d^.u"=nu"~^d^u, for every rational value of 7i. 
 208. Theorem, d,. {-] = -.{— - 
 
 For, d. 
 
 i^{)=dr.(uV-^) 
 
 209. CoR. d 
 
 =v-Kd^u+u.{-l)v-'d,v, (203), and (207); 
 u fd^u dj.v\ 
 
 If \ U V ) 
 
 VrVrf P,.V, { P,7/,. P,.t 
 
 210. Theorem, dj" . x" = |w . a-" - '" . 
 
 For, d,,.x''=nx''-\ (20?) and (197) ; 
 d,^^..v"=n.d^.x"-\ (196); 
 
 &c. = &c. 
 
72 
 
 211. Theorem. dJ'.(uv) = S,nr^^^-^^.d;'-"' + Ui.d:"-'v. 
 
 For, d,,.(uv) = vd,u+ud,v, (203); 
 
 = {d,^+^d,^)uv, where 'd,, belongs to v, (l6l). 
 ••• d;'.{uv) = {d_,+^d;},,uv 
 
 = (d,+'d,yuv, (162); 
 
 ... t 
 »i-l 
 
 Or thus 
 
 £,(«»)= (S.j^.</, 
 
 ")(s-^-''-"-'")' cs^'^ 
 
 sj^'-'-s S'"-"'"": , m- 
 
 m-\ 
 
 \n—in ■ 
 
 \n '"\n-m+l \m-l 
 
 (201); 
 
 and (/;' ■ (?/ iJ) = Sm I '" ' • d," -'"^^u. dj" 'u. 
 |m-l 
 
 212. Cor. rf,.P,w,,= ('d,+-rf, + ...+"'d,)^<l^<2. ..?<„„ (205). 
 . • . dj' . P, w , = ( ' c/,, + ■</,, + . . . + "'(/.,)" w , 7/, ...u,„, ( 1 64) ; 
 
 = (kS,'t/,)".P,?/,,, wliei'e 'f/, belongs to ic,. 
 
73 
 
 213. Theorem. If ti is such a function of a; as may 
 be expanded in positive and integral powers of x, then shall 
 
 CO oi'K - 1 
 
 For, assume M=S„a„_i.<:»?" '. where a„_i is either zero or 
 some finite quantity; 
 
 05 
 
 then c?;"M=S„a„_i.[w-l ..J?'""""', (191), (196), and (210); 
 
 = S„ «„ _ 1 . |»^-1 . af """' + '?•,„• [m + S„ «,«+ „ |rra+n • .^'^ (9) ; 
 = «,„.[ w+S„ff'„,+„.|m+w. a?", since J7i-1 = 0, (^^^). 
 •■• <=o-'" = «»-H» and «„,= T— .<!=nW, ("^L). 
 
 Also (?f)^^p=ap; and therefore, f/,„-i= , .d'^,~j2i, QZ\)i> 
 
 and u==S,ni^-d"::;.n. 
 \m-l 
 
 214. Theorem, c?,, (/^ 7^ = f/„ (/,, v ; .t' and ?/ being independent 
 of each other. 
 
 For, put Dx=h, and Dy=k; then 
 
 and rf,rf,,w= 5 A-' . D,(k-'D,,?i),^,},^, 
 
 = \h-\{k'\D^.D^u),^,\,__,, (137); 
 
 = 5A;->.A-'.A^A,,4,^^„^^.^^„ (170); 
 
 = ;A;-'.AX^'-^-A^)S.=o..=o, (137); 
 
 = JA-'.A.(/r'.Z),..0,=oS.=o 
 
 = dyd,u. 
 
 K 
 
74 
 
 215. Cor. dj".d,;'u=d,j'.dj"u. 
 
 216. Theorem. 
 
 CO CO 7,«l-l^,«-l 
 
 ^{w+h, y+k) = S,n Snr ^^— -. d/->.rf/-^^Gx', y)' 
 
 Oil — 1 \71' 1 
 
 For, (j)ioj+h,y) = S,nA^.-d/-'.(p{x,y), (198); 
 
 and 
 
 h'"- 
 
 (b (.r +k,y+k) = S,„ i • dj" -'.d)(w,y+ k) 
 
 \m — l 
 
 CO Am-l CO i.»-l 
 
 = S.,^.d/-.S„f— -dZ-'.^C^,^), (198): 
 Iw-l lw-1 ■ ^ 
 
 — 077i On 
 
 h"'-\k' 
 
 -. d,:"-\d--\cp{.v, y), (191), 
 
 (196), and (6). 
 
 217. Cor. 
 
 CD m J^m -nj^n-l 
 
 (j>{.v+h,y+k) = S^Sr,] i --d,r-''.d,;-\(p(w,y), (18). 
 
 ' \m—n\n—l ■ ' 
 
 218. Theorem. If u is such a function of .v and ?/, 
 that it may be expanded in positive integral powers of a;, 
 and 2/> then shall 
 
 "=s-s.££.d.-.rf;:.'.- 
 
 For, assu me u = S,„ '^''" ~ ^ • S« ?/" ~ ' • «« - 1 , « - 1 ? where o„ 
 is not infinite; 
 
 then y^^^=S,„a^'"-^a,„-i,,_i, Czi), (213); 
 
 
75 
 
 And, ;&5> =«,.->,„-„ crj, cii). e^i3); 
 
 r-l. w-l 
 
 \m—\ .\n-\ 
 
 
 and ?^- = S,«S„ 
 
 1-1 .U^-l 
 
 dZJ.d'l_}u. 
 
 219- Theorem. If ;jr is a function of cV and ?/, 
 
 then shall d, { d^z .(j)(z)\=dy{ d,z . (^) | . 
 For, rf„.d,|/.0(^)}=d,.f/4/,0(^)}, (214); 
 .-. dj^dyz.d^.{,cp{z)\=^d,{d^z.d.^.j:<p{z)\, (202); 
 or d4^^^.(j^(«)}=<?yjtZ^^.^(jjf)j. 
 
 220. If y=^\x+cv.(p{y)\, where « is independent of 
 X and y; then shall 
 
 /(y)=/^/'0^) + S.r^.rf/"-'S(^x^(^ir.4./v/>(^-)S- 
 
 For, f{y)= 5/(^)i..o+S.|^. Co/(2/). (213). 
 
 Fut z+iv.(l)(y) = u; 
 
 d^.y d,,.yl/(it) 
 
 then 
 
 rf,7/ d..\j/(7l) 
 
 dj.u.d,^.\l/(u) 
 d^ii.d^^.\\f{n) 
 
 (202) ; 
 
 d J, n 
 dji 
 
 (l){y)+oo.d,.(p{[i) 
 l+x.d^.(p{y) 
 
76 
 
 .-. \l+x.d,_.<p{y)\d,y=\(p{y)+a;.d,.(p{y)\d,y, 
 and, cancelling identical terms, 
 
 d^y d^.(t)(y) 
 d,y=d._y.(p{y), since -^ = —^^ , (202). 
 d.^y d,.(p(y) 
 
 Also, d,.f{y)=d,,y.dj(y), (202); 
 = (izy(p{y)d,,.f{y); 
 dj.f(y)=d4d,y.cj>(y).d^^.f(y)\ 
 
 =d4d^y.(p(y).d^.f(y)\, (219); 
 
 = d4d._y.<p(y)\~.d,.f(y)\; 
 and, similarly, dj" .f\y) = dJ" -\\ d._ y . (p(y) \"' . rf, .f(7j) \ 
 
 =d--K\^^\d,.f(y)\, (202). 
 
 ••• d::,.f{y) = dj''-^.\^i4^'''.d^,fi.(z)\, 
 and f(y) =/v/,(..) + k^.d--'\ ^W)]'" ■ <- -.W^) \ 
 
 221. Cor. If y=z + .i!.(p(y), 
 
 then f(y)=f(^) + S,nT-.dJ''-\l(b(z)\"'.d.^,nz)\ 
 
 222. Theorem. 
 
 fr(uv)=L(-ir-'.d.;'''^u.j:;''v+i-iy.f/^dj'7,.f:'vl 
 
 For, p^dj^-'u. f,!" - ' « ( = f/;" - ' 7/ . /'" f» - /, \ d/' n . fj" v ] , (2O.'0 , 
 
 =(-iy"-'d,"'-hi.f,r"'r+{-iy\f,\d;"2c.fj"v\. 
 
77 
 
 AX-^y''~'-dJ"-'uJj''v + 8,n(-iy'.l\d:'u.f:''v\, (4) and (5); 
 Uuv) + sU-^rfr\dJ"u.f,"'v\, 
 
 + (-!)" f^ld/u.fj'v], (9). 
 
 .-. Uuv)=8,n{-iy"-'dj''-'u.f;"v+(-iy.f,\dfzt.f:^v\. 
 
 223. Cor. 
 
 /.w=s,„(-ir-^T-.rf/->w+(-i) ,,,,, 
 
 ig.c^ 
 
 (197) & (210). 
 
CHAPTER VII. 
 
 ON POLYNOMIALS. 
 
 224. The symbol 8,.+sCf',) denotes the sum of every 
 term that can be formed with the following conditions: each 
 term is the product of m quantities in which r has the values 
 of the successive natural numbers, while s has any m values 
 such that their sum shall be w, zero being admissible as a 
 value of s, and repetitions of the same value of that letter 
 being allowed in the same term. Thus : 
 
 225. C6r. s"+.Ca,)=K"''"''«ls,, + ,C«.)- • 
 
 226. Theorem. ^ (S,.«,)"=S,,+,?^- 
 For, J-.(S,.«,.)"=8! "'"7' 1?"' ^ W' 
 
79 
 
 Suppose, therefore, 
 
 I m-l m~l,n f, ^ 
 
 p.(S,-»,)- = S„.^: 
 
 1 m 1 ,n-l 
 
 then i— . (S,.a,)"= t- (f/,„+S,.«r)" 
 
 n+\ a n-t+l m-l 
 
 \n-t+\ \s 
 
 = S"+A, (225). 
 
 If, therefore, the law were true for m-l, it would be true for 
 m ; but it is true for 2, and therefore for in. 
 
 1 TO m,n Sj ^ II 
 
 227. Cor. ^.d;'.P,?,,.=S.+.-^, (212). 
 \n 'If 
 
 228. The symbol '3r^"'0(a) denotes the coefficient of ai^ 
 
 in the development of 0(S,„«,„_ia?'""'), which coefficient may be 
 called the m!^ polynomial coefficient of (p(a) taken with respect 
 to a. In this symbol the index subscript of nr is the letter 
 according to the indices subscript of which the different powers 
 of X ascend, and the quantity following the functional symbol 
 
 is the term independent of x in the series S,„aa-i'^"'"'*- If the 
 index subscript of "W is omitted, that letter is understood which 
 
 Throughout this Chapter a is put for Aq, for the sake of brevity. 
 
80 
 
 immediately follows it, and if the function is a power of the 
 polynomial, the parentheses including the first term of the 
 polynomial may be omitted : thus 
 
 *zjr'"0(a) denotes the coefficient of cT?'" in S„0,„_i(a)<2?"'"', 
 
 *Z3""'«" |S„,a„_ia?™-ij% 
 
 •2r'"a,"' 5^'«"'+"'-i'^"'"'S"- 
 
 229. CoR. •2r„".0(«) = 0(a). 
 
 230. Theorem, -zzr'" a'' = S,. I ri .«"-'' . — j — — , for every 
 value of 01. 
 
 X CO 
 
 For, (S,„ Q„,_ 1 -t?™ - 'y={a+a; . ^,a,x'-y, {[)) and (6) ; 
 
 = S,„-r^.«"-'" + ^.t^'"-^(S,.a.cr'-ir-', (92); 
 Im-l 
 
 \n 
 
 = S»r-^2:^i--.«"-'" + '..3?'«-^S,-^^'■"'.•^^^'■ "'«/"-', (228); 
 
 = S™-^™ ' Sr I '" ^ . • «" '"^'■.•zjr'' -'.«,'"-'■, (6) and (18). 
 
 w-r+1 
 
 m— r+1 
 
 ^+''-''ar'-ia,'»-' + i, since '5r'".a,o=0; 
 
 =s,-7— •«""'' •'23""'~''- ^^r> (8). 
 
81 
 
 231. Cor. 1. 'Z«r"'a,"=S,"^— •«/"'''^^"""''a,+l• 
 
 232. Cor. 2. If ii is a positive integer, 
 
 •ZtT . o,, ^ a, • ' 
 
 =s. 
 
 n-r 
 
 ■m+r - 
 
 \L 
 
 and 
 
 w-w+r-1 . TO-r+l 
 
 233 From this last value we may deduce any number of 
 
 terms of the expansion of 
 
 TS" a 
 
 much more readily than from 
 
 the general expression for that expansion. We hi 
 
 73-'". a" 
 
 n-m \m \n—m-\-l \m-l 
 
 + &C. 
 
 \n-m+2 . \m 
 Hence, putting m=l, and n=m — I, 
 
 a" 
 
 TfT a 
 
 \n-m+l \m-2 
 
 \n-m+2 . \m—2 
 
 ■&c. 
 
 And, putting m=2, and n=m-2, 
 
 \n-m.\m \n-m+l . \m-2 
 
 |n-m+2'\ |TO-4 [2 | m-3 ' J 
 
 W-Wi + 3 
 
 + &C. 
 
82 
 
 And, proceeding in the same manner, we shall obtain suc- 
 cessively the following terms ; 
 
 n \n-m. 
 
 n-m+l \m—2 
 
 • «2 
 
 \n-m+2\\m-4! \2 Im-s' J In-m+s' [Im-G is 
 
 \m-5 w-4 J- ?i-m+4Hm-8 U \m-l 3 
 
 ai'"-^ /«3 \ a{'-'> 1 a»-™+5 f aj"' - '" . «/ 
 
 |m-6 V|£ 7 \m-5 \ \ n-m+5 \ \m-lO \5_ 
 
 a., «! 
 
 «3+ |- 
 
 fi'-a • r— + 1 — • Oj 
 
 («3a4 + a2«5) 
 
 -9 [3 '""" ' 1^-8 V""' [2 [2 ■ V \m-l 
 
 |w.-6 *j Iw— m-i-6 ||m-12 |6 Iw — 11 U 
 
 «i'"-'" fuo-.a.,; a.? \ a^"'-^ (ai ar^ \ 
 
 + 1 -p — f- + T^ • «4 + 1 J— +a2a3«4+ 1 — • «5 
 
 |m-10 V|2 [2 [f 7 [m-9 V|3 [2 / 
 
 + i I — +03a5+a2«6 + I •«-} + I -<i v- 
 
 m-8 \[2 / m-7 j [7i-m+7 \ m-14 [7 
 
 %+ 
 
 ^ |m-13 [5 [rri-l2\\3 [2 "" [4 ' "^ 
 
 + 1 «£ • T— + I — • «3ff4+ 1 — •- O5 
 
 |m-ll V [3 [2 ^ ' [3 7 
 
 a,'"-^" lai ra~ ~\ ai \ 
 
 |m-10 V|2 ' ^L[2 'J [2 V 
 
 W'-9 Uw.-8 I 
 
83 
 
 a." a, ' . a., 
 
 |M-m+8 [ |m-l6 [8 \m-\5 [6 
 
 «i "" fa.;, a/ a.; 
 
 + 1 I ," , • + T^-a 
 
 «," "* la.r.a-/' a.; 
 
 + i — asai+ r— -fls 
 
 1 7^^-14. V Lf [2 " [5 V | m-13 V [2 [3 ' [3 ""^^ [^ 
 
 \m-l 
 
 
 L! 
 
 [m-ll V [^ [^ li ^ 
 
 m-lO \ 2 \m-9 j 
 
 Whence, giving to m the values of the successive natural 
 numbers, 
 
 In [w-: 
 
 a (I ~ . a-r a 
 
 \n \n-2 [2 \n-\ 
 
 73" . a" rt" -^ . a^ « ~ « 
 
 + I r/i«2+ T 7 . «.v 
 
 1^ |_n-3 [3 [w-2 " | w-l 
 
 U_4 4 W-3 2 -^W-2\2 ' l \n-l 
 
 \n 
 
 a"'" . a," a . a^ 
 
 [n \ n-5 [5 \ n-4> [; 
 
 \n-s\ [2 [2 '\ 
 
 ««- . a" ^ 
 
 Ui-2 ■ ^ \n-\ 
 
84 
 
 + 1 • «o • 
 
 -i-ai ^+a2«i + I -as}- + i \a^ai + aoa-^+aM 
 
 VI 2 / 2 j 1^-2 •^ 
 
 + 1 :-«7- 
 
 [7*-5lL2[3 [3_ [4 7 
 
 rt"-' (a./ «./ af (a.r \ a{^ ] 
 
 ^l^r-5-^li"'^+^'^''^'^''^^'^'-^''^^l2;-««l 
 
 [7A-2 ( 1 2 j [^-1 
 
85 
 
 234. Since ~ (S,„a,;.-ry'"-')"=s'r +. ''\^ — -, n being 
 any positive integer, (226); 
 
 05, n ff ^ 
 — Or. + s \ • ^*' 
 
 .-. — 1 = the sum of all those terms of Sr+s-f-^ in which 
 
 {r-\) = m. 
 
 This equation may be thus written: 
 
 S " r - 1 
 ,-, + ,, + s(r-l)-7-— • 
 
 235. Theorem, -sr'" . « - ' = a - ' . A + ,. ( -' ) . 
 
 For, (S^a,„_i.r'"-')-^ = S„a^""'V""'a-', (228); 
 .-. l = (S,„a,„_i.2?'"-^)(S„ci?"-V"-'«-') 
 
 = S,„a'""'.S„a,„_„.'Z3-"-'a-', (18); 
 
 = 1 +S,„.'P"' . S«a,«_„+i .-za-"-' r/ - ', (()). 
 
 >n+l 
 
 = 8„a,„_„+i-'Zjr"~ '«"' + «. 73-'" «-', (()); 
 
 = S„(-^).^"'-'V/-', (s) 
 
86 
 
 •• ■3r'V/.-i = (^".r/-^).A^.f^V (65); 
 
 236. Theorem. 
 
 Sf, /n"* - 1 CO m >i - 1 / _ /) 
 
 =S™.r'"-'-S„»,„_„/>-'.A^ 
 
 SnK-,^V^' 
 
 (^) 
 
 For, ^;^"'-^-^"' ' =(S^ff,_,.r."-^)(S„fi„_,.i-"-')-', subject 
 
 to the condition that all the coefficients after «,_], and ^^.i 
 vanish ; 
 
 = S,„.t?'"-'.S„«„-„.'Z3-"-^&-', (228), and (2()); 
 = 8,„*^'"-'.S„a,„_„./>-'.A+,(^], (235). 
 
