21 illustration of this method, we shall apply it to find a few terms in the expansion of sin a + b - + d 1 . Z J. A O +&C Here A. = sin a, B = 8. sin a = b cos a, C = S .. b cos a = c cos a - i 2 sin , D = 5 . C = d cos a - Bbc sin a fc 3 cos a, E = 5 . D = c cos a - (kbd + 3c i2 ) sin a - 6b 2 c cos a + b 4 sin . Arbogast s theorem has been treated somewhat differently by Pro fessor De Morgan. Thus, suppose 0( + a 1 a; + a 2 a^. . .+a n x n + &c. ) = A + A^x + A 2 ar. . .+ A,,a; n + &e. ; then, if we differentiate with respect to a,,, we have da n " Hence we infer that, if m be less than n. we have -" = 0, da n also dA, ! Ul. m -i-n lllln , and j-^~ = - , &c. da m da The values of Aj, A 2 . . . A,, can be hence calculated (see De Morgan s Differential and Integral Calculus, arts. 214-220). 56. Lagrange, in addition to having been the first to place Taylor s series on a satisfactory basis, also enlarged the powers of analysis by a remarkable theorem which contains Taylor s as a particular case. This, which is commonly called Lagrange s Formula, first appeared in 1768 in the Berlin memoirs, and may be stated as follows : It z be connected with x and y by the equation then the expansion, in ascending powers of y, of any function F(z) may be thus written : This result can be deduced from Maclaurin s theorem, as was shown by Laplace, thus : - Let zi = F (x), and we may write -, fdu f d-u , , , , f du d-u where I - ) , I - ... represent the values of , - . . V dyJ 9 dy 2 J dy d>f- when we make y after differentiation. We plainly have u = F(cc). Also, it is easily seen by differentiation that ?fl _ 7 du _ 7 du dy dx dy dx writing Z for <j>(z). Again ( d ^} = f (z^V-f ^y dyj dy dx J dx Hence we can deduce in like manner and in general dy* dx dx d"u _( d "~ l / 7n du y n dx) dx) dy du If now we suppose y = 0, since Z reduces to 4>(x), and to F (x), we get X-**^-. ).-{ ^^ } from which the series immediately follows. For example, let z = x + -- ( - I ) ; JU then the expansion of z becomes Again, from our equation we get ~ V ~ Hence =n _ o Consequently wo have If we write this expansion in the form (1 - 2xy + 7/->) - 1 = 1 + XT/ + X 2 7/2 . . . + X7/ . . . we have * X n = -
2"
The class of functions represented by X n was extensively studied by Legendre, to whoso works the reader is referred for further de velopment. An expression for the remainder in Lagrange s series in the form of a definite integral will be given further on. 57. Taylor s series admits of ready extension to two or more variables ; thus, if we change x into x + h in the equation u = <j>(x,y), we get, by Taylor s theorem, du 7i 2 d-u h 3 du If now we change y into y + k, u or <j>(x, y) becomes u + k - + + &c. ; dy 1.2 dy* 7 du , . du h -7- becomes 7i^ rf* rfa; d a u ^ ^ 1 . 2 and accordingly we have du f; : ft6 + +< du By aid of Lagrange s theorem in 46 we can obtain an expres sion for the remainder of the series. In like manner, if u = (p(x,y,z), we get du , du ,du h* d 2 u k* d?u I 2 d 2 u ,. rf 2 <4 7 cZ 2 ?t tPtt + =,7; ^-5+^-7^ Tr5 + M j-T- + ^ -TT7- + Wi ^-^- 1.2 rfy 2 1.2 d3 2 dxrf?/ dydz dzdx The method can be readily extended to a function of any number of varial iles. 1. As an example of Maclaurin s theorem, the first three terms x 3 2x 5 in the expansion of tan x are x + -^- + -y^- (2) Trove that tan- 1 (a; + /0 = tan- 1 a; + Asin z ~-(h sin z} 2 S1 " z + (hsmz) 3 sinSz - . . . &c. o where z cot - l x. 3) If a; _^ + -J/ + ? / = o , y may be easily expanded in terms of x dx* dx * by the method of indeterminate coefficients. (4) By similar methods the first four term in the expansion of j_ (1 +x) x in ascending powers of x are found to be x llx 2 7x 3 James500 (talk) + (5) Find the development of x sm in ascending powers of sin x sin 2x x, the coefficients being expressed in Bernoulli s numbers. (6) Prove that Legendre s function X,, satisfies the differential equation dx* dx also that Indeterminate Forms. 58. Another important application of the infinitesimal method is to the determination of the true or limiting values of indeterminate expressions. For example, if the fraction
- ^ becomes of the form , or
<p(x) when x = a, the fraction is said to become indeterminate for that value of x. In fact, the method of the evaluation of indeterminate forms may bo regarded as the foundation of the differential calculus, since the determination of the derived function of any expression f(x) reduces when 7i = 0. to finding the limiting value of ll 1 This remarkable expression for X, t is due to Jacobi (CreHe, ii. p. 223).
