EQUATIONS IN FINITE DIFFERENCES. 93 



which contains an arbitrary multiplier; but it is plain that the ex- 

 pansion of ^^., being unknown, S«, is also unknown, and the successive 

 difFerential coefficients would similarly be expressed in forms of un- 

 known functions; and therefore the expansion by the immediate ap- 

 plication of Maclaurin's Theorem would be impracticable. 



Taylor's Theorem in Finite Differences gives the identity 



now to find A"z<„ in this case, we must have recourse to the theorem 



xn , w(« — 1) w(w — 1)(h-2) 



A"«o = «„ - ««„_, + \ g . «„-. - ^^;g — ^ . «„_3, &c. 



the coefficients thus found being obviously more complicated than the 

 function itself. 



Try again to find u„ by a series arranged according to the powers 

 of X, and containing indeterminate coefficients, that is, put 



u, = A -^^ Bx + Cx' + Bx"^ + he. 

 and since u, = log. m,+ , ; .-. u^ = log. ?«, = log. e = 1 ; .-. A = \. 



Then the equation «^+, = e"- becomes 



1 -I- B{x-vV) + C(A-+iy^ + D{x^\)\ &c. = e.6^'.e''^\6^-'\&c. 



and equating like powers of x we get, 



1 + ^ + C -h Z> -1- &c. = 6, 



B+ 2C+ 3Z) + 4^ + &c. = ^.6, 



C+ 3i)-|- 6£+ 10F+ &c. =—.6 + C.f, 



1 .2 



&c. &c. 



which clearly show that the coefficients cannot be found but by the 

 resolution of equations of infinitely high degrees. 