 ^ d,".(h(u) " , , , ^ T*r" "'ft'" , fL'"fi 
 
 For, <l>{u+D,n) = S.^^^-~ -(.D-u)'-', (198); 
 
 4.5^^*w.,-.(s,„/,-.''p)"", w/ 
 
 Jn-1 V [wv. / 
 
«7 
 
 where a„_,= , (228); 
 
 \m ^ 
 
 S„h"-'.S,n~-. V^^.^-ia-™, (26). 
 
 hi-m 
 
 d".d)(u) «+i dJ'-'" + '(h(u) , , ^ 
 
 • P-^=S>„~ ri_Z.^™-i «»- + ', (201): 
 
 = Sm-^^^ ^-^-^.•z3-'"-^a«-"' + S Since 'Zcr"a''=0; 
 
 n-m + l 
 
 = S,nd,r(p{?i).—r^^, (8). 
 
 238. Theorem. 
 <^(S,„ff,„_,,r'"-') = 0(a)+S„A^".S,„d/-'" + '0(f/), 
 
 For, 0(S„,a,„_,.^^'«-') = 0(f/+A^S,„a,„.^"'-O 
 
 , X V ^ d'ld)(a) , ^ 
 =0(a)+S„^^(*>.S.«,„.^"'-O% (198): 
 
 „jn-l „ n-m + l 
 
 \n—m+\ 
 
 = <p{(i) + S„ j^ . dl . (/)(«) . S,. .1^'" - ' . -22r"' - ' a,", (228) ; 
 
 
88 
 
 1 " 2m / _j \ 
 
 239. Theorem. =A'-'-i+S„,.i?'-"'-^ A+J r -K 
 
 For, 
 
 -T-7= S.r- , (106) and (9): 
 
 ■-x-K^rnX""-^ .'sf''-^ a-\ when a™_, = T— , (6) and (228); 
 
 ■■^moo'"' ^.a- 
 
 \^r{-^\ (6) and (235) 
 
 =8..^'" 
 
 -^aI.i 
 
 l[r+l) 
 
 0= 
 
 
 
 =S..cr™ 
 
 -^&™-., 
 
 suppose. 
 
 But 
 
 1-, 
 
 1 
 
 6'^ 
 1 
 
 
 6^- 
 
 1 1-e-' 
 
 
 1 
 
 1 
 - + 
 
 = S„62.-3(^""-'-'V""^') + S™6„„_.(.r^'«--+.^''^'«-^), (14): 
 =2S„6,._2.a^^'"-% (6) 
 =2feo+2a,6.„,.r2"', (.9). 
 
89 
 
 Hence -i-=.r-^-l+S,„.v'". A+r fr^), (9) and (64): 
 e'^-l \ |r+l / 
 
 2»i+l / — 1 \ 
 
 240. Cor. 1. A+ J j =0. 
 
 \\r+lj 
 
 2m / _1 \ 
 
 241. The number A+rli is called the (2m-iy^ 
 
 number of Bernouilli. In order to deduce the successive 
 values of these numbers, we have 
 
 2m / 1 \ m 1 2n—2 / i \ 
 
 2n-l / _i \ 
 
 +s. 
 
 2m-2n+2 
 
 m _i 2n-2 / _J \ j 
 
 2m / -1 \ 
 
 and, putting, A+r (nr~j) =^2m-n 
 
 Ci =-T-=,083 
 
 €3=-7g=-,00138 
 
 M 
 

 
 90 
 
 ^5 = 
 
 1 
 
 6 [7 
 
 ,000033068783 
 
 ^7=- 
 
 3 
 
 -,000000826719 
 
 €9 = 
 
 10 
 
 Hi ' 
 
 ,000000020876 
 
 eu=- 
 
 691 
 
 -,000000000528 
 
 gl3 = 
 
 1 
 
 ,000000000013 
 
 12 [is" 
 
 ei.^=- 
 
 3617 
 30|l7~ 
 
 -,000000000000. 
 
 242. Theorem. 
 
 
 
 D,''u=h-\ f^u-"^ +^X2u-xh'"'-\d^''^-'u. 
 
 For, assume D^ \u=^„Ji"' ^a;„_2, where a^_o, is some 
 function of w, and independent of h. 
 
 .'. ««=S.A'"-~.i),.«„_2, (136) and (137); 
 
 = Lh--'.k~.dJ'.a^_,, (199); 
 
 = S./^'"-.S„^^^^^, (6) and (18); 
 
 =4.a-, + S,„A^sl^5fl^^, (9). 
 
91 
 
 ;=t/^.cr_i, or a_i=f^.7C 
 
 m+l ^ n 
 
 and 0=Sn-— 
 
 \n 
 
 m A n-y\ 
 
 ••• 4- «,«-!= -S„ i , (9). 
 
 A little consideration respecting the form of this de- 
 velopment will convince us that d/'w is a factor of d^. «,„_!. 
 
 Assume, therefore, d^ . a,„_ i = d/ 7^ . — j . 6,„ ; 
 
 | m+l 
 
 then d^.a_i=?*.(-l).fto? and =u. .-. &(,= -!. 
 
 /_jy« + l m (-1)'"""+^ 6m_„ 
 
 Also d/'w.-V-^^ — .&^=-S„c^/.dr""«*.V-^ — n r. 
 
 Im+l TO-y^+l [n+1 
 
 by substitution; 
 
 -4'"^<-S. i^ ^^"' "•^"'"" , (6). 
 w-/^+l m+l 
 
 (-ir&. _s (-ir-"fe.-.-(-i) 
 
 *■ [m+l | m-y^+l [w+l' 
 
 and a,_, = d/-'z<.A+,.(,^ 
 
92 
 
 Hence A"'«=^"'-/.^+S„^'""'-4'"-'w. A+, f^) 
 
 =h-Kf,u--+hji''"-'.dr'''-'u.A+r(r^\ (240), 
 
 ■■h-\f,u--+SX2.-ifi""-'.d,'"'-'ii, (241). 
 
 243. A..-'u=f,u--^8mCzm-.-dr-'u. 
 
 n+l 2 L__ 
 
CHAPTER VIII. 
 
 ON THE DIFFERENTIATION OF EXPONENTIAL AND 
 CIRCULAR FUNCTIONS. 
 
 245. Theorem. d^.e''=e'. 
 
 For, e^=l + S«f-, (106) and (9). 
 
 rf..e^=S™T^— -, (193), (191), and (210); 
 m— 1 
 
 = 6% (106). 
 
 246. Cor. l. d/.€'=e'. 
 
 247. Cor. 2. 4.e"=e".rf^?^ (202), 
 
 248. Cor. 3. d/.e" = e"^rt". 
 
 249. Cor. 4. d/.a'^=d/.(6''°'^'") 
 
 =6"'°e.«.(log^a)« 
 = a^(log,a)«. 
 
 250. Theorem. d^.(\og^.To)= - . 
 
 For, ci'=6'°«^" 
 
 .-. l = e"'s^^4(loge*), (197), and (247); 
 = ct;.4.(log,a). 
 
 ••• 4(loge<r)=-. 
 
94 
 
 251. Cor. l. d^.(\og,u)=—, (202). 
 
 252. Cor. 2. d,.(log„w) = d, ^°^'^ 
 
 logj« 
 1 
 
 rf,.(logeW), (196); 
 
 loge« 
 
 1 d^u 
 loge a u 
 
 253. Theorem. 
 
 For, 4.(logecx)=-, (250); 
 
 
 1+cT"-1 
 
 71 being any number; 
 
 =.^"-^{s.(-lr-'(a^"-lr-^+(-l)^^;^|, (12); 
 
 (191) and (207); 
 
 nnc 
 
 + const. 
 Hence 
 
 iog.i=-.s.,(-.r-'.<i^:^+(-.)'. f <^:^+comt. 
 uog..= l.s.„(-.r-.^^^^+(-.r.L^f:^. 
 
95 
 
 254. Cor. l. U x"^l<i. 
 
 1 » ^r" — A" 
 
 then log,.r=-.S,„(-ir-^^:^-^ 
 n 7)1 
 
 255. Cor. 2. If .j; < l', then 
 
 (12). 
 
 log.(l-aO=-S.-: 
 
 1+J7 
 
 and loggj— =loge(l+j7)-log,(i-a?) 
 
 =s.(-ir- 
 
 ■s.- 
 
 ^s.K-ir-^+iS-, (5); 
 
 =^-S'«i;^' (^^)- 
 
 1 " if"' " 1 
 
 256. Theorem. logga?=-.S 
 
 For, loge W=- .{ loge loff , > 
 
 (-!)> 
 
 (i^TT-)"] (^^'^^ 
 
 n \ m(,i'"+l)"' m(a'"+l)'"j 
 
 _ 1 » <f""'-l 
 
 n 'm{x"-\-\y 
 
 (5). 
 
96 
 
 257- Theorem. 0(O=S;;,£— ^ • {0(l + A)i 0»->. 
 For, (^(60 = 05 l+e'-lf 
 
 = S„^£:M(,..1)«-., 098); 
 
 1^2-1 
 
 -1 |y^-i 
 
 =S„%^.S.(-iy-'.f^.e^"-^'^ (86); 
 \n-l I?'— 1 
 
 !w-l 
 
 CO W"~l A\(t\ n I » -vi'n-l 
 
 =s.^^-s,(-.)'-.^.s.(«-r-.^,(.o6); 
 
 CO ~,TO-1 0= jn-\ ^(J.\ n I 
 
 =S«irT-S.^[^^-S.(-.)'-.j^.(-r-, 07); 
 
 =S.,^.S.!?^-^.A-.0.-. 050); 
 =S.,-^^^.50(1 + A)SO"-', (198). 
 
 .n+m-1 
 
 + !«-! 
 
 258. Theorem, (e' - 1 )" = Sm i : • ^" • 0' 
 
 ^ ^ n+m-1 
 
 For, (e'^-l)"=S,„r^^-(l+A-l)".0'«-\ (257); 
 
 CO '»?'"" ' 
 
 =S.T^ -.A"o™-\ 
 
 lm-1 
 
 = S.T^— . A"0"'- + S. r 7 • A". o-'«-\ (9) 
 
 \m-i \n+m-l 
 
 = 8. ,'''" "* .A-O"-^'"-', (146) and (l38). 
 w+m— 1 
 
97 
 
 259. Theorem. 
 
 r' + l m-1 
 
 A"-^0" 
 
 For, -^=S,„^^ — ■ — :--W-\ (257); 
 
 TO-1 V2 + A/ 
 
 w ^TO-i CO A"-^0'""^ 
 
 A"''.0"'-^ 
 
 'S"u^-s.(-')-- ... 
 
 , (146) and (138). 
 
 260. Theorem. -^ = S™ t^^ . S„ ( - 1 )" " ' . '— 
 
 e^-l \m-\ n 
 
 For, 
 
 X ^ logs e ' 
 ''-I ~ e'^-l 
 
 "^"•Im-ri 1+A 
 
 loge(l+A)K™ 
 
 0'"-', (257) 
 
 = S„ 
 
 cT?'""^ [logg(i + A) 
 
 jlog,(i + A)j 
 
 A"~^o'"' 
 
 =S.i — -.S„(-i)"-'. 
 
 \m-\ n 
 
 {^5b), (146), and 
 
 (138.) 
 
98 
 
 261. Cor. Since— — =i- - +Sm^t^"'-Q2m-i, (239) and 
 e — 1 2 
 
 (241) ; 
 
 .-. S„(-i)"-.^— :^— =0, 
 
 2>ii + 1 A ''! - 1 Q2m 
 
 S„(-iy-'. = |2m-€2.-„ and 
 
 ^=,_%s.^:s:(-o"- ^"■'' 
 
 6—1 2 \2m n 
 
 262. Theorem. Dj'Ti=S,n^^ — ' ./i"+"'-'.d;'+"'-'M, 
 
 For, D/u=(€'"^--l)\u, (201); 
 
 = S,„1 ./i"+'"-i.d/+"'-iM, (258). 
 
 m+m-1 "^ 
 
 263. Theorem. \\og^(l + A)\".0"'=0, {m%n) ; and 
 {log,(i + A)J".0" = [w. 
 
 ^«**' §'»|^l-{loge(l + A)}".0"'-^ = (log,e')% (257); 
 
 .-. 5log.(l.-A)S".0'«-=0, (^>.), andM^il^M:^' 
 
 \n 
 
 264. Theorem. \log^ (l + A,,) \ u=d^u. 
 For, ;iog,(l + A,)}?*=S,„(-l)'"-'.-^— , (255); 
 
 = S. ^^^ \ U'^^-iru, (201); 
 
99 
 
 = S,„ ^—^— . S„ j .f//' + »-' w, (258) ; 
 
 m \m+n-l 
 
 " rf^""?^ ^ (-1)'"-" 
 
 = S. i—Sn — ^^ . A"'-" + '0'% (6) and (18) 
 
 [m m-n+l 
 
 = S„^.S„(-1)-'.^^, (8) and (13); 
 
 = S.^.{log,(l+A)fO"', (255); 
 
 = 4w (263). 
 
 265. Cor. dj'u^ {\og,(l + /\,)Yu. 
 
 266. Theore^n. f^) =1, and f^^) =1. 
 
 For, 
 
 tan A' siHci? 1 
 
 /tanAA /sincVN 
 
 Also, for every finite value of v, tancV>cV, and sinA'<cV. 
 
 And these two relations can only be made to agree by 
 
 the equations 
 
 /tan.i'\ , /sin.2?\ 
 =1 and =1. 
 
 267 Theorem. dp.sincV=cosct'. 
 
 _ , . fsin(cX'+/i)-sin.vl 
 
 ror, d^,sintV={ > 
 
 I ^ h, 
 
 s(.4).. 
 
 h\ . h 
 sin- 
 
 2 
 
 = cos*, (266). 
 
100 
 
 268. Cor. l. f4.cos.i'=dr.sin f jc\ 
 
 (I-..). (30.) 
 
 269. Cor. 2. d/"-^sina?=(-l)"-'coscj?, 
 
 d/". sina7= (- 1)" . sino,', 
 
 cZ/"~^coScC=(-l)".sm.r, 
 
 t//" . costr=( - 1)" . cosa?. 
 
 270. Thewem- dv . tan a" = (sec x)-. 
 Jbor, dr.ta,na'=a^. 
 
 COStt' 
 
 (cosij?)-+(sina?)" 
 (cosa)^ 
 
 = (secci')^. 
 
 271 . Theorem, d^ . sec a' = sec x . tan d'. 
 For, d y . sec a = rf^ . (cos cv) " ' 
 
 = (-l)(costr')~^.(-sina7) = 86001?. tan .^?. 
 
 272. Theorem. dr.sin~'a?= ^ ==. 
 
 For, .!■ = sin (sin - ' .r) . 
 
 .-. l=cos(sin"'ct?).rf,sin"\T;, (202); 
 
 = y/l-w'. d.r . sin ~Kv. 
 
 1 
 
 v.. 
 
101 
 
 273. Theorem, d^.cos- 
 
 -1 
 
 For, .r'=cos.(cos"''a;). 
 
 . •. 1 = -sin (cos' x) . d^.cos~^ x 
 
 = - v^ 1 -w'^ . dj, . COS" ' a;. 
 
 .-. d^.cos'Kv= — 
 
 274. Theorem. d^.tan-Kv= 
 
 l+x' 
 
 For, A' = tan . (tan ~ ^ a?) . 
 
 .-. 1= ^sec.(tan~'cZ')p.d,,.tan"',r 
 = {l+x^). d^ . tan" ' X. 
 
 .•. c?^.tan~'a?: 
 
 l+,'»^ 
 
 275. Theorem, d^.sec ^x- 
 For, ir=sec.(sec~^i7). 
 
 .VVc^~— 1 
 
 1 = tan (sec ' x) . sec (sec ~ ^v) . d^ . sec ' 
 
 .r^ — 1 . .r . d,, . sec ^x. 
 
 rf,,.sec 'cr= 
 
 xy/ oir- 
 
CHAPTER IX. 
 
 ON THE EXPANSION OF CIRCULAR FUNCTIONS. 
 
 276. Theorem. 
 
 n _ n n 
 
 P,.(coSci?,.+ V - 1 . sinc^r) =cosSr*, + V - 1 . sin S, -^•,• 
 
 For, (cosA,4-v-l .sina'j)(cosa;2+ V -1 .sin do) 
 = cos (x^+x.^) + V - 1 . sin ('Ci+x.,). 
 
 And the introduction on the first side of the equation 
 of a new factor of the form cosa?,-!-^/ — l-sin^v will increase 
 the arc on the second side by the quantity a?,. Hence the 
 truth of the theorem is manifest. 
 
 277- Cor. l. Put .v,-x, then 
 
 (coscr+'v/-! .sin.r?)"=cosWir+ V -1 .sinWiTj w being any posi- 
 tive integer. 
 
 Again, (coScr-i-\/-l.sint77)(costr-v -l .sinA) = l. 
 .•. (coStT+v-l .sin,r')"'=costr;- V -1 .sinx 
 
 = cos (-.1) + -v/- 1 . sin (-ci), 
 and (cosA + V-l. sin ,?;)'"= |cos(-,7,) + '\/-] .sin(-cX^)|'' 
 =cos(-w.r) + \/-l.sin(-WA'). 
 
103 
 
 Hence, (coScV+ \/^. sin<r)*"'=cos(± W2 a') + \/~^\ . sin(=t m.v), 
 m being any positive integer ; 
 
 ?i being any positive integer; 
 
 = {cos(i'^..r)+v/3,.sin(±^.,.)j" 
 
 .•. (cosa?+- V— 1-sina;) "=cos| i — .a' 1 +v — l.sinf ±— .ii'j . 
 
 Consequently, whatever rational value w has, 
 (cos,T?+ v -1 .sin*')"=cosw.x'+ V -1 .sinw.r. 
 
 278. CoR. 2. 2coswa'=(coScV+'\/^.sina?)" 
 
 + (coSci;±'v/-l .siuct')^", 
 and 2 V^-l • sinwcr= (cos .37+ V -1 . sin A')"-(cos.r± v"^ . sinci)"^". 
 
 n—2m+2, 2m-2 
 
 279. Theorem. cosSr'^,■=Sm(-l)'"~'.Cr,,(cos.^^,.sinc^',), 
 
 " 55 n—2m+l,2m-l 
 
 and sinS,.^■,-S,„(-l)"'-^C,,s(coscl;,.sincr,). 
 
 For, cosSrA\+v -l-sinSrA\. 
 
 = P,. (cos c37,. + v - 1 . sin ,%\) , (276) ; 
 
 CO n—m+l,m 
 
 S,„ C„4(cos*v)(A/-l.sincr,)|, (67) and (13): 
 
 CO m— 1 n-TO+l,m-l 
 
 = S,„(-1) '^ C,,s(cosav.sin.i',), (55): 
 
 CO w— iini+a, 2»J— 2 
 
 = S;„(-l)™-'.C,,,(cosa?,.sincX',) 
 
 «— 2n^+l, 2m 
 
 + \/ - 1 . S« ( - 1 )"' - ^ C,,, (cos .r, . sin v,) , (14) and (6) . 
 
104 
 
 Whence, by equating possible and impossible parts, we 
 obtain the above theorems. 
 
 280. Theorem. tSLX\i^,cV,=: 2^;:=^, • 
 
 s,„(-ir-^a(tan^,) 
 
 For, cos S,.a?, + v-1 • sin Sy^r 
 =P,(cosa?,) . P,(l + \/^l . tan.r,), (276) ; 
 =P,(cos.iv).S.(-l)^ ".C','(tan.xv), (68), (55) and (13); 
 
 =P,(cos,r,){S„(-l)'"-^C,(tan.r,)+A/-l.S,„(-l)"'"'C,.(tan.xv)5, 
 (14) and (6). 
 
 Hence, equating possible and impossible parts, 
 
 n n a 2m-2,n 
 
 cosS,ct?,=P,.(coScV,) .S;„(-l)'"-'.C,.(tan.?7,), and 
 
 n n ~ 2m—l,n 
 
 sin S..^^,.= P,. (cos w,) S,„ ( - 1 )'" - ' . C. (tan x,) ; 
 and, by dividing the second of these equations by the first, we 
 obtain the theorem sought. 
 
 281. Theorem. 
 
 \n-m 
 
 cos n,v=ln . S^ (-1)'"'^ 'r^~ -(2 cos ,i-)"--"'+-; n being any 
 
 -^ jm-l 
 
 integer. 
 
 For, 2 cos nx= (cos .r + v - 1 . sin .vy 
 
 + (cos x-y/ -\. sin .7?)", (278) ; 
 ^ cos w<r 2. (cos ci' + V - 1 . sin ii)" 
 
 .-. -2.S„ — i7-=-S„-^ 
 
 nt" nf" 
 
 (cos x-\/^-l . sin a?)" 
 
 ■s« 
 
 nf 
 t being any quantity greater than unity, (i) and (6). 
 
105 
 
 , / cos .t+a/ -1-sin A'\ , / 
 
 :l0g, (^1 -^ -j +l0g, \l 
 
 (255) ; 
 = log, Jl-ri.(2cos v-f-')] 
 
 " (2 cos x-t-^y ^ ^ 
 
 = -S„- -, (255); 
 
 s A^-V -1-sin A' 
 
 ')' 
 
 = -S„— -S. 
 
 :i (-1)" 
 
 nt" \m-l t'"-^ 
 
 )SA')"-"' + S (86) and (13) 
 
 \n-'i 
 
 = -S„-.S.(-l)"'-^^2i^.(2cos.xr-^'"-^^ (6) and (18). 
 
 Hence, equating coefficients of t' 
 
 —2 cos nx 
 
 _S^(_ !)'«-' .iL:2 _^2 cos *•)"-""+' ; 
 
 .-. cosnx=\n.^„X-\r'\f^.{2co^xy-''-^\ (13) 
 
 282. Cor. If n is even, every term will vanish after the 
 
 (iw+])% and 
 
 ^n+m—2 
 
 cosnx=ln.{-\f" .'^.{-^Y"' -TT^ (2coScV)' 
 
 (8) and ((i); 
 
106 
 
 iw+m-2 
 
 and -^n.-^ 
 
 =i'» 
 
 ^n+m-2 
 
 ^n-m+1 ^ \2m-2 . \^7i-m + l 
 
 ^n+m-2 
 
 = in.l2i;^ 
 
 \2m-2 If^ ^ ^ 
 
 = £-fe±A-Lfczi 
 
 , (40); 
 
 \2m—2 
 
 ■V.\{\n^r){\n-r)\ 
 
 \2m-2 
 
 -Vr{{\nf-^^\ 
 
 \2m-2 . 2^ 
 
 -.P,(^^-4r^); 
 
 in+l p („2_A^2\ 
 
 If n is odd, every term will vanish after the ^(n+iy^. 
 
 and 
 
 iin+i) \ i(n-l)+7n- 
 
 i(w + l)-m 
 
 (2cos.r)2'"-', 
 
 (8) and (6) 
 
107 
 
 pnd 
 
 H,i-\)-m i^^ - ^ ■- 2m-2 
 
 (w + l)-m 2m-l . ^(n+l)-m 2m-l 
 
 |l(n-l)+m-l . \i(n-l) |l(n+l) . [i^-O 
 
 2m-l 
 
 2//2-J 
 
 p,.Ki^+>-4)(i^^->'+i)i P>K4^r-0-i)1 
 
 |2m-l 
 "PrJ/i^'-(2r-l)^( 
 
 l^- 
 
 2m-1.2=^ 
 
 
 283. Theorem. 
 
 n 
 
 (smc^7)»=(-ly^2-"+^Sm(-l^^'•^^=^^^•cos(w-2w^ + 2) 
 
 [m-1 
 
 L« 
 
 ill 
 
 (sm.^?)'' = (-ly'"-'^2-" + ^S,,X-0'""'•■i^~•sm(w-2;/^+2)^; 
 
 according as ?i is even or odd. 
 
 For, 2 \/ - 1 . sin ,t= (cos ,v+ \/- 1 . sin ,v) 
 
 - (cos <!■+ v-^ . sin d!) ~ ' . 
 
108 
 
 (l) If w is even, then 
 
 2U-iy\isma;y=S,n{-iY'''-T^^^^^Ucos.v+y/-l.sma;y-'"'^' 
 \m-l ' 
 
 \n 
 
 (cos 
 
 O! + v^ . sin xf - ^"'+^ Ij w 
 
 (90) 
 
 .ir, in 
 
 ^g^(_l)'"-i "'-I .2cos(w-27?i + 2),y + (-iy"-^, (278). 
 
 (sin .r)" 
 
 __/_,y« o-" + i Q (-1)'"-^ """^ .COs(M-2m-h2),r'+2"".-ip— . 
 
 (2) If w is odd, then 
 2«.(-l)^".(sin.T)" = S,„(-l)™-'--r^U<^«s,x'+V-l.sin.77)"-2'»+' 
 
 ^ |, 
 
 (cos,r+A/^.sin,r)"--'"+'-^J 
 
 I- 
 
 (90); 
 
 :S^(_l)'«-i._::Lj_.2-y-l.sin(w-2w+2).r, (278). 
 
 (sin.»')" = (-])^'"-'^2-" + '.S.(-l)"' '•p~-^in0*-2'«+2).r, 
 
109 
 
 284. Theorem. 
 
 (cos a')" = 2 " + ^S« 
 
 \m—\ \^n 
 
 (cos cr)"=2~""^^.STOr~-^--'C^'^ (^~~^+2)'^'' according as n 
 
 \m - 1 
 
 even or odd. 
 
 For, 2 cos OD = (cos x-\-\/ -\ . sin ct?) + (cos d' + v - 1 . sin x) 
 
 (1) If w is even, then 
 
 71—1 
 
 m—1 
 
 2" (cos ci')"="S,„ I "'"^ I (cos .77+ \/-l . sin .v)"-~"'+' 
 
 (89) 
 
 \n 
 ^ J 1 _tr_ 
 
 "^ (cos. r;+\/-l. sin *')"-"'" "'J [i^ 
 
 = Sm,"'~^ .2 cos (w-2w? + 2).'P+^^— , (278) 
 lm-1 \-^n 
 
 , \fi' [** 
 
 .-. (cos.??)" = 2-" + ^S,«-r^^^^^-cos(^-2OT + 2)A' + 2-".-±L_. 
 [m-1 \in 
 
 (2) If w is odd, then 
 
 \n 
 i(«+i) L_ I — 
 
 2". (cos cv)"=S,r. I "'"^ Ucos cr+ V -1 . sin 0!)"-'^'"+- 
 \m — l [ 
 
 + y^ 1 , (89) 
 
 (cos,r+'\/-l.sma)"-'^'"+~J 
 
 *(«+., [^ 
 
 = S„,~^.2cos (w-2m+2).r, (278) ; 
 
 m-1 
 
 (cos .-p)" = 2 - " + ' ■ S,„ I "'~^ ■ cos (ri-2w+2)*'. 
 m-1 
 
110 
 
 285. Theorem, sin a?=S,„(-l)'""^ 
 
 2m-; 
 
 For, sin *'= Sm r • C=f/ • •'^in a', (213) ; 
 
 m-1 
 
 cc ^2m-2 CO ^2m-1 
 
 =S-^^-<"-i"-+S.y5^, .C.--in.., (14); 
 (269); 
 
 ce ,vi2?n — 1 
 
 =S,«(-i)" 
 
 2m-l 
 
 286. Cor. 1. cos .^■=(/,.sin ,i', (26?) 
 
 CO ^.2m - 
 
 =rf..S.(-i)" 
 
 2m-l 
 
 =S,„(-i)" 
 
 2m-2 
 
 287- Cor. 2. 
 
 '^i.S.(-i)" 
 
 2m-l 
 
 2m-2 
 (14) and (6); 
 
 =cos cT+ V -1 .sin i??, 
 
 and e"''^-^ = cos {-<v) + y/ -I .sin (-a) 
 
 =cos cT'-V -1 -sin .r ; 
 
 ... 6'^V~l^g-.rV-l^2COS.«7, 
 
 and 6 ' ^~i - 6 " ' '^ = 2 \/^l . sin a-. 
 
 288. Cou. 3. 6'~'"-'>^'^-v'^ = (-i)'«-'Vri, 
 and e''"-'''^^^~i = (-l )"'-'. 
 
 (106), 
 
Ill 
 
 2w-2m+l '^' v|2r/ 
 
 289. Theorem. 
 
 For, put^=-ct^^ then 
 tan ,^' = sin ^r' . (cos .t") " ^ 
 
 =-S.^.(S.^,) . (.85) and (.86) 
 
 =cr.S„r- . S,n^""'-'sr'"~'-a' 
 
 \2n-l 
 
 =^.S„^"-^S„ 
 
 where «,„_i = 
 
 2m-2 
 
 (228) ; 
 
 •Z3-' 
 
 «-'«-' 
 
 , (18) 
 
 |2n 
 
 -2m+l 
 
 in-l 
 
 <^ 
 
 1 
 
 
 ^'"\2n- 
 
 -2m.+l ' 
 
 ce w 1 '"-1 / - 1 \ 
 
 =s.(-i)-...-'.s,g^3^^.A..y. 
 
 290. Theorem, sec cr = S„(-l)"-'-^^'"-'- A + , (i^l 
 For, sec <r = (cos ct?) ~ ^ 
 
 = s. 
 
 2w-2 
 
 (286) 
 
 :S„«"-V"-^a-\ where«„_i=T^^^, (228); 
 
 = S„i?f"-^a-^ A+, (-^■) , (235) ; 
 
 =a(-i)«-.^--.A,,.(^J. 
 
112 
 
 291. Theorem. 
 
 cot 
 
 ce « 1 "'~* / 1 \ 
 
 ^"'^ ^ ™[2?i-2m \ |2r+l / 
 
 For, cot .r' = cos ^r . (sin w) ' ' 
 
 = §^^!lL. /^.S,„^-f^V\ (285) and (286); 
 
 l27i-2 
 
 "'^"^«"-=^' (-*'^ 
 
 --'■s-^'-'-S-^^' <'«' 
 
 CO "1 '"~^ / tt \ 
 
 ^S«(-)"-.-^"--S.^^^3^.a-.A.,.(^-),(235): 
 
 :S„(-l)"-^.^''^"-^S,«T:-— — .A,.. 
 
 2n-2m' '^'' V[2r+l, 
 
 292. Theorem, cosec .x'=S„(-l)"-'..i?'"-'. A+r fi-^^ I 
 For, cosec w = (sin w) " ' 
 
 =^-'.S„s?"-^'2r""^a"', where «„_i=T , (228): 
 
 = .^-^S„^''-'.rt-'.A + ,.(— V (235); 
 
 =S-(->""-^""-"'^-(|-i77I 
 
113 
 
 293. Theorem, sin k=x . P, | 1 - [— ) I . 
 
 For, the roots of the equation 
 
 0=sin X 
 are a'=0, andcr=±r7r, (J^^)- 
 
 CO 
 
 .'. sin .r=cfa-\Pr {(cX'-r7r)(ci'+r7r)^, where a is independent of <»; 
 
 =...p,(-.v^).p,{.-(^)] 
 
 05 
 
 = a.2?.P,.(-»*'7r^). {l+terms in a?^|. 
 But sin a? =.r+ terms in a;'') (285); 
 
 .-. a.P,(-r'-7r-) = l, 
 and sin a^rr.T-.P,. <1- I — I >. 
 
 294. Theorem, cos .^=P,|l- f = — ji. 
 
 For, the roots of the equation 
 
 O=cos ,v 
 are ,r=±(2r-l)7r, (|^"x)- 
 
 .*. co^x=a.^Ax'^-{- .TTJ ?, where a is independent of A' ; 
 
 = a . P,. I - I . TT J Ml +terms in x^ \ . 
 
114 
 
 But cos x=l + terms in or, (286) 
 
 and coSc'g=PJl- I L — ) >. 
 
 295. Theorem, log^ sin cT=logg cr-Sm ( — ) • — • S^w"-"'. 
 
 \7r/ m 
 
 For, sm.v=x.vAl-(~] |, (293); 
 
 log, sin ^=log, x+S„ loge 1 1 - y—j I 
 
 » CO / ,r \ ^'" 1 
 =\og,a;-S„Sj—] .-, (255); 
 
 CO /r\^"' 1 *" 
 =loge.x'-Sj-) .-.S„n-^ (17) and (6). 
 \7r/ w* 
 
 CO /2x\^"' 1 " 
 
 296. Theorem, log£Cos.r=-S^ — • - ■Sni^^-l)'^"' 
 
 V TT / TO 
 
 For, cos.r=pJl-(=:^.— I i, (294); 
 
 log, cos 0?= S„ loge 1 1 - f — ^ -) J 
 
 = -Sn S. . - , (255) and (6) ; 
 
 \2n-l.'7r/ ^ 
 
 = -k, {—) .-.a,(2n-l)-^'% (17) and (6). 
 
115 
 297- Theorem. 
 
 OS / '> t/X 1 " 
 
 log,tan^^=log,a'+S,„ \^~) .- -S,, (-lr-'•'^■""• 
 For, loggtana;=logeSina?-log6COsa; 
 
 CO to 7>\ '^"^ 1 ==■ 
 
 ^ Vtt/ w 
 
 Vtt/ m 
 
 CO /O t\ '^"' I '^ 
 = loge.i' + SJ-) . — S„(-0"-'-^"'"'' (I*)- 
 
 298. Theorem. 
 
 For, «^,.tan-^^;= -, (274) 
 
 1 -{-CC^ 
 
 =s.(-ir-'-^^"^'"-'+(-ir-^:^,, (12); 
 
 •. tan-..= S.(-l)'"-.^— ^+(-ir-j:..J— :, (l^O & (210). 
 
 209 Cor. If .r < 1, tan"* .r=S,.(-l)'' 
 
 2m-l 
 
116 
 300. Theorem. 
 
 r=,s.s.(-0"-'.^^-S.(-0- 
 
 2m-l ' ^ (239)"™~*(2w-l) 
 
 For, put tan~^-=a; then 
 5 
 
 2 tan a 5 120 
 
 tan 2a= — ; = — , tan 4a= , and 
 
 l-(tanay 12 119 
 
 TT 
 
 tan 4a-tan — 
 
 7r\ 4 1 
 
 tan I 4a = = 
 
 4 / TT 239 
 
 1 +tan 4a. tan - 
 
 4 
 
 TT 
 
 — =4a- I 4a- 
 
 4 V 4 
 
 ,1 ,1 
 
 = 4. tan — tan 
 
 5 239 
 
 :4.S,„(-1)" 
 
 52™-^(2m-l) 
 
 ■S-<-)-- (.39f.-'(.^-,) ' (^9* 
 
 1*"' jir, =4.x{,2r-' = ,8x(,04)"-'. 
 
 .•.-=.».s,.(-o-'.^^-s..(-o- 
 
 (239)-'"-'. (2m -1) 
 
117 
 
 H 2m— I : — 
 
 301. Theorem. .r"+l=P„,(a-6 " '' )• 
 For, put 0=.x>" + l; then x"=-\ 
 
 ^g(.'»-i);rV=i^ (288). 
 
 2m— \ I — 
 
 .ttV-I 
 
 .-. ^■=e " , 
 
 2m- 1 , 
 
 and the n different values of e " ' ~ are the roots of the 
 equation 
 
 = ct>"+l. 
 
 2TO-1 
 
 Hence the n different values of {x-e " ' ) are the simp] 
 
 iL , , 2OT-1 / — - . 2m-l . , 
 .i?"+l=P™5'^-(cos .TT+V-l.sin .tt)}, (287). 
 
 factors of (ci?"+l). 
 
 302. Cor. l, 
 
 2OT-1 / — - . 2m-l 
 
 -TT+v -l.sin 
 
 71 n 
 
 303. Cor. 2. If n is even, then 
 
 i« 2»i-l , — 2H-2(n+l , — 
 
 .x-+l=P„{(a^-e— •''^^)C.-e-^^-'^^-^)|, 
 by inverting the order of the latter factors, (31) ; 
 
 \n 2m -1 , 2m- 1 , — 
 
 =P„,5*--.t'(e^-"^-^ + e""^-"^-^) + i;, since e^'^^'-i = l, (288): 
 
 i " 2 W — 1 
 
 =P;„Gt^''-2,r.cos~ .TT+l), (287). 
 
 If 11 is odd, then 
 
 i(«-l) 2m-I ;_ i(n-l) 2w-2m+l _. 
 
 a.''+l=P,„(a;-6— •'^^)(^+l).P.Gr-6~^^-'^^), 
 (32), (288), and by inverting the order of the latter factors, (31) ; 
 
 ^(ra-l) 2m-l , — 2m-l ,_ 
 
 = (.^.+l).P„.{..^-..(e— •'^^Ve- — •'^^-^) + lS, 
 since €^'"^-^ = 1, (288) ; 
 
 i(n-l) 2IW-1 
 
 = (cr + l).P;„C^'-2.t.cos TT+l), (287)- 
 
118 
 
 n 2m— 2 / — 
 
 304. Theorem. w"-\=V,n{^^-i~^' \ 
 For, put 0=ct'"-l; then of = I 
 
 = e^^-"^S (288). 
 
 2m— 2 , — 
 
 and the 7i different values of e " "'^ ~ are the roots of 
 
 the equation 
 
 0=*'"-!. 
 
 2m — 2 
 
 Hence the n different values of {w~e " ' ) are the simple 
 factors of (a?"-l). 
 
 305. CoR. 1. 
 
 A'"-1=P;;; {.i'-(cos . TT + v " 1 • sin"^^ --tt)^ 
 
 306. CoR. 2. If n is even, then 
 
 in 2m-2 
 
 . /— - -zn—'Mi I — 
 • Tr\J—l .7rV-l 
 
 a?"-l=P,„{(c'P-6 " ' ~ ){x-€ " " ~)|, by inverting the 
 order of the latter factors, (31); 
 
 i«-l 2m /— 2re-2/K _w— : 
 
 = {oj-l).V4(,x-e-- ){w-e » ■ '''')|(,r^ + l),(32),&(288); 
 
 in—\ '2m , — _ 2m /-^ 
 
 = {ci^-\).Vm{ai'-w{e^^' ~'+e ~-"^-') + \\, since e'^W^i^j; 
 
 = 02?--1).P„OX'2-2.V.COS^— .TT+l), (287). 
 
 If w is odd, then 
 
 i(«— 1) 2m^ ^— 2n— 2to ^ ^/-^ 
 
 <T'"-l = Gt?-l).P„, j(.x^-e " * )(<??-e " )!, by inverting 
 
 the order of the latter factors, (31); 
 
 i(?i— 1) ^JI^sPTi -^^^\l~\ — 
 
 = (.v-l).P,„5'^''^-'*^(e " +e » ) + l|, since €2'^^-i=l; 
 
 (.i-l).P„,(f'-2.f.cos^^-^^ +1), (287). 
 
 n 
 
119 
 
 307- Theorem, (l +e. cos.r)"= 1 +S.i-fii- • {he) 
 
 +2cos?WcX'.S,- 
 
 \n 
 
 \m-\-r—\ . Ir-l 
 
 (¥)"■ 
 
 For, (l+e.cos<i)"=l+S;„-p-.(ecosc2?)'", (92) and (9): 
 
 t 
 
 t 
 
 !m 
 
 2W 
 
 l2w-l 
 
 = 1+S„ i '"'-^ .(ecos.i-y'-V-^:^. (ecosa^)^'"h (14) and (5); 
 '|2m-l ^ [2w ^ I V / 
 
 1+Q i ^"'-^ .e-'"-^ .'^.^^^^ cos(2r-l)c 
 
 2w 
 
 2m 
 
 2"' -2™' ^_.S,f^— •cos2r,'p+2i — .-i-jj-, (284) and (8); 
 
 +^:::_.e^ 
 
 
 t 
 
 i2m- 
 
 +S„-r^^.2.(ie)'^'"-^S,.^i^i; cos(2r-l).r 
 
 \9,m- 
 
 1^ 
 
 + a„T^^.2.(le)^'«.af=^.cos2r.i; 
 [2w ^ |w-r 
 
 ■ S„52cos(2m-l),?;.S,- 
 
 l« 
 
 
 |27»+2r- 
 
 -3 
 
 2„i+2»-— 3 
 
 r— 1 
 
 
 \2m+2r- 
 
 -3 
 
 |r-l 
 
 
 ney"'+2r-^ 
 
 2m+2r-2 
 
 +2cos2ma7.S. i '"'^"'" 
 
 2m+2r-2. [r-l 
 (19), (17) and interchanging m and r; 
 
120 
 
 \m+2r-2. lr-1 -^ ^ ? v yi 
 
 ='-s.|(^.(i^r 
 
 +2cosmA^S,.^ "-^^^, .(le)'"+^'-4, (5). 
 
 \m+r-l . Ir-l ^ '^ J 
 
 308. Theorem. If 2/=^+,r.sin?/, where z independent 
 of X. then shall 
 
 y=x 
 
 2m -1 
 
 CO flxAS'n-l m I 
 
 ^'S.(-0"-'.^,|^.a(-l)'-.;=^.(2r-l)--.sin(2r-I) 
 
 
 + 2S,„(-ir-'.^fT^.S,.(-ir'.7i;=^-(2r)^'"-^sin2r;^. 
 
 For, 2/=^+S™T^^^^.d/'"-^(sin;^)^'«- 
 
 05 ^2m 
 
 + Sm-, .rf/"'-^(sin«V^ (221) and (14); 
 
 2m -^ ^ ^ J-> 
 
 =^+S.- 
 
 2wi 
 
 ^■-■1 
 
 (-4) 
 
 ^.•^'(-ir-'.^j^—.sm(2r-l)z\ 
 
 2m ^- 'i^'" ^ 
 
 ■4)" 
 
 cos2r^ 
 
 + T^-*'l' (283) and (8): 
 
121 
 
 \2m-l 
 
 -"S,.g ^(.,),.. .a(-.r-.^.(-o-.(..-i)-- 
 
 .sin(2r— 1)^ 
 
 so y,2m Q 
 
 +s.g^jEip-a(-T-'.^(-.)'.(2^r-'.sin.,.., 
 
 (269) and (202); 
 
 = {1 t\-"' - 1 »' 
 
 =^+2.s.(-ir-'/-^^^.s,.(-i)' 
 
 2m-l 
 
 2m-l 
 
 \m—r 
 
 {2r-iy"'-\ 
 
 sin(2r-l)^ 
 
 |2m Im-r 
 
 309. Theorem. In the same case, 
 cos 2/=cos^-a,\ (sin^)'- 
 
 +S.(-ir.^f-^.S„(-l)"-'. |"'-"^^ ■(2y^-l)-^'"-'.cos(2y^-l> 
 
 [2m ^ [m-w + 1 ^ ^ ^ 
 
 ,j x2)« + i„, |2w + 2 
 
 + S.(-1)'".^^^^.S'(-1)"-^ "'-"^^ (27zf".cos2^^, (6). 
 2m+l in-n+\ 
 
 For, cosy=cos2;+S,„^.rf^'"-'.5(sin;y)'«c?,.cos;?}, (221); 
 
 = cos sr - S,„ r- . d:" - ' . (sin zy" + \ {268), (l 96) and (6) ; 
 
 =cos z-x. (sin %f-S,n r d :~"" \ (sin zY'" + ' 
 
 |2w* 
 
 -S-«^ .4~"'.(sin^)2'«+^ (9) and (14); 
 
 2m+l \ / -> \- J \ j^ 
 
122 
 
 = cos;^-,j?.(sin^)~ 
 
 |2m+l 
 
 CO -v,2?n f 1 m+1 I I 
 
 CO ^,2m + l f o ?H+1 
 
 [2m+l [(-4)'"+^ 
 
 l2m+2 
 
 ..cos2w; 
 
 m—n+1 
 
 |2m+2 
 
 :2ii .4-<'"+^>L (283) and (8); 
 
 \m+l J 
 
 .r^'" 1 
 |2w'(-4) 
 
 2w + l 
 
 CO ^.2m I TO+l I 
 
 m— w+1 
 
 cos(2w-l)^ 
 
 1 2m+2 
 
 o: yfim + \ Q m+l I 
 
 i2»^;^f, (196), (191), (269), (202), and (igs); 
 
 = cos»— a7.(sin^)^ 
 
 CO (1. rV"' "i+l 1 ___!__ 
 
 +S.(-i)'«. ^f-^.S„(-i)"-'.r^^^^ 
 
 1 2m [m-n+1 
 
 (2>i-l)~"'-'.cos(2w-l)2r 
 
 .1 n , 2m+2 
 
 +S.(-ir- i!H^-Sn(-l)"-'- """^^ , {2ny'".cos2nz, (6). 
 
 I2m+l 
 
 m-w + ] 
 
CHAPTER X. 
 
 ON THE INTEGRATION OF CERTAIN DEFINITE FUNCTIONS, 
 
 310. Theorem. f y^ =\og,{a;+y/w'^l). 
 
 For, put a;~±l=u'; 
 
 then x=ud^u, 
 and oc->rU=ii{l-\-dtU). 
 
 1 \-^d,,u 
 
 u ,v-\-u 
 
 and ri^rifk" 
 
 =log,(.r+w), (251); 
 
 C 1 /-. — 
 
 or / —===\og^{x^\/x~^\^. 
 
 /I X 
 =loge > 
 ^crVliA" l + Vlicl'-^ 
 
 For, /— L==/^^ 
 r <^...^^-'. 
 
 = - loge (.3? ~ ' + VA' - =^ ± 1 ) , (310) and (202) ; 
 
 =^°S^~~i — 7=1T7 
 
 X ' + Vci?-^±i 
 
 ,77 
 
 =log. 
 
 1 + 
 
 \/\^x 
 
124 
 
 1 
 
 — — =loff,.tan- 
 
 1 sin-z- 
 
 For 
 
 sin*' l-(cos*)''* 
 
 = l.sin.^| + > 
 
 "^ (l-cosa? l+cosa7J 
 
 ^\ l-COS<t' l+coso? J ' 
 
 .-. f- — = i{log,(l-cos.r)-log,(l+cosaO}, (251) 
 
 , /l-cosa? 
 
 = l0ffe X/ 
 
 l+COScT 
 
 :loge.tan- . 
 
 r 1 T (it a, 
 
 313. Theorem. I = logj . cot . - - - 
 
 1 COScX' 
 
 Foi 
 
 cos.T? l-(sin<j7)'^ 
 
 = A^.cosa7< : — + : — > 
 
 ^ [l+sma? i-sma'J 
 
 _ f 4-sinc^ d,.(-sin.r) \ ^^^^^^ 
 ^\l+sincX' l-sina;' j 
 
 -. r^_ = i^log,(l+sin.i?)-log,(l-sina?);, (251); 
 
 /rcos.-r 
 
 =log, 
 
 /l + sinA' 
 
 l-sin-r 
 
 /7r -rX 
 
 = loge-Cf>t • 
 
125 
 
 314. Theorem. 
 
 n-1 
 
 af-~'- 
 
 m,-2 r, -2 
 
 For. 
 
 aA-^ 
 
 .-. /'-^l==-.r"-^A/l-ct?2+j;(w-l).»"-^'yi-a?^ (222); 
 
 —?=, (196). 
 
 r x" J r ^"~^ 
 
 and / , \/l-.t?-+ • / / - • 
 
 ■■>^Vl^' y n-2.(m-iy '""j ^1 w-2.(s-l) J 
 
 i!^ (n-l-2.(s-l)] r •^"'" / X 
 
 x"--'' 
 
 (6). 
 
 315. Cor. If n is an even positive integer, then 
 
 Iw-l |l 
 
 ^^« in I . ^ . ' 
 
 (272); 
 
 ll^2 g^:2:i^.,v"-'^'" + ' + i^.sin-'ir, (40) and 
 
126 
 
 and if n is an odd positive integer, then 
 
 .\/l-<a;- ^ ' "* 1^ 
 
 316. Theorem. 
 
 f— 
 
 \n-2 \n-2 
 
 '—— i Qj '»-i.-2 __!__ I '-.-2 r 
 
 -a?2 
 
 _ r if 
 
 For, w. / —7= 
 
 •^.r\/l-cr 
 
 r a?" .t?-*"-" / 71-2 r x-^"-^^ 
 
 ••• / - y VI -.1?-+ • / y • 
 
 '" 4' " 
 
 
 ■(n-2-m-\-\) 
 
 J Hw-l-2(s-l)J 
 
 |>^-2 l^w-2 
 
 z^ q; "'-'--2 ^ , '■.^^2~ r ^ 
 
 (6). 
 
127 
 317- Cor. If w is an even positive integer, then 
 
 , in L 1 
 
 VI 
 
 m,—2 
 
 
 and if n is an odd positive integer, then 
 
 \n-2 ll 
 
 w,-2 i(n-l),2 
 
 (40) and (311). 
 
 318. Theorem. 
 
 . [n-l [^ 
 
 iX^inxy = -cosa?.S,r. "'i ^' ^ - (sin .x?)" ""'" + ' + lij^ — . /^ (sin .??)""-'' 
 \n_ 
 
 71 
 
 For, (sina?)"=-(sincr')""^d^.cosa,', (268). 
 
 .-. j^(sina;)"=-(sina?)"~^cos<r+^(/^-l)(sin<J7)""^.(cosa?)^, (222) 
 and (267); 
 
 = -(sin^)"-^cos.'^?+(»^-l).^{(sin.»)"-^-(sinc^?)"5, (196). 
 
 .-. ?2.^(sinci7)"=-(sina?)''"^cosa?+(w-l).j^(sincr')"-~, and 
 
 r . . ^ (sin<j?)""^ n—l ., . , „ 
 
 L(sin.r')"=- -^ <- .cos .17+ . L(sinA')"-^. 
 
 n n 
 
 n ' ^» C f (sin^)"-^<'«-'>-' l»-ir^_2(s-i)-n 
 
 .-. L(sina?)"=Sm< .cos.z'VPJ ^ — -—\ 
 
 ^"^ ' '"I w-2m+2 J '\ ^-2(s-l) J 
 
 jn-1 [w-1 
 
 = -cos/».S«-2T^^.(sinct;)«-'''' + ' + I:pi^./,(sin.'r)«-% (6). 
 
128 
 319. Cor. If n is an even positive integer, then 
 
 ^(sina?)"=-cosjr.Sm-^j^^. {dwxf-^^'^^+^.w, (40) and (197): 
 
 m,-2 in,2 
 
 and if 7i is an odd positive integer, then 
 
 m,-2 
 
 320. Theorem. 
 
 \n-2 \n-2 
 
 /,(sin,i-)-"=-cos.r.S,„-p-f. . . ! , + j^^^.X(sin.y)-^"-^'>. 
 1^-1 (sin ct)" "-'" + ' w-i "^ ^ 
 
 For, w./,(sin,x')"=-(sind?)"-'.cosr/7+(w-l).j;(sinc^)"-2, (318). 
 .-. -w./,(sin.i?)-"=-(sina>)-"-'.cos.r-(*i+l)/,(sincr)"""", and 
 
 w./,(sinc^)-'' = (sina?)-'" + '>.cosj7+(w+l)./,(sinc^)-<»+2>. 
 .-. (*i-2)./,(sin.a7)-*«-2>=(sin.77)-<"-i>.cosA^+(w-l).j;(sin.r)-% and 
 
 /,(smaO-"=-- -— -^ + .X(sina^)-<"-^'. 
 
 '"[ n-2(m-iy^ ' (sin.vy-^^'"-'^-'] '\n-l-2{s-lj] 
 
 ^j,/ .-2-2(.-i) | .,^ 
 
 [W-1-2(S-1) j •'^ ^ ' V 7' 
 
 W-2 
 
 |w-2 
 
 [71-1 (sm.x')" --'" + ' w-i •'^ ^ ' V >» 
 
129 
 
 321. Cor. If n is an even positive integer, then 
 
 X (sin aj) - " = - cos A' . S,n !" '' ^ • -r- — 
 •'^ ^ \n-\ i^mx 
 
 (sin a;)"-''" 
 
 and if n is an odd positive integer, then 
 
 |n-2 
 i(«-i)L___^ 1 
 
 £,(sin<.x')-"= -cosa?. S,n ?' '' > ■;^^ .„ om-^-i 
 
 ■^^ ' \n-\ (suia?)"-2"'+i 
 
 Li 
 
 + |i2;;ii:^.log,.tan.-, (40) and (312). 
 
 322. Theorem. 
 
 \n-\ 
 
 \n-\ 
 
 f^ (cos a?)" = sin w . Sm "'. ^' " (cos a')" ' "'" + ' + ^ /, (cos on)" "''. 
 
 For, (cosa?)"=(cosc7?)"~^d,..sin<a?, (267); 
 .'. £(cosa?)"=(coSc^)""^sin*+j^(w-l).(cosA')"-^(sina')-, (222) 
 and (268); 
 
 = (cosct)" - '.sina?+(w -!)./,{ (cos -1?)"'-- (cos. v)" 5 , (196); 
 
 .-. /i. j^, (cos a?)" = (cos ct?)" ' \ sin .^+(^-1)./^ (cos cf )""'-, and 
 
 -, , (cosaO""^ . 71-1 ^ . .„ , 
 Ucos.r)"= ^ — .sina;+ . L(cosa?)"— ; 
 
 71 71 ' 
 
 |y^-i 1^-1 
 
 = sinA^S.T^-(cosc^;)«-'~''«-^' + if^— .j:(cos*)»-^ (6). 
 
 fn,-2 r,^ 
 
 R 
 
130 
 
 323. Cor. If n is an even positive integer, then 
 
 \n-l \}_ 
 
 /.(cos.r)"=sin.^;.S.^•(cos.^^)"-^'"-^ + |f ..^, (40) and (197): 
 
 and if w is an odd positive integer, then 
 
 w-1 
 
 /,(coscx0"=sin T. Sm ^T^- (cosa-)" •''« + ' 
 
 324. Theorem. 
 
 71-2 
 
 /,(cos.r)-«=sin.r.S.j^- (eos..)— ' ^ fef ^.(cos.r)"--' 
 
 For, n.X,(cosa?)" = (eosA')"-^sinj? + (w-l).X,(cos.r)"--, (322). 
 .-. -w.j;,(cosd?)-"=(cosa?)-*" + '*.sina?-(w+l)./,(cosa^)-'''+'*, 
 and w./,(cosci^)-"=-(cosciO-'"+^sina^+(w+l)./.(cos.r)-'""-'*- 
 .-. (w-2).j^,(coScT) -*"-''= -(cosaO-*"-^*.sinA'+(w-l)./,.(cos,r)-", 
 
 (cOSif) "*""'* 
 
 and f^{cosw)' 
 
 =s 
 
 sin 07+ ^./,,(cos>r)-*"-^\ 
 
 91-1 w-1 ■' 
 
 7i-2(m-l)-l (cos,r)"-'~-'"'"''"' 
 
 j^| 7Z-2(^-l)-2 | 1 
 
 '\w-2.(s-l)-lj "•''(cosa?)"-^'' 
 
 '■\w-2(s-l)-lj 
 
 (51); 
 
 W-2 
 
 O ffl-l.-2 ^ , '-.-2 r (fi) 
 
 "^"jir^ (cosaO"-^'" + ' |9i-l V,,(cos.r)"--^'' ^ ^* 
 
131 
 325. Cor. If n is an even positive integer, then 
 
 _^,(cos.^)-"=sin<2?.S,fEkEl 
 
 n-l (cosa7)"-2'"+» 
 
 and if n is an odd positive integer, then 
 
 \n-2 
 
 f,. (0080!?) " " = sin a-' . Sm j-^ 
 
 1 
 
 n-l (cosa?)"-^'"+' 
 '■^ . loge cot ( ^ - ^ j , (40) and (313). 
 
 i(n-i),2 
 
 326. Theorem, /^(sina?)" 
 
 ' I wi-1 w-2m+2 
 
 ^2-"JJi-.je, or 
 
 w 
 
 ■ / ty(n+i) .-...f^^g^ ^y„-,_ bl _ cos(yi-2m+2)a^ _ 
 ' I ra-1 * w-2m+2 
 
 according as 71 is an even or odd positive integer. 
 
 \7l 
 in L_ 
 
 For, (sincT)"=(-l)*".2-"+'.Sm(-l)"'~'-"r^^^^^-cos(/z-2m+2)a? 
 
 \» " 
 
 +2-". 7^, or 
 
 \7l 
 i(n+l) L_ 
 
 ,(_l)i(n-i),2-«+i g (-_iy/-i..^:i;:i_.sin(w_2w+2)c 
 
 according as 71 is even or odd, (283). 
 
 Hence, by (196), (191), (268), (202), (197), (267) and 
 (6), the truth of the above theorems is manifest. 
 
132 
 
 327. Theorem. 
 
 1^ . ^ \n 
 
 ■'■'^ ^ \m-l W-2W + 2 i7» 
 
 1^ . . 
 i^'o^* ^Ti sm(w-2w+2).r 
 
 '"Im-l* 7i-2m+2 
 
 according as 7i is even or odd. 
 
 in t t 
 
 For, (coSc2?V=2-"+^"Sm i "' ^ •cosfa-2TO+2).-t?+2"".rT^, or 
 
 i(n+l) L_ 
 
 =2"''+^Sm i """^ 'COs(n-27n+2)wy 
 \m-l 
 
 according as n is even or odd, (284). 
 
 Hence, by the same articles, the truth of the theorems 
 is manifest. 
 
 328. Theorem. f^(a\u) 
 For, ^^*.a^ 
 
 =k(-^r-'-d-'u.fr-a'+(-ir.f,\(d:u).f:a^\, (222); 
 
 =a^S.(-ir-^(log,a)-'".4'"-'w+(-l)«.(log,a)-"./,(a-.d», 
 (6) and (196). 
 
 n+l 
 
 329. CoR. /,(a^.T")=a*.S™(-l)'"-'.[M^.A'"-'" + ^(log,a)-"'; 
 
 ni-l 
 
 7i being a positive integer. 
 
133 
 
 330. Theorem. f~=\og,x+S 
 
 J J. X 
 
 (logoff)'" 
 
 (.rloge«)" 
 
 For, a-=l+S« 
 
 (107) and (y). 
 
 a* 1 " (logea)™ 
 
 •• - = -+S / I ^ •^^•"'-S and 
 
 r-=log,.^+S,„^^^-^f^, (250), (lyi), (196) and 
 
 (210). 
 
 331. Theorem. f^{a\u) 
 
 =a\8rn{-ir-'.Qog,ay'-Kfj''u+(-iy.(\og,ay.f,{a\fj'u). 
 For, f^a'.u 
 = ^A-ir-\dJ'^-K{a^).fj''u+(-iy.f,\dJ'.(a^).f;'u\, (222); 
 
 = S,„(-ir-^(log,<-^a-./:"'«+(-l)»./4(log,«)".a^//7.i,(249); 
 
 =a-^.k(--^r-'-(}og,ar-Kfj"u+(-iy.i\og,ay.f,M\fj'u), (6) 
 and (196). 
 
 332. Cor 
 
 ra^ n-1 ^^,-« + ™ 
 
 . _(-=a'.S.(-l)-'.(log,«.)"-.(-l)«.i^^ 
 
 a'(-l)"-'..);-' 
 
 +(-l)-.(log.ar-./,(5^1^ 
 J ^y-« + '« (logea)""' ra^ 
 
 (210); 
 
 = -a^S:.(log.a)^-.^^^7^. r , (6) and (196); 
 
 •«^S,„(log,a)"-'.-i— -^ + 
 
 j .»-"+'" (log^a)""' 
 
 6«) fi a O^-logett) 1 
 
 ^ -— • { log. .f + S,„ T K 
 
 /i-1 ( m.\m ] 
 
 (330). 
 
134 
 
 then 
 
 333. 
 
 Theorem. 
 
 
 =log/a^+S. 
 
 m.lm 
 
 For, 
 
 put \og,x-- 
 
 = t(,; 
 
 
 
 I x = 
 
 e", and /-— — = / - 
 
 
 
 
 
 
 -, (202); 
 
 
 
 
 
 
 
 
 
 = ^°S^" + ^'"^.^' 
 
 , (330); 
 
 =log/cr?+S,: 
 
 {\og,.vr 
 
 334. Theorem. 
 /.log, (l+e.cosc't?) = -c^^log, 52e-^(e-^-\/e-^-l) J 
 
 + 2.S,„(-l)"'-'.(e-'-Ve-'-l)'". ;;— 
 
 2 / 
 
 For, put e= ; then /(;=e ^-ve"^-!, and 
 
 2 
 logf ( 1 + e . cos tr) = log^ ( 1 + - — — — . cos a) 
 k+k 
 
 = -logt (l+A;''')+logj(l+2A;.cosa;+/(;''') 
 
 = -loge(l+/r)+log, J(l+/f.e'^^)(l+A;.e-^^^)|, (287) 
 
135 
 
 = -log, (l+A;'^) + log,(l+/(;.e''^-0+log,(l+A;.e-''^-0 
 
 = -log,(l+^=) + S,„(-ir-^ — (e'^'-^^+e-^^^^i), (255) and (5); 
 
 -log, (l+A;-)+2.S,„(-ir-^ — .cosOT.r, (287) and (6). 
 
 pog,(l+e.cos.i?)=-A'.loge(H-A'') + 2.S,„(-l)'""'- — sinm.r.-, 
 
 -,v\og, {2e-^{e-^-\/e-^-l)\ 
 :.S,„(-i)"'-^(e-'-x/e^^^y 
 
 (267). 
 
 sm nix 
 
CHAPTER XI. 
 
 ON GENERATING FUNCTIONS. 
 
 335. The symbol Sm«m will be used to denote the sum 
 of the series formed by giving to w every integral value from 
 n to T both inclusive ; zero being also taken as a value if n is 
 either zero or negative. 
 
 — oi.es 
 
 336. Definition. If (p(t) = 8„uj% then (pit) is called the 
 generating function of m,., and is denoted by GfU^- 
 
 337. Theorem. Gt{u.,.+v,) = Gt.u,,+ Gt.v^. 
 
 — 05,05 
 
 For, Gt(ur+v^) = ^x{u^+v^v)t' 
 
 — CO, cc —a, CO 
 
 = 8.rUj-''+S^V^r-t% (5); 
 
 = Gt.tf''+Gt.t\^. 
 
 338. Theorem. Gi(cr?/^) = cf.Gt.Wr; a being independent of 
 t and X. 
 
 -05, CO 
 
 For, Gi{au) = ^^au^.t'' 
 
 = a..8.rUj% (6), 
 ^a.Gt.u^.. 
 
137 
 339- Theorem. If Gt.7/^,= Gi.t'r? then shall ^^^^=1;,. 
 
 — 05, CO 
 
 For, S^u,rt'-'=Gi.u^ 
 
 = Gt.Vj,, by hypothesis ; 
 
 .-. Wr=v^, by equating coefficients of ^^ 
 
 340. Theorem . /*" . G, . m,= Gt . ?/,.^„ . 
 
 For, t'"\Gt.u,=t".Ki('.vt' 
 
 =Ku,.f^", (6); 
 
 =srt,.^„.#', (9). 
 
 341. Theorem. (r^-l)".Gt.?f,-Gt. A/.m^. 
 For, (r'-l).Gt.«*,= G,.w,,^i-6?i.M,r, (340); 
 
 = Gi.A,w., (337). 
 
 Hence, (t-'-iy.Gt.ti^={t-'-l).Gt.A,u^ 
 
 = Gt.A/u,; 
 and, similarly, (r'-l)".Gt.M,,= Ge. A/?*^. 
 
 342. CoR. r.(t-^-iy.Gt.u,= Gt.A"Ur-r,>- 
 
 S 
 
138 
 
 343. Theorem . f S„, ~l\ ■ Gt ■ w,,= Gi . S,„a« - 1 • u,, 
 For, (a„^V^'-^-Sl^-G.-^.. (6); 
 
 = S,„a„,-:.Gt.^f.+»-i, (340); 
 
 n+l 
 
 = Gt.S„a,_i.?Ar+,«-i, (338), and (337). 
 
 n+\ 
 
 344. Cor. 1. If V''w.=S„,a,„-i.V'"'Wr^.,„_i, and v''?<^=Wr, 
 
 then will ( Sl^l .G,.u,= Gt.V'Ur. 
 
 345. Cor. 2. 
 
 (r^-1)^ ( S,/^j .f.G,.w,= G,.A/.V"-^.-r. 
 
 
 346. Theorem. A/w,,= S„(-l)'"-'.-^i^.w.+«-m+i 
 
 For, Gj.A/?f,= (r^-l)".G,.?/„ (341); 
 
 "S,„( 1)""' ""^ f -(«-»- + !) ^. ./ (86) and (6); 
 
 tm— 1 
 
 w— 1 
 
 A;'w..=S„,(-ir-'-r2;^-«.r+«-,«+n (339). 
 
 [wi-l 
 
139 
 
 347. Theorem. i^:,+„=Sm , "' ^ -A/'^i/'.,- 
 m-1 
 
 For, Gj.Zf,+„=r".Gj.w,„ (340); 
 
 m— 1 
 
 = Gt."s,«-^^^^^-Ar"''-'f.5 (^41), (338), and (337). 
 
 I m-1 
 
 \m-l 
 
 „ m+wr-1 
 
 348 . Theorem, u^ + „ = 7/.^, + n . Sm -^^^^-^ A/" Wj; _ ,« , ; 
 
 I m 
 
 r being any integer. 
 
 9 
 For, put ^"' = 1+—; then 
 
 t-''=l+S,nP-d:L\'{z-\n.z'^''], (221), (207) & (197): 
 
 \n+mr-l 
 = l+n.S,n^^^, — 0'", (196), (210), and (6); 
 
 \n+mr-l 
 
140 
 
 \n+mr-l 
 
 ••• Gt.u,^,= {l+n.Sm^^^^, r^(r'-l)'"}G,.w,, (340); 
 
 \m 
 
 \n+mr-l 
 
 A/«*,-„4, (6), (342), 
 
 
 = Gt{u^+n 
 
 
 
 (338), and (337). 
 
 
 
 \n+mr-l 
 
 .•. u 
 
 [» 
 
 349. 
 
 Theorem. 
 
 
 A/' «*,_,„„ (339). 
 
 «..(«.osl^^i^M.,.-.,.,, 
 
 .„.S.P^).A,-^„,., 
 
 2m-l 
 
 For, let G and - be any quantities less than unity ; then 
 
 - 0" 1 
 1+S„- = ^, (12) and (9); 
 
 1-- 
 t 
 
 1-- 
 t 
 
 l-0t 
 
 i-et 
 
 \-9(2 + z) + & 
 
 -, where z=t(t-'-iy; 
 
141 
 
 i-et 
 =(i-^0-a^^^' 02); 
 
 =(i-et).Sne''-'.^''-'.S,nf^^^^.e''-\ (92) and (39) 
 
 \29i,-2m + 2 
 = (l-et).SnO"-'-kn'"~\' ;g"-'", (i6) and (26); 
 
 2 m 
 
 =(i-et).s„0''-'.s,n^^^^^^-^'"-', (8). 
 
 \2m \m+n-l 
 
 But 
 
 \n-m \n—m 
 
 (40); 
 
 2w— 1 . 7i— m 
 
 m+72-1 
 
 2TO-1 
 
 /" I 2 m - 1 
 
 2'/w-l 
 
 2m-l 
 
 n+l ** + *^ „ /i + TO — 1 
 
 2m- 1 ^m-1( 
 
 2m-l 
 
 \2m-\ 
 
 , (9), ((^). 
 
 and (5). 
 
^^^, \n+m 
 
 .^'"-•-^S.^ 
 
 142 
 
 H — 1 
 
 2m-l 
 
 \27n-l \2m-l 
 
 2m-l 
 
 w+m— 1 
 l2m-l 
 
 (340), (342), (338), i?,?,!) and (339). 
 
 But [^* + m=|w+??z.(ri + l).lw— m+2, (41) and (40) 
 
 2m-l wjT] m_i 1 
 
 = (w + l).Pr |(w + w-r+l)(?i-w? + )-+l)^ 
 
 = (w+l).P,{(w,+ iy^-(m-r)2 
 
 :(M+l).P,{(/i + l)-r^}, (31). 
 
 u, + „ = (w + 1 ) . S™ S^^ ^ • A /'" ' w,. - ,„ + 1 
 
 .„.iP4!t!!).^»-.„,_„ (fi, 
 
 tm-1 
 
143 
 
 350. Cor. Put -=c„-t.c„_,, where c„=S™~^^ -'""'' 
 
 t" \2m-l 
 
 1 1 c„_ 
 
 then — =c„_i-if.c„_2; .-. ^„ = — 
 
 and - =c„-c„_i. + (^ ^-i)c„-i- 
 
 n—l <v'« - 1 
 Now, C„-C„_^ = Sm 1— (|nH-«J-|w + W-2)+2W..t:"-^ + ;^" 
 
 (9) and (5). 
 
 And, I W+7W - [w+w-2 
 
 211^1 2m- 1 
 
 = \n+m-2 \{n + m)(n+m-l)-(n-m)(n-m+l)\, (41) 
 
 2m-3 
 
 =2w.(2m-l).[w+m-2 
 
 2m^^3 
 
 = 2M.(2/w-l).lw + m-2.:^i.[w-m + 2, (41) and (40); 
 
 ])i^ ~ m-2, 1 
 
 ?n-2 
 
 =2»2-.(2m-l).P, 5(w+^^-^'-l)(^*-*'*+^+0? 
 =2w~.(2m-l).P, |»^~-(m-r-l)~J 
 
 = 2?i^(2m-l).P,(7r-r'), (31). 
 
144 
 
 \2m-2 
 
 \2m-2 
 
 since : .Frin^-r")=n; 
 
 r 2TO-2 
 
 4-i^"+(i+0(r^-i).^.S.|^.P.(^-O 
 
 r»-2 
 
 \2m-2 
 
 m— 2 
 
 + *i-0»i — i • > '-»,r "■•T-iii + lT^^^.r 't'.r-mi 
 
 2m-l 
 
 351. Theorem. 
 
 CO m M-r+l 
 
 ^here V''^*:r=Srar-i- V'"^^^+r-i? and &,._i=a^-r+i, C=m+i)- 
 For, put ;?=S,.-^; then a-s?=-S,ar^"'% and 
 
145 
 
 1 S,.ff,.0"'-'-(i-^'-^") 
 
 1 S,.a,.0'"-(i-a'ro (3^^^,^j(,)^ 
 
 ^-^*'' aa,.0'"-'~e'".s,.o,.r 
 
 S,.«,.0'"-^S, 
 
 
 p 
 
 — , suppose. 
 
 , (11) and (9); 
 
 1 o: nt+l 
 
 Then -=S.-'-^e"'<*-'^(S.«,„-,.+le^-^)-^ (12) and (S); 
 Q 
 
 :S„e"-^S.^-^•^^^"-'"''-'^-^6-n 
 
 7^- 
 
 and P=S,.0™-". S, 4^, (22); 
 
 ^s..0'-.s,^, (8). 
 
 since •23-'"«"=0, for every negative value of w; 
 T 
 
14G 
 
 ••• r"=S,.S,^^-S.^~^-'-^"-^^'-"''^-"-^- 
 
 m 1 TO— »•+! 00 
 
 =§'-^-"s,\+.-'Ss^'"'-^'''''''"'-^~^ («)' 
 
 C6 »t ^*-' m—r+\ 
 
 = SsSr-^- Sp a,,,_,.7Jr^ — +^6-^ (6) and (17). 
 
 352. Theorem. G,iu,).Gt{v,;) = G,.GXu,,.v,,), ?/, being 
 the coefficient of s'', and v^ of ^''. 
 
 For, G^Gi{Ux'^x) denotes a series of the form 
 
 S'm S„ «,„.„. s"'.r, 
 
 such that the coefficient of s\f\ shall be ?f,i',; and the product 
 
 will be such a series. 
 
 .-. G,Gi{u,v,) = GXur)Gi{v,). 
 
 353. CoR. 1. s-'".r».G,OO.G,(^g = G,G,(7/,+,„.7',,,.,). 
 
 354. CoR. 2. (s-'-i)'"(r'-l)".G,(w,).Gt(0 
 
 = G,.G,.{A;'(w,).A/(r„)|. 
 
 355. CoR. 3. (s-V-'-l)".C;,(7/,).C7,(n,) = G,G',.A,".(/<,,«,). 
 
147 
 356. Theorem. 
 
 A;'(w>i'x)=S,„ri^^^.A;'-'""'(«.r.™-,)-A;"-'.Jv 
 
 \m-\ 
 
 For, G\G(.A;'(tA,u,) 
 
 :5(*-'-l)+^-'(^-^-l)f"-G,0O.G,(iv) 
 
 ^S.ri^.(^-'-i)"-"'"'-^-*'"-"-(^--ir-^-G.K)-G,K),(86): 
 
 m-l 
 
 :S,„r^-^^-(V""''-^.^+'«-0-<^*-A;"-H>„ (6) and (342); 
 
 ^G.-G,.S»-r^=^-A/-"'^'K-4-,«-i)-A;"-'»., (352), (338) and 
 
 I m-l 
 
 (337). 
 
 A;(M,r,) = S»r^-A;'-"'+Hwr+„,-,)-A/-'t'.., (339). 
 m— I 
 
 357. The symbol G„f.?/,,^ will be used to denote the 
 double series Sr- ^yS^t'^.u^^^. 
 
 358. Cor. l. s''^ J'" .G,,i.u.r,y=G,,t.u^+„,y+„. 
 
 359. CoK. 2. (s-'-i)".(r'-l)\G,,,.w.,,= G,,,A/A/w,,,. 
 
 360. Cor. 3. (s-7-'-l)".G,,e7/,,y=G,,e. A^w,,,^. 
 
148 
 
 361. Theorem. 
 
 \m . \n 
 
 m + l n + l 1 I . 
 
 For, Gs,t.U^ + ,„,y + n 
 
 = \l+s-'-l\"'{l+t''-l\"-G,,t.ti,,y, (358); 
 
 = S g ^1 v-i .G^,.A/-'A/-^%,,, (6) and (359); 
 
 
 and (339). 
 
 362. Theorem. 
 
 \n_ 
 
 A:„«*.,=s.(-i)"-'-p:^-w..n-.+.,.H 
 
 = (s-'r>-l)".G,,tW,,.,, (360); 
 
 =S.(-i)'--p^-(^0-'"-'""''-G^u^f..' C^6); 
 
 m-l 
 
 = S.(-l)'"-^r^:^^.G,e. «.,„_,„+,.,.„-,„.„ (6) and (358); 
 m-l 
 
 ■■• A"«..,=S.(-i)"'-'-r=T-«-t.-«,,,*.-,.+„ (338). (337) 
 
 ■' ^ m— 1 
 
 and (339). 
 
NOTES AND ADDITIONS. 
 
 25,1. Theorem. If a,„_i=6,„_i-f/„, 
 
 then ao=S«(-ir-'.6,„-i + (-l)"-««- 
 For, (-l)'"-^a,_, = (-l)'"-^6,„_l + (-l)'".fl,„, 
 
 ••• s,„(-ir-^a.-i=s,„(-ir-^&»-i+s,„(-ir.«., (4)&(5). 
 
 w-l 
 
 ai 
 
 nd ao+s,(-ira.=s„x-ir-'-?>™~'+s„,(-ir •««+(-!)"'««' 
 
 (9). 
 
 .-. «o=a«(-ir-'-^«-i+(-ir-««. 
 
 cancelling identical terms. 
 
 Page 16. 
 
 42,1. Coil. This theorem being true for every value of 
 a, and w, we may put «=0, and m=-l, and we shall have 
 1 = 1 ; which result will be found of perpetual occurrence. 
 
150 
 
 Page 18. 
 48,1. Theorem. If «„+i,,„ + i = «rt,„,4i + ff„ 
 
 L» 
 
 "''" \m 
 
 For, «, + i,,„ + ! = «,■.»« + l + «r,m- 
 
 •■• ««,« + i = «i,m + i + S,.a,,„,, (24.); 
 
 = ^rCLr,ml sinCC ffi,„i + i=0. 
 
 Hence, putting m=\, 
 
 ««,. = Sr«,-,, = Srr=|^, (48). 
 
 Also, putting w=2, 
 
 «.,,=S,a„=S,f- = -?-, (18). 
 
 Li ti 
 
 Suppose, therefore, a„,„=i2_; 
 
 then «„,,„ + , = Sr«r.™ = Srf-=-:-:iii^, (48). 
 [m lm + 1 
 
 If, then, the law were true for m it would be so for w+1 ; but 
 it is true for 3 and therefore for m. 
 
 48,2. Similarly it may be shewn that if 
 ^n+i,m=««.m+«5„,m-n ^/h,(.= 1? '?i,i = 1, and fl„,„+,.=0, tlicu shall 
 
 m-1 
 
151 
 
 Page 20. 
 54f,\. Problem. To find the number of terms in C,n, ■ 
 I'l't 6„, „, = the number sought ; then 
 
 ^„ + i,,« + i = 6«,m + i + &«,,«, (54), b„^ = n, and 6„_„+, = 0. 
 
 ••• &„.,«= f-, (48,1). 
 \m. 
 
 Page 22. 
 GO,i. Cor. 2. If c is independent of.?, 
 
 ni,w— m m,n—tn 
 
 then C,,, j(«')(^*^OS=^''"'"-Cr..(«r^.)- 
 
 See Art. 55. 
 
 Page 28. 
 
 74. The theorem of this Article may also be proved as 
 follows : 
 
 It will be readily seen that we may assume, 
 
 [0 + 6= S,„ f „, m - 1 • [_« • L^ ■> 
 
 n,r n-m+\,r m-\,r 
 
 where c„,,„_i is independent of n and 6; 
 
 then [a+6 = (ff + 6+wr). la+6 , 
 
 r>+\,r nTr ~ 
 
 n+2 
 
 or S„e„ + ,.™_,.[« .\h 
 
 n-m+2,)- m-\,,- 
 n+1 
 
 = S,„e»,,„_i. |« • U> (a + w-w + 1 •>"+& + w-l •■»')• 
 
152 
 
 7J+1 
 
 = S,„c,,_, ([a ^. [^+ |« . \b_) 
 
 n—m+2,r m—l,r )i—m+i,r m,r 
 «+!,)• n-m+l,r m,r n+l,r 
 
 also we have e„,o=l, ^'i.i = l? ^^<i c„,„+,.=0. 
 
 ••• c„„„_i=_2i^, (48,-2). 
 jm— 1 
 
 The theorems in Articles 86 and l6l may be proved in th( 
 same way. 
 
 Page 31. 
 Cor. 7. (S,J« •] -)=S,«|wa.T^ -; 
 
 83,1. 
 for every rational value of n. 
 
 Page 37 
 
 102,.. Cor. 6,1. (s,„^^^p^)''=S: 
 \ m-1 / 
 
 " ^ (^zo)'""^.,??" 
 
 w being any rational quantity. 
 
 Page 6l. 
 177. Or thus: 
 
 Since, Z),r ' 1 a;" ~ H«) • -t>,r~ *'" ''^Ej''-'v\ 
 = DJ" - ' (71) . D,-" EJ" -'v-D,-'{ DJ" (u) . D,-^E:'v ] , (1 76) ; 
 
 \D;{n),Dr.E;{^^)\, ('2.5,,). 
 
153 
 
 Page 52. 
 152,1. Theorem. Dj'\x =|m./i" 
 
 n^h n 
 
 For, Z),,. \x = U+/i - \x 
 
 = {x+h) \^ - \x (x-mh + li) 
 
 m-l,-A m-l.-A 
 
 m-l,-/, 
 
 2 m-2, -A 
 
 Similarly, Dj' \x = \m. h". \,:v 
 
 Page 53. 
 1 55,1 . Theorem . 7/ = S„ !""'''" . -^^=^ ; where ^ = Z).r. 
 For, put t<=S,«U'^ •«)«-!!> where «,„_i is some finite 
 
 m-l,-h 
 
 quantity independent of x; then 
 
 D,J'-^u=8,n[m-l.h''-K[x .«,„_!, (136), (137), and (l52,i); 
 
 n—l m—7i, —It 
 
 = S,„[m-l./t"-'.[.v .ff,„_i+[w-l./i""^ff«-i 
 
 + S,„ [w+m-1 . /i" - ^ [.V . a„ + „,_!, (9) and (42) ; 
 U 
 
154 
 
 + 8m \n+m-i .k"-' .\^.a„+,n-i, since \m-l =0, C=i_i)- 
 
 n-l m, -h n-i 
 
 1 D'^.-'u 
 
 and cin-v- 
 
 \m—l ' A'""^ 
 
 
 Page 71. 
 
 n n n n;m 
 
 207,1. Theorem. d^.P,.wA = Pr(Wr"^-')-Smam<^^w„-Pr«*r; 
 a,, being independent of ,v. 
 
 n 
 
 For, — ;^ =Dm > (205); 
 
 = S.-^, (207). 
 
 "■to 
 
 .-. d..P,..V'-P,.(«A-')-S.^^^^^^.P,.^^,., (fi); 
 
 = Pr (Wr"' " ' ) • S,n a„, . d, U^ ."P,. W,. • 
 
 Page 73. 
 213. This theorem may also be deduced as follows: 
 
 We have ;,=§„- f^^^- ^^^ ' '^^^'"^ ^=^''- ('•'^•^'')- 
 
155 
 
 This being true for every value of h, we may put h=0; 
 
 then ?^=S» 
 
 m-\ \ h" 
 
 •). 
 
 =S«r— T-<="^. (188). 
 \m—l 
 
 Page 76. 
 222. Or thus: 
 Since f,{d--^u.fj"-^v\=d--^u.fj"v-f4dj''u.fj"v], (203). 
 
 ,-. f^iuv) = k{-'^r-'d--hcJj"v + i-iy.fAdJ'uJf^h (2V). 
 
 Page 94. 
 
 253. The symbols C w, L^u, and (/.-/^J?* are equi- 
 
 valent to f^u-l,^a^i- 
 
 Page 100. 
 270. Or thus: 
 
 tan^r+tan^ 
 Since tan (,r +70 -tan '^=73^^^,^^-^^^ 
 
 tan h . (sec xf 
 ~ 1 -tan X . tan h 
 
 -tan ^ 
 
 ( 
 
 tan ( a; +70 -tan cT? ^ 
 
 ftan7j (sec xf \ 
 
 ^\ h ■ l-tan.r.tan7ij^ 
 
 = (sec.7?)% (266). 
 
156 
 
 Page 100. 
 
 270,1. Cor. 1. d.,.cot x=dj..{tana;)~^ 
 
 = (-l).(tan a!)~~.{secosy 
 
 ( sec x\ ~ 
 Vtan ,vl 
 
 = — (cosec .x')~. 
 
 271,!- Theorem, f^. cosec <r= -cosec <r-. cot x. 
 For, rf^.coscc cr=c/t. (sin a)"' 
 
 = ( - 1 ) . (sin ci) ~- . cos ct? 
 
 1 cos a: 
 sin 0) sin w 
 
 = -cosec .I'.cot ,v. 
 
 Page 105. 
 
 281. Every term in this series will vanish in which zero 
 is a factor of \n-m , that is of 
 
 {7i-m){n-m-\) .. . {n-2m-\-^){n-2m+S). 
 
 (1) Let n be even; then n-2m-\-'^=0 if m= — +2; and 
 
 o 
 
 \n-{\n^l-\-r) =^ |^l^-l-r=0, if r>lw-l. 
 
 in+l+i---2 in-l+<- 
 
 (2) Let n be odd; then w-2m + 3 = 0, if /»= +1; and 
 
 \n-\^{n + \)+v \=^\- ^{n-\)-r = 0, if r>>l(w-l). 
 
 i(»+])-|-r-2 H"-l)+»-l 
 
 Hence, the number of significant tirms will be 7\?z + l, or 
 -^(w+1), according as n is even or odd. 
 
157 
 
 It will also be observed that, although n appears as a mul- 
 tiplier of the whole series, yet the coefficient of the first term 
 
 \n-l 
 
 beinff ^V— = - ' (^S) and (42,i), the first term will become 
 
 lo n 
 
 1.(2 cos x)". 
 
 Page 116. 
 
 300,1. Theorem. 
 Iog-=:2wlog4. + 4log[w-2log[2n-log (27i + l), (w=co )• 
 
 Fo,-,si„,.=.,P,{(.+-;^)(.-^)j, (^93); 
 ,^^ (2r+l)(2r-l) 
 
 •. -=P,. 
 
 4r^ 
 4?-- 
 
 (2r-l)(2r+l) 
 4^([ny 
 
 ([0'-'-(2W + l) 
 
 , (W = X): 
 
 
 
 !i,2 
 
 
 
 
 [2w 
 2".[w 
 
 
 but 
 
 »-,2 
 
 
 
 TT 
 
 ■ 2 
 
 4".([n: 
 
 r-.4"([ny^ 
 
 
 - {[2ny 
 
 .(271+1) 
 
 
 
 4-", 
 
 .{\ny 
 
 ([2yi)-.(2w + l) 
 log - =2 wlog 4+ llog [w-2 log 1 2w "log (2 «+ 1 ), (w = X )- 
 
2m- 
 
 158 
 
 Page 117. 
 301. That there are not more than n different values of 
 6 " ~ may be shewn as follows: 
 
 For m put 7ir+m, when r is any integer and m<n\ then 
 e « becomes e " =e2»-^v-i.g „ 
 
 2m-l / — 
 
 = c-^-^-, (288). 
 Page 118. 
 304. For m put nr+m^ when ?• is any integer and m<n\ 
 
 2m-2 /— 2„r+2m-2 ,— 2m-2 ^ .-r 
 ,1 _.7rv— 1 , _.7rV— 1 „ /— r .TV— 1 
 
 then e " becomes e » =62»tv-i .g n 
 
 = c— •'^^-\ (288). 
 306,1. Theorem. 
 
 x-"-2cosd..v" + l=\*,„{a;~-2a\cos +1). 
 
 n 
 
 For, put cv^" - 2 cos . a" +1=0; 
 
 then .r" = cos ± v^ . sin 
 
 = cos (m-1.2 7r+0)±'\/^.sin (m-1. 2-^+6), 
 
 , wt-1.27r+0 , . m-1.27r+e ^,„. , , 
 
 and .r=cos ±V-l.sm , Cl ), (277). 
 
 n n 
 
 .'. ,t-«-2cos0..'p"+l 
 
 T> rf / W-1.27r+0 y — . m-\.2'K-^Q\\ 
 = 1,J <,r-lcos +V-l.sni Ifx 
 
 f / W-1.27r+0 J . m-1.27^+0^1^ 
 
 f-r^ — n — ^/^•^'"■ — ;: — )]\ 
 
 f, ^ ., Wi-1.27r + ^ 
 = !*„, (.1- -2 .X . cos + I ) . 
 
159 
 
 Page 132. 
 
 327,1. Theorem. 
 
 r /'ton >An-2m + l 
 
 £(tan ..)"=S.(-ir-^ \^_^^^^ +(-l)'-./.(tan .^•)«- 
 
 For, (tan ,r)''=(tan cr)"-^(sec a'|'--l) 
 
 = (sec cr)^.(tan .r')"~~-(tan .r)"~^. 
 
 .-. (tan ,r)" = (sec aif . Sm ( - 1)'" ~ ' • (^^" ^r)"""'" + ( - 1)' . (tan cr)"--', 
 
 (51) and (6). 
 
 .'. /.(tan ,.)" = S.(-1)'"-^- ^ 7 ^ +(-l)'./.(tan.O"-", (270). 
 
 327,2. Cor. 1. 
 /.(tan .)-=S„(-.)--. 5— ^ +(-.)"..■, or 
 
 according as w is an even or an odd positive integer. 
 
 327,3. Cor. 2. 
 
 (cot .1?)" = (cosec .r)^ S;„ ( - 1 )"" " ' • (cot .17)" - 2-" + ( - 1 )^ (cot .r)" -^ 
 
 r TcOt -pV ""'""*"' 
 
 .-. /.(cot..)"=s,„(-ir. \^1^^^^ H-^y.u^-ot.vy-"; (270,0. 
 
IGO 
 327,1. Cor. 3. 
 
 
 /.(cot.r=l(-ir. — 4^^ +(-iy".-. or 
 
 i(n-l) font ^AM-Sm + l 
 
 according as w is an even or an odd positive integer. 
 
INDEX TO THE THEOREMS. 
 
 4. If a,n=b,„, Qj„), then S,„a,„=Sm&m- 
 
 6, If b is independent of m, then S;„«m&=&S„,«»«- 
 
 « )• J!— r 
 
 9. S™ a,„=S,„a,„+ S,„ «,.+«! • 
 
 10. s..x--=V^. 
 
 1-a? 
 
 11. =Smn"-'\v" 
 
 a-x 
 
 1 " .1?" 
 
 12. =Sm'^''"~'+-^ — ) and 
 
 \-x l-x 
 
 If c'P<l, then =Sm^v"'"S 
 
 and -JL=S,„(-ir-' .'»'"-l 
 l + ,r ^ ^ 
 
 X 
 
162 INDEX. 
 
 ART. 
 
 13. If a„,=0, for m>n, then Sma,„=S,„a,„. 
 
 2«-l » w-1 
 
 S„i a™ = Sm ^2™ - 1 + Sm «o,r, j and 
 
 16. (s,„«,„)x(s„6„)=a„ff„s„&„. 
 
 17- If »■ is independent of w, and * of w, 
 
 then S„, S„ «».. „ = S„ S,„ «,„, „ • 
 
 X CO CO ni 
 
 CO m cc » 
 
 19. S,„S„a,„,„=S,„Sna^+„_i.„. 
 
 CO TO— 1 05 X 
 
 20. S,„ S„ «,„, „ = S,„ S„ a„, + „, „ . 
 
 OD r »■ m » >■ 
 
 21. S,„S„fV»=S^S„a,„_„+i,„+S,„S„a,«+r-«+i,n- 
 
 m w H! m-n+1 
 
 22. S„S,.«„.,.=S„ S.a„+,._,. 
 
 and =S^ S,«„+r-i,«- 
 
 2to w to » '• 
 
 23. S„Sran,,=S„S,(ff2n-r.i-+Cf2„_,.+ i,,.)+b,„Cr2n!-r+l.r+l 
 TO-1 m-n 
 + 0„ Or (^2m-r+l,r+n + l + ^2m-r+l,i+n + 2)5 ^nU 
 
 2m-l M TO-1 M TO-1 
 
 S„ S,.ff„,r=S„ S,.(a2„-r.r + a2n-r+l,r) + S,a2m-r-l,r+l 
 TO-2 TO-n-1 2TO-1 
 
163 
 
 24. If «,„ + i = «„,+/>„„ (;":!,_,), then «„=rti + S™6,„. 10 
 
 25. If rt,„ + i = ca„ + &,„, CZl-i)^ then rt„=c''-'ai + S,„c'"-'t,,_,„. 
 
 25,1. ifr/,„_,=6„_j-«,„, (;;;:;;), 
 
 then ao=S„(-t)'"-'.&,„_i + (-l)"«„. 149 
 
 26. ( S,„ o.,„ _ 1 .i''" " ' ) ( S„ h„ _ 1 .?'" " ' ) = ^« ^"i - 1 •^'™ " ' • S» ^n - 1 • *'" ~ ' 
 
 = S,„.^^'"-'.S„«,„_„.6„_i. 11 
 
 27. (S™a,„-i.ci>'"-')-=S,„.i"'"-^S„«,„-„.«„_i. 
 
 29. U a„ = b,,, Cll), then P,„ «,, = P,„ 6,„ . 12 
 
 30. If 6 is independent of m, then Vm{f'mb) = h".V,na„^. 
 
 31. P,„«,„=P,„a„_„ + i. 13 
 
 32. P.«,„ = P,„K).Plrr,„„. 
 
 »• H n+c 
 
 33. P,„ («„) •?'.«,+,.= Pm«m- 
 
 
 
 34. P,„«,„=l. 
 
 -n 1 1 
 
 35. P»a,„= -, = li • 
 
 »— 1 
 36". If «,„ +! = «„;. 6,„ , then a„=ai. P„, 6,„ . 1 4 
 
 w-l n-1 
 
 37. If «,«+o=«,„.6«, then rta„=«..P,„62;„, and «2„_i=«i.Pm&am-i- 
 39. |^rt6=6".[« . 15 
 
164 
 
 ART. 
 40. 
 
 rt = a+7i-l .m. 
 
 \a =\a . 
 
 42. a =1. 
 
 43. 
 
 44. 
 
 [0=1. 
 
 \a ^ — ^ 
 
 L — \a-nm la— 
 
 46. ^ 
 
 \n In 
 
 m TO— « 
 
 jm \n-m 
 
 47. 
 
 48. 
 
 48,1 
 
 48,2 
 
 \m lm + 1 \m + l 
 
 If w and m are positive integers, then 
 
 "'+^ ,=S^..!!i— , and -f— is an integer. 
 \m+l \m [m 
 
 PAGE 
 
 16 
 
 149 
 Ig 
 
 17 
 
 18 
 
 If a„4.i,,„+i=«„,,„ + i+««,,«, a,„i=7i, and «„,„+,=0, 
 
 then «„ m=-r~ • 
 \m 
 
 If a,,+j,,„=tt„,,„4-a„,,„_i, a„.o=lj ''i,i=l? and a„^„+y=0, 
 then «„.,„_! = 
 
 150 
 
 m-l 
 
 m-1 
 
165 
 
 PAGE 
 
 50. S«^+6,„5...j6'=8^a,„.P,,6,+c.P,ft,. 19 
 
 m m+l n+1 
 
 51. If «„=6„+c„.a„+„, 
 
 then a„=S,6„+,,_,)a-PtC„+,f_i)„+a„+,„„.PtC„+,t_,)„. 
 
 52. If o,.,„=^',„,„+c„,„.a,„+„,„+6, then 
 
 + a/,. + r«,« + ,e-PtC,„4-(t-l)a,n+(t-l)/3- 19 
 
 54. Cr«r=Crffr + «a+l-Crar. 20 
 
 [n 
 
 54,1. The number of terms in C,.ffr is-f — . 151 
 
 \m 
 
 m,n m,n 
 
 55. If b is independent of r, then C,(«,i') = 6"'-Cr«,- 20 
 
 m,n l__. 
 
 56. If a is independent of r, then C, («)= r — ""• 21 
 
 I m 
 
 57. v^-^ = Crar"'. 
 
 O.ra 
 
 58. Cr«r=l- 
 
 7n, n— m m,n 
 
 60. If&,=6, thenC,.,(a..6,)=6"-'".C,«,- 22 
 
 60,1. If c is independent of s, then 
 
 "''a™ iK)(^>.c)} =c''-''\a"(«r-6.)- 151 
 
 ra— m+l,n n—m,m n—m+\,ni—l 
 
166 INDEX. 
 
 ART. 
 
 n n n-m n «i-l 
 
 63. A+,.«, = S™fi„.A+,.a,=S,„a„_,„+i.A+,a,. 
 n 
 
 64. A+,ff,= l. 
 
 65. If a«=S„j«„-m.^»i> then a,=aQ.A+rb,.. 
 
 66. If «„=c„+S«cf„_™.6,„, 
 
 n ,«_l „ 
 
 then a„=S«c„_„+i.A+,6,+«o-A+,6,. 
 
 « >i+I w-7«+l,m-l 
 
 67. P,(a,-f6,) = S„ CAttr-h,). 
 
 68. P,(a^+a,) = S,„c??"-'.C,a,. 
 
 69. If a,, is the r'^ root of the equation 
 
 1+1 n — tn+1, 11 
 
 0=STOam-i-'2^™~S then shall ffm-i = C,.(-ar). 
 
 70. If h,_, = kn<'-'-Xm, C::), then cc,= ^^„V''^'^~'''^ 
 
 P,(at-«,.) 
 
 71. If //-'=s,„a/-'..r„, c;;;;), then .t^,=p, ^ * ''■ 
 
 72. — - =S».^•'" ^S,.«,«+,-i-6 + 
 
 at—a^i 
 
 PAGE 
 
 23 
 
 27 
 
 73. If 6 is a root of the equation 0=S,„a,„_i.cr''"''', then 
 the second side of the equation is divisible by x-b. 
 
 74. |ff + /) =:S„,~-^ • [« • \b_ 
 
 28 & 151 
 
 75. \a-h^^A-iy^-\-^.\a . 
 
 29 
 
167 
 
 77. 
 
 
 30 
 
 78. 
 
 79. 
 
 
 integer. 
 
 
 la 
 
 -6 
 
 ^^,.-1 
 
 m-] 
 
 ,»• 
 
 
 
 h 
 
 -1 
 
 80. S.^^:^ 
 
 m-1 
 
 ■Sm^^^ 
 
 81. (s..l„..|^j =k 
 
 — .a 
 
 ^ m-1 
 
 82 
 
 83 
 
 83 
 
 /^ [a_ a^^-^X-" ^ I -na w""-^ 
 
 ' V^"" "•->"• -j^j =^'«-^"- -j^ZT' 
 
 — .a . i 
 
 n \m-l 
 
 '*• (S„l„^.j^j=S.L^.j^; ..rational. 
 /< la a7'"-i\ /* 16 x^-'X 
 
 ^0.« ^^^rr^ • I 7 + terms in {v\ 
 
 31 
 
 152 
 
 31 
 
 Stain ville. 
 
168 
 
 ART. 
 
 85. 
 
 PAGE 
 
 /« \a x""-^ Y * \na a^'"-^ . , 
 
 0,n kriTr • I 7 =^'n bir^ • i 7 + terms in x\ 32 
 
 V \m-lj ^''^ \m-l 
 
 86. (a+bY=Sm , " ^ • a"""'+^6"'-^; w a positive integer.* 
 
 (a+&)" "+i o"-"+i . 6'"-i 
 
 33 
 
 jw ln-m+1. w— 1 
 
 or =Sm-r^^^-iab)"'-'(0'"~^'"^'+b''-'"'+'); according 
 \m—l 
 
 as n is even or odd. 
 
 90. (a-by=Sm{-'^y''~'--f^^^^-iaby"~'{a"-^"'+^+b"-'-"'+~) 
 
 n 
 
 + (-iy"._i^(a6)i», 
 
 i(n+l) 
 
 or =S.(-l)'"-'.T^2=i-.(«6r-Ha"'""'''-6"-"""''); 
 
 according as w is even or odd. 34 
 
 I 
 
 -; n and r positive integers, 
 
 91. (1+aO '=8 
 
 r \m 
 
 and cr?<l. 
 
 92. (] +j?)" = Sn, | "' ' .cy'"-^ »^ rational, and .i7<l. S.-j 
 
 Newton's Binomial. 
 
169 
 
 93. (i+a')'-=s,„f^; , 
 
 35 
 
 94. 
 
 ^ \m \n) 
 
 95. (i+w) "=s.(-ir-'.f^ ^ 
 
 \m-l \n 
 
 
 36 
 
 (a+6)™-'.j?" 
 
 w-l 
 
 V Im-l / |m-l 
 
 V m-1 J 
 
 \m-l J ^'" I 
 
 ; ?i a positive integel*. 
 
 99- S,„-j — =S,n{-) ., 
 
 m-1 / Vw/ m-1 
 
 100. 
 
 V Im-l / V \n-l J \m-l 
 
 101 
 
 (-a)™-i.a?™-i 
 
 I ^>n i I J ~Om" I 
 
 V [m-1 / [m-1 
 
 [m-l / [m-1 
 
 37 
 
 isj:^ =§ 
 
 K , « (±^.a)"-'..r"-' 
 
170 
 
 ART. 
 
 102, 
 
 
 I .g 1 — i. ; n rational. 
 
 rw.-l 
 
 103. I S 
 
 _g 1^ — i ; n rational. 
 
 m-l.l '" lm-1 
 
 104. e' 
 
 ^ » Crv^)" 
 
 f=S, 
 
 Im-l 
 
 cV rational. 
 
 PAGE 
 
 152 
 37 
 
 105. e^ =S™f --■> AMrrrational. 
 
 \m-l 
 
 38 
 
 106. €'- = S»: 
 
 107. a''=S„ 
 
 [w-l 
 
 (.17. log, g)'" 
 iw-l . 
 
 .r' any quantity. 
 
 39 
 
 40 
 
 108. (a±fe)2=a-±2a6+6". 
 
 109. (S,„«,„)'=S.a^+2.S,„o>„'s%.+, 
 
 110. 0"'(/)"(w) = 0'"+"(«^)- 
 
 111. (I)\ic)=u. 
 
 112. (()-". (p"(u)=u. 
 
 118. If (^,(m), <^o(?0, 03(w), &c. and \//i(m), >|/2(w), >/^3(w), &c. 
 are all distributive functions, and commutative with 
 each other, then shall 
 
 i+l n-m+l,m-\ 
 
 K<^v+V/,.)S u=8,nl C,,, (^r-^.)\n. 
 
INDEX. 171 
 
 ART. PAGE 
 
 119. (cb+f)„u=S,n-r^^^-(l>'"""'V''0'')- ^3 
 
 ^ ^ \m-l ^ 
 
 120. ((j) + f)n={(p + ^ly)".ti. 
 
 =5 CO 
 
 121. If v//O^) = S,„«,„_i.0,„_iOO, then >//''(?O = (S„,a«-i-0».-i)"-?<- 
 
 122. If \l/{u) = 8,na"''^(p"'-hc, then \|/"(?/) = (l-a.0)~".?^. 
 
 123. If 0^ denotes such an operation, performed with re- 
 spect to o!, that <p.i{ti + a) = (p^{u); a being independent 
 of a;, then shall 
 
 where c,„ is some quantity independent of x, and is to 
 be determined, in any proposed case, by the conditions 
 of the problem. 44 
 
 128. D^u=E^u-Uy E.,M=u + D^2t, and u=Ejc-D^u. 45 
 
 129. E,.(p(x) = (l)(x+Dx), D,(l){.r) = <p{w+Dx)-(p{.v), and 
 
 130. d;\d;'u=dj''+"7i, dj'u=u, ej'.ej'7i=e;'"-"u, 
 
 E^u = ti, and E,r~"-E/u=u. 
 
 131. EJ'.(l){x)=(p(.v+7iDx). 46 
 
 132. E,-'(p{,v) = (l)(.v-D.v). 
 
 133. E^'\(l){x) = (p{x-7iDx). 
 
 134. £/^(M+tj)=£/'?^+^/'t;. 
 
 * With the same Ihnitations as in 118. 
 
172 INDEX. 
 
 ART. PAGE 
 
 135. Ef^{ciii) = a.E^^u; a being independent of .t\ 47 
 
 136. D,(u+v) = D^u+D^v. 
 
 137. Dr(au) = a.D^^u. 
 
 138. D^{u-\-a)=D^u, and 
 
 Z>^~"Z)/?/=t<+S„,c„,Z>,,~^"""'*.l, c„, being independent 
 of X. 48 
 
 139. D^-\u + v)=^D^-hc + D^-^v. 
 
 140. D,,-^{au) = a.D^-^u. 
 
 141. E,.D,u=D,E,u. 
 
 142. E,.D,-'u=Dr'E,u, D,E,-u = E,-'D,n, and 
 
 E,r~'D^'hi=D,r'E,-'u. 49- 
 
 144. i>/?^ = S„,(-l)'"-^7-^2;;i— £/-'" + ^?^. 
 
 I w-1 
 
 ... t ~" 
 
 145. £/w = S,«~— ./)■"- '«^ 
 
 Im-l 
 
 146. D,,\v"=[n.(D.vy. 50 
 148. A"..r"' = S,(-l)'-'.^^.Or+w-r + l)"'. 
 
 14.9. S,(-l)'-'.7^.(.r+w-> + l)"=[/i. 
 150. ^\o"' = S,i-iy-'.~^.{n-y+l)"' 
 
INDEX. 173 
 
 ART. PAGE 
 
 151. i>/. «"=«'. (a''- 1)". 52 
 
 152. D^. ]^a+bcV =bnh. \a+b.(a;+h) . 
 
 n, bh n-1, b/i 
 
 153. /^^.^a;=n.^. 
 
 71 n—l 
 
 154. D^( \a+bx) -^=-b7ih.( \a + b,v )'\ 53 
 
 n,bh n+\,bh 
 
 1 -n 
 
 155. A^.— = r — • 
 
 if. n'+^ 
 
 n w+1 
 
 156. i),.P,0 {A^+(r-l)/iS = 5(/)(,x'+7i/i)-<^Ci')S P,(^C^+r/i), 
 
 157. i).[P.0^^+(r-l)/^n"' 
 
 n + l 
 
 158. i),. P,K) = S. 'Cm (m,-^.w,). 
 
 159. D^{icv) = u.D,v+D^{u).E^v. 
 
 160. D,.-= ^ ^ ^ -^ 55 
 
 V V.ErV 
 
 161. (cpf+(p,^j.,),.lcv = S,,,J^^.<p"-'''■''.(pr-'{^t)^i^'''"'^'fl'"-'iv): 
 
 where (p{ii), (pi{u), ^{v), and \//i(y) are distributive 
 functions, commutative with each other and with a 
 constant factor. 
 
 162. {^^■V(p\^^\)nUV = {(p\\/-\-(p^\\/^)"llV. 56 
 
174 
 
 ART. 
 
 163. 
 164. 
 
 165. 
 
 166. 
 167. 
 
 A 
 
 168. 
 
 169. 
 170. 
 171. 
 172. 
 
 \m—l 
 
 \n_ 
 
 D/ (Ml Wo) = { (1 + 'A) (1 +^A) - 1 S " Ml w„ and 
 
 E,EyU=EyE,zi. 
 E_r:Eyii=u+D^u+DyU+D^D^u. 
 
 
 59 
 
 lis. D:,^u=s,n(-ir-'.—h.E: 
 
 m—\ 
 
 175. 
 
 176. 
 
 177. 
 
 60 
 
 174. E:^u=S^j^.DI-^u. 
 
 \m . \n 
 
 EJ" E,fu==Srks r^ .~\ .Dr'D;-^ n. 
 •' \t-s\s-\ 
 
 D,-'{uv)=u.D,''u-D,-'\D,{u).D,-'E,v}. 
 
 D,-'(7iv)=k,X-^y"-'-Dj"-'(u).D,-'"E;''-'v 
 
 + (-iy.D,-'{D;{2().D,-'E/vl. 61 & 152 
 
INDEX. l^S 
 
 ART. PAGE 
 
 178. If, for some value of r, 2>/?^=0, then 
 
 D,-'-(uv)=S,n(-lT-'-j^.D;''-'{ii).D,-^''^"'-'^Ef-'v. 62 
 
 «■'■ 
 179- D^~^.a^=—f^ — +const. 
 a —I 
 
 180. />,-".«' = «■'. (a"-l)-« + S,„Z>,r*''-"'*.c,„. 
 
 181. Z)^-'.(a^M) = aMS^(-l)'"-i.a('"-')\(«"-l)-»'.i)'"-iM 
 
 Ya+h.{x-h) 
 
 182. D-Ha + hx=^^ 
 
 ^ A 
 
 ITh hh{n+\) 
 
 183. 
 
 - ^+1 
 
 184. 
 
 ^ 1 1 
 
 '^^a-y-hx hh{n-l)\a+bx 
 
 
 n,bh }i—l,bh 
 
 185. 
 
 , 1 -1 
 
 "' "1^ (w-i)cr-r 
 
 186. 
 
 ,,+1 ^™-i.o" , 
 .^"=S.-| --Iv, and A-^ 
 
 64 
 
 t 
 
 187. Sm«,«=(A;/-A;Jo)««+i- 
 
 189- dj\d/ti=dr""^"u. 66 
 
 190. d," ?/ = ?<. 
 
1^6 INDEX. 
 
 ART. PAGE 
 
 193 
 195 
 196. 
 
 197 
 
 198 
 
 d^.(u + a) = dj.u ; a being independent of x. 67 
 
 4.a=0. 
 
 d^.{au) = adj,u. 
 dr-x=\. 
 
 cp{^v+h) = S,n^^-d.:"-Kcl>{a^). 
 
 68 
 
 199. E,u=Sm-r -dj^'^u, and D^ri=Sm] — .dj''u. 
 
 [m-l [m 
 
 200. i)/7* = (e'"''-l)''w. 
 
 201. d/.0(cv) is the coefficient of I — in the expansion 
 
 of (p{.v+h). 69 
 
 202. If?* is a function of a', then d^.(p{u) = d^(p{u) .d^u. 
 
 203. dXii'v)=vd^ii+ud^v, and f^uv=u f^v-f^{d^u. f^v). 
 
 dJuv) d^u d^v 
 
 204. ^ — ^ = _^_ + _i_ . 
 
 .05. ^-:^-"-4.^. 70 
 
 VrUr ^'" 
 
 206. 4 . VrU,= 8,n d,U„ . P,'.W,. 
 
 207. d,..u"=nu^~^dj.u ; w rational. 
 
 * Taylor's Theorem. 
 
177 
 
 AHT. , PAOE 
 
 iiOS. (/,._ = --"-—. 71 
 
 V V \ u V J 
 
 VrUr PrW, ( " </,?/, " d,,V, 
 
 (I- a = i; \br~ 
 
 -op. ,/^. :ji_^ = ^r::- s. ^^ -S,. - 
 
 P.r, P, 
 
 210. dj\:v"=ln..v"- 
 
 '^Ti 
 
 211. r//'. (/H') = S„,~^^ •(//"'" + ^^-(^;"^'"- 72 
 
 m - 1 
 
 212. d;.V,u,= {^/d,)\V,u,. 
 
 21.'^. If M is such a function of .v as may be expanded in 
 positive and integral powers of .r, then shall 
 
 u = S„, .^ . d"^:^ . 7/ .* 73 & 1 54 
 
 214. d ,d,^u = d,^d,.ii\ a' and 7/ independent, 
 
 21.'") . dj" . (/ ,/' u = dy' . d, '" V. 74 
 
 lm-7i.\n-\ '' ^ 
 
 218. If u is such a function of a; and y tliat it may be 
 expanded in positive integral powers of w and ?/, then 
 shall 
 
 ■ -«=s,„s„-^ — ^^ — ,d::: .d!:zin. 
 
 Vm-\ . \n-l 
 
 = • "!/ = () 
 
 31aelaurin's Theorem. 
 
 z 
 
178 INDEX. 
 
 ART. ^AfiK 
 
 219. If ,r is a function of .1 and ?/, then 
 
 d^.\d^z.(p(^)\=d^{d^z.(p(z)\. 75 
 
 220. If y='*^\z+x.(p(y)^, where z is independent of x 
 and y ; then shall 
 
 221. If y=z+x.(p{y\ then 
 
 /(y)=/(^)+S. ^ . d--'- \^)X- d.-K^) \ -t 7« 
 
 222. /,(wi))=S;„(-i)'"->.d/-'M./;"t)+(-i)"./,(d;'«.//v). 
 
 76 & 155 
 
 223. /.?«=S,„(-0""'T--^/"'^*+(-i)"./r{r--^'/^4- 77 
 
 225. a. + .C«,) = Sr' + 'fC8~+.sC«-)' 78 
 
 226. r-.(S.«.r = S,.,.,sT^. 
 
 22p. •2r„".0(o) = 0(a). 
 
 ni <7t"' ~^ fl ^ 
 
 2^0. -ST'" . ff"= S J w . r/"- ' -U- ; w rational. 
 
 80 
 
 [r 
 232. If w is a positive integer, then shall 
 
 73-"'. a," ™ a,"-^•s^'«-^a'',+, ?; <'-'" + ''- '.■Hr'-i.a7^V+' 
 
 = bri -1 =br- 
 
 \n '^[n-r . Ir ' ln-w+r-1 . 1 w-r+1 
 
 " liaplace's Theorem. t liagrange's Theorem. 
 
179 
 
 234. 
 
 W .0' =*'";'" a 
 
 r~ ■ =or,+«,+*(i-i) — 
 
 zoi). "sr -'i =(f 
 
 A„ 
 
 -Or 
 
 \S6. 
 
 
 :S,„.r"'"^S„ «,„-„•/> '-A^ 
 
 ,'r /-'', 
 
 CI 
 
 8() 
 
 237^ 
 
 T =^m«« • (p (U) . r- ; 
 
 \n ^ ' \m 
 
 d"' u 
 
 i238. 
 
 '-139. 
 
 = (p («) +8„ci''' . 8,„ C -" + ^ (/)(«) 
 
 73- '.ffi 
 
 W— 7W+1 
 
 
 87 
 
 88 
 
 2i0. 
 
 X,(r=^)=0. 
 
 89 
 
 241. 
 
 =cr-'-l + 8,„g.._,..i"'-""-'. 
 
 242. 
 
 D,-'ii = h-\j,u- - +S™e,,„_i./r""-'. (//"-'?/. 
 
 90 
 
 243 . A,, 'u=frU- -+ S,„ g„„ _ 1 • (//™ " ' ^^ 
 
 .92 
 
 244. zl-'.-r^ 
 
 24.K r/,.e' = , 
 
 ;i+l 
 
 + S,„ g..,r. - 1 • I >* • 'V"'-'" *" ' + const. .92 
 
 im—l 
 
 9'^ 
 
180 INDEX. 
 
 ART. I'AGK 
 
 LM6. (!;.€' = €''. ij3 
 
 2i7- d,.e" = €".d_^u. 
 
 248. d;.f:"' = e"\a". 
 
 249. d;.a' = a\{\og,ay. 
 
 250. (/,. (logj .r)= -. 
 
 d^ii 
 
 251. f/,.(loo-^//)= — . ()4 
 
 233. iog..,,=i.s,„(-,r-.t'-ir,(_,y. /• <_^:zi):. 
 
 254. If .i"^l<], lo-,.r=i.S„(-0"'"' ^'''~'^"' 
 
 W Wi 
 
 55. If a' < 1 , log, (1 +.r) - S,„ ( - 1 )"' - ' . — ^ 
 
 log, (1 -.?•)= -b,„ — , and 
 m 
 
 log^ -- =2b,«^ 
 
 " l-.i' 2m-l 
 
 1 ", ,7,'""'-l 
 
 256'. log, x= - . b™ , „ ,t;;, • 
 
 » ^,IH - 1 > 
 
 257. 0(€') = S,„r^^.l<^(i + A)5o"'-'.* <}fj 
 
 CO ,>,« + Wl - 1 
 
 J5S . (e '■ - 1 )" = S„: ,-'- • •^" • 0" 
 
 n + w-1 
 
 Hcrschcrs Tlieorcm. 
 
n-I nrn-l 
 
 '^59. -T—^S.i -.S^C-i)"-' 
 
 26"0. 
 
 2()I. 
 
 262. 
 263. 
 
 2G4. 
 2G5. 
 
 26(). 
 
 267. 
 268. 
 269. 
 
 270. 
 
 270,1. 
 
 271. 
 
 e' + I 
 
 -T~r=^'n\ r-S.(-ir-' 
 
 A"-'.0'" 
 
 A"-'.o" 
 
 7. ,, = ->,2"' 2m+l A"-' n'-" 
 
 - 1 2 2 /?/ 7^ 
 
 - A"0"-"' 
 
 \n+ 
 
 in - 1 
 
 .h'' + '"-\r2" + "'- 
 
 |log, (1+A)|".0"' = 0, (m^7i); and 
 |log,(l + A)j".0"=[w. 
 
 sin .X' 
 
 I, and 
 
 tan ,v 
 
 181 
 
 PAGE 
 97 
 
 98 
 
 99 
 
 f/r-sin cr=cos cr. 
 
 rfj. cos cr= -sin a-. 100 
 (//"-'. sin , T = (-1)"-'. cos cr, c//".sin cr=(-l)'\sin .r. 
 
 <//"-'. cos cr=(-l)".sin x, and rf/".cos ci' = (-l)". cos a\ 
 
 d,. tan ,r=(sec wf. 100 & 155 
 
 </, . cot X = - (cosec xy. 1 56 
 
 f/f.scc ti'^scc f.tan t. 100 
 
182 IXDEX. 
 
 ART. PACK 
 
 271,1 </j.cosec .j = -coscc a.cot a'. 156 
 
 272. dr . sin " ' a - . . 1 00 
 
 273. (/^.cos"' d;= -■ ^ ■ ■ - . 101 
 
 274. d,.tan-'j?= 
 
 275. (ij,.sec~'.r= 
 
 x^yar — \ 
 
 270). P, (cos x\+\/ -\. sin ,x\) = cos 8, a', + ^ - 1 . sin 8, .r, . 1 02 
 
 277- (cos X + \/^ . sin a;)" = cos /i x+ \/^ . sin « x ; n rational.* 
 
 278. 2 cos w.r=(cos a'+ \/-l .sin cr)" + (cos x^ \/-l .sin .?.')^''. 
 
 103 
 
 2 a/^I • sin «a'=(cos ci+ \/-l .sin .i')"-(cos .ri \/-l . sin xy. 
 
 y/l-ar ' 
 
 -1 
 
 x/1-,.^ 
 
 1 
 1+a- 
 
 1 
 
 279. cos 8,.iV=S,„(-l )'"-'. Cr,, (cos.xv.sin.r,), 
 
 sin 8,. .r, = 8,„ ( - 1 )'" - ' . C, , (cos x, . sin ,r J . 
 
 = 2m— 1, n 
 
 MO. tan S,..-,= g- <-"""'■ '^^ ft°""-' ,04 
 
 ' '^ CO 2m-2,n ' A"* 
 
 S„(-l)"'-'. C,. (tan.iv) 
 
 281. cosw,i = iw.S,„(-l)"'-'.-y^-2 .(2cos,x')"--"'^-, or 
 
 m— 1 
 
 = -i»i-S,„(-l)"'-'.f^2 (2cos.r)"""" + '- 104 & I.^rS 
 
 [w/-l ^ 
 
 * Demoivrc's Theorem. 
 
laa 
 
 282. cos 
 
 M.^=(-ly^n^s,n(-lr-^ ^^ ^ .(cosaf"--'. 
 
 2m-2 
 
 105 
 
 or =(-i)^'"-'>.7z.s.(-ir-'. "^ J;_\ '^ ^ (cos.^r-\ 
 
 in I 
 
 283. (sinA'y'=(-iy"-2-"+'.Sm(-l)'""'.r^^^^i--cos(w-2w+2)a,' 
 
 m— 1 
 
 ^ 
 """•^' "' 
 
 107 
 
 t 
 
 i(w+l) 
 
 :(-iy'"-".2-"^^S,„(-l)'""^, '"'^ .sin(?z-2y?/+2),r. 
 
 ^ 
 
 284. (cos.vy' = 2-" + '.S,« i" ' ■cos(M-2m+2).r+2-".77^ 
 
 1^^-^ Li!' 
 
 i(n+l) L_ 
 
 or = 2 "" "^ ' . Sn, 1 "' ^ • cos {n -2 in + 2) .t. 
 m— 1 
 
 109 
 
 185. sin.r=S„,(-l)"'-' 
 
 !0''"- 
 
 l2w-l 
 
 tlO 
 
 286. cos.r=S,«(-l)" 
 
 2m-2 
 
 287- e*''^^=cos.r±\/^.sin.r, 
 
 2 cos .r=e''^^+e-^^^i , and 
 l.sin .r=e*^^-e"'''^'-i . 
 
184 
 
 ART. 
 
 288. 
 289. 
 
 j(2m-i)i'rV-i = (_i)'"-iy^_l, and €'"'-'>'' ^'-' = (-1)'""'. HO. 
 
 "^ ' 2w-2m+l V 2r/ 
 
 2.00. sec.r=S 
 
 > "7> / - 1 \ 
 
 291. 
 
 29£ 
 
 29.'3. 
 
 294. 
 
 cot.r=S„(-i)''-'-.'''"''"-'-S™r— ^^ — .A. ,.(7-^ 
 
 "^ ' [2?i-27« ^[2r+l 
 
 cos.c,r=S„(-l)-'..t'--.A.,(i^). . 
 sin ,T'=.r.P; 
 
 ?95. 
 
 29f). 
 
 297. 
 
 
 log, sin cr=loo-, .t-S;„ ( - J • ^^ • S.^-"". 
 
 ll.S 
 
 114 
 
 M / O ^.\ -'Hi 1 OS 
 
 ^fee '^"" '"— '"C36 
 
 298. tan- 
 
 2m -1 ,^57, 
 
 299- 
 
 If .v<l, tan-'.r=S,«(-ir''- 
 
 2 W - 1 
 
185 
 
 300. ^=,8.s.(-ir-'.^^^-s.(-ir-. 
 
 4 2m — 1 
 
 116 
 
 300,1. log— =2wlog 4 + 4log ln-2log I 2«-log (27i + l), (n= co )• 
 
 157 
 
 301. .r"+l=P,,(cr.'-6 " ■'''^'). 
 
 " O ffl — l 
 
 302. .v" + 1 = P„ j .t?- (cos 
 
 117 & 1.'58 
 
 — _ 2m-l , 
 V -l.sin — .tt) V. 
 
 71 ' 
 
 *" 2r» — 1 
 
 303. If n is even, .r" + i = P„ (.v'-2.v.cos .tt+I); 
 
 n 
 
 i(»-i) 2m — 1 
 if M is odd, ..x'" + l = (cr7+l).P,„(,t?^-2cr.cos^^^ .tt+I). 
 
 304. .??"-! =P„,(,i'-e'^^'''^^'). 
 
 305. a?''-l=P,J.t?-(cos — 
 
 n 
 
 118 & 158 
 
 , 2m-2 
 
 TT+V -1. sin — - — .7r)| 
 
 306. If w is even, .'r''-l = (,i'^-l) .P,„(.??--2a,'cos^^^ +l); 
 
 71 
 
 2 711 IT 
 
 n 
 
 4{n-i) 2m7r 
 
 if w is odd, .r"-l=(,y-l).P,„(.r--2.r'.cos^^ +1). 
 
 n 
 
 306,1. a?'^"-2cos0.,r" + l^P„(.v--2.x'.cos ^~^'~'^^ +1). 158 
 
 ZOl. (l+e.cos.t?)"=l+S„,<i-p^-.afiy" 
 
 -2cosm.»' 
 
 • S, 
 
 m+r-1 . r-1 
 
 (i^)"-- 
 
 HP 
 
 Machin's Theorem. 
 
 Aa 
 
186 
 
 AHT. PAGE 
 
 308. If y=z+.r.smy, where z independent of .r, then y=z 
 
 |2m-l 
 
 (2r- !)-'"--. sm(2r-l).5r 
 
 ^ ^ 2?^? \m-r 
 
 309. In the same case, cos ?/=cos;:r-.r.(sin;?)'' 
 
 \2m + l 
 
 (W 
 
 + S.(-ir. ^f^-S„(-l)"-'- '"""^ ■(2n-]f—.cos(27Z-l).g 
 
 ?w-n+l 
 
 (i«^)^ 
 
 2 7W + 2 
 
 + S,.(-ir-^^f^^— 7--S.(-0''""'-|"'~"^' , (2w)-'".cos2n;y. 121 
 
 2w+l 
 
 tn—n+l 
 
 3\ 
 
 312 
 
 3 1 0. f = log, (,r + x/,r - ± 1 ) 
 
 '^.rcX'V l±.:l- 1 + Vl±.t" 
 
 ^r sin .V 
 
 /-^ = 
 
 /j cos .t? 
 
 - ^" . r L. I - iV 
 
 tan 
 
 : loo;^ .cot 
 
 4 2 
 
 12.^ 
 
187 
 
 PAGE 
 
 
 n-\ 
 
 1.-2 ^-.n-'im + X , i».2 
 
 
 a."~'"""^' + -i--ii.sin 'a?, or 
 
 :]()'. 
 
 
 TO,-2 
 
 [w-2 
 
 317. 
 
 x\/\-,v^ ' ' '"U-1 .:l'" "" + ' 
 
 \n-2 
 
 125 
 
 126 
 
 \n~2 1 
 
 J(n-1)L . I L ^^ 
 
 127 
 
 318. 
 
 [w-1 
 /,(sin.i')" = -cos.r.S,„-^pi^. (sin.r)" 
 
 • ./,(sin.i')" -'. 
 
 w-1 
 
 :i.9. /,(.sin.f)"=-cos.p.S™-2^. (sin,r)"-^'" + '+ i2:£..f, or 
 [n_ [2_ 
 
 "1,-2 in. 2 
 
 W-1 
 
 -coscr.Sm "|~''~^ . (sin ,<■)""-"'"' 
 
 ■IT- 
 
 128 
 
188 
 
 W-2 
 
 320. /,(sin.T)""=-cos.r.S„ 
 
 n-l (sin<x)" -'"' 
 
 \n-i 
 
 r,-2 
 
 /,(sin.r)-'''-^'>. 
 
 128 
 
 321. /|, (sin .!■)-"= -cos A'. S;„p^ 
 
 71-2 
 i n 1 I 
 
 71 -I (sin a)" -" 
 
 = -cos.r.S.f^- , . „,^^ +fg^^.log,.tan-. 129 
 
 w-1 (sin A')' '^"' + ' 2 2 
 
 w-1 
 
 322. /',(cos.r)"=sin3?.S„, "! '' " • (cosa')"-""^' 
 
 m,-2 
 
 [w-1 
 
 J",, (cos, I')" "■'^' 
 
 323. j^ (cos .vy = sin r . S,« "', '' - • (cos a;)" ~'" + ^+ i±L, a;, or 
 
 m,-2 in, 2 
 
 i(«+l)l - 
 
 iin .V . S„, "!~''"""^" . (cos .i)" 
 
 130 
 
 lw-2 
 
 321.. L (cos cf) - " = sin .1- . S,« f^i^ ^^ 
 
 •'^ ^ '"[n-l (cos,^;)"-'^"'^' 
 
 m,-2 
 
 [w-2 
 
 + -— -.;(cos,r)-'«-^^ 
 n-l 
 
189 
 
 1,-2 
 
 325. f (cos ,f) - " = sin ,f . S,« f---7 • 7 r 
 
 •'^ ^ \n-\ (cos/j') 
 
 \n-2 ll 
 
 i{n-l) I 1 I /tt ,|,'\ 
 
 :si,i.x. S -^^^-^i^. + iiii^iLi.Wcot --M. i.'u 
 
 *" I- sin(w-2/??+2).?? 
 
 J'^ ■> ^ -> ^ ^ \m-\ n-2m + 2 
 
 In 
 
 o-" .-^"^..v, or 
 
 tl 
 
 ^ ^ ■~ "^ ^ 'Im-l n-2m+2 
 
 1" L sin ('«-2m + 2)-r 
 
 527. i;,(cos..)" = 2-"-h.^. ^_2^^^— 
 
 + 2-". -if — .r, or 
 
 Li!! 
 
 ^-» + /'o" »,:=1 sln(»^-2m+2).r ^^^ 
 
 '^ •^"'iw,_r W-2W+2 ' 
 
 327,.. /-(tan,.)-=S,„(-.)"-'.^^^;^^^ +(-ir-/ (•»"•')"-"• 
 
 159 
 
 *» , (tan.i!)"-""' , ,.„ 
 
 327,. /,(tan..)-=S,.(-l)-- „_,^^1 +(-l)-'..r, or 
 
 1'S:;(_,)"-'. <^^'-^^ +(-l)"»*".log.cos,.; 
 
 "^ W-2TO+1 
 
 '■ (cot .rV'""'"'^' 
 
 307,x /.("'"')"=s„(->)"--43,krr +<-')' -J'*™' •''""■"■ 
 
TOO 
 
 AUT. 
 
 327,4. 
 
 328. 
 
 320. 
 
 330. 
 
 *" , (cot ,r)" -'■'" + ' 
 _/,(cot,r)"=b„,(-l)". i- +(-l)^".a, or 
 
 W-2AW+1 
 
 ^<"-i> (cot.r)""'"'^' 
 
 n—1m+\ 
 
 4-(-iy'"-".log,.sin.7T. 
 
 332. 
 
 333. 
 
 /, (a '■ . ?/) = a ' . 8.,, ( - 1 )'" - ' . (log, <i) - '" . (//' - ' w 
 
 +(-i)'Oog.«r-/,(«'.(//w). 
 
 j;(a^,^■") = ff■.S„,(-l)■''-'.[w .. I'" -"" + '. (log, r/)-'", 
 / — =log, ,r+S,„ ^ • 
 
 j;.(«.z/)=a'.s,„(-ir-'-(iog.<7r-'.//'w 
 
 ffi" /-\, .,„_, ^r-"+"' (log.o.)"-' ra' 
 
 /^.i-" ^^ [n-l \n-l J^oD 
 
 — \i^ce - />i_L "^ — 
 
 l60 
 
 132 
 
 133 
 
 
 V: 
 
 134 
 
 334. jj log, (1+e COS j;) = -.i'.logj jse"' (e"'- \/e---l) | 
 
 337. 
 338. 
 339. 
 340. 
 341. 
 342. 
 
 mr 
 
 Gt(aii^) = a.GtU^; a being independent of t and .i\ 
 
 If Gi.u^=Gt.v^, then shall 7f, = (^^. 137 
 
 (fr'-i)".Gt.ii,:^Gi.A/.u,. 
 r(/-'-i)".GV?/^=GVA"7/,_,„. 
 
IXPEX. 191 
 
 ART. PAGE 
 
 344. If yMx^S^Om-i-V'^'^'x+m-i' ^"^ V'^it.-ff'.r, then 
 will 
 
 /n+l d \ »■ 
 
 345. (r'-l)^(sl-^::;)^^^G,.7..= G,.A/.v'•^'^^ 
 
 346. A/?/,,=S.(-l)'"-'-r^^^-^^-+«-'"+i- 
 
 w — ] 
 
 347. U,^^=S,n-^'^^^^'<lJ"-'Ur. 139 
 
 l?w-l 
 
 348. 7/,,^„=w,,+w.S,„-^-^ . A/'Wr-m.-; ^" being any 
 
 integer. 
 
 349. r^,..„ = (^+l) S, /^^ ! -^ ^ • A/™-7.._„,, 
 
 2m-l 
 
 TO-l 
 
 2W-1 
 
 
 
 2m- 1„ ? 
 
 143 
 
 2m -I 
 
192 
 
 ART. 
 351. 
 
 352. 
 353. 
 354. 
 
 355. 
 
 ?/.,+„=S,. S,V* ^u.v+r-i-^pf''r+p-\-T!r"''""'^^-b~' ; where 
 
 m+l 
 
 y'?/, = 8,ffl,-i.V'"'w,+,_i, and 6,_i = ff,„_,+ ,, C-lL+O- 1^4 
 GXu,)-GtM = G,.G,{u,.v,). 146 
 
 «-"' . r « . G, (w,) . Gi (v,) = G, G, (n,^,,, . v,^,). 
 
 (6^-'-ir.(r^-i)«.G,0O.G,K) 
 
 = G,.G^.\X"'M.AJ%v^)\. 
 (s-U-'-i)\G, (u,,) . G, (i',) = G, . Gt a;'. {u,v,). 
 
 \m-i ' y +»' ij .V 
 
 358. S-"'.t-\G,j.2l,_y=G,,f'>i.r + m,,, + n- 
 
 359. (^•-'-ir.(^-'-i)".G,t.?f,,,^=G,,,A;".A/M,.^. 
 
 360. {s-'r'-ir.G,j.u,,^=Gs.tA:^.u,,^. 
 
 ••^(^l. ^f...„.. = S, S,-^--^.A/-'.A/-'..„.,. 148 
 
 362. 
 
 \m-l 
 
Lately puhlished, hj the same Atitlwr, 
 
 8vo. Pyice 6s. in cloth. 
 
 A Short GRAMMATICAL INDEX to the Hebrew Text of 
 the BOOK OF GENESIS; to which is prefixed a Compendious 
 HEBREW GRAMMAR: designed to facilitate the progress of 
 those who are commencing the study of the Hebrew Language. 
 
 Preparing for the Press, 
 A TREATISE ON ALGEBRA. 
 
 Being a complete Course of Abstract Analysis. 
 
 This Work will comprise, in addition to the ordinary Algebra, 
 the Calculus of Finite Differences, the Diffei'ential and Integral 
 Calculus, and the Calculus of Variations, 
 
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