156 EXPANSION OT F [X + ll). 



and, by continuing this process, and substituting for ^" [x . h), ^' {x . h), ^ {x . h) 

 &c., their values, we should get easily the common development of F {x + h). 



But let us suppose that 4'' (a; . 0) is infinite, that is, that as h approaches 0, 

 9' (x . h) increases without limit according to a law depending upon the form 

 of the function. 



Whatever this law may be, as functions vanishing with h admit of an infi- 

 nite diversity of form, it seems obvious that there must be some one fh, such 



that — ^^ — I shall increase with the same rate as 9' {x . h), ^" {x . 0) being 



finite. We may write, then, 



h.¥{x.h) = ^ ^" {x . h). 



As fh is inferior to h in degree, since h. ^' [x . h) vanishes with h, and as 

 the powers of h admit of every shade of magnitude, and diminish towards 

 with every possible degree of rapidity, it appears evident that for very small 

 values of h,fh may be replaced by h" where e; < 1. Hence 



where a = 1 — v, and Q' is so chosen as to agree vriih. Q"" [x . h) for very 

 small values of h. By similar reasoning we have 



q = ^" {x .0) + h^. Q". 

 Putting Q for ^ {x . h) we may write ( 1 ) under the form 



F {x + 70 ^Fx + h. Q. 

 And if we replace ^ {x . 0), *' {x . 0), &c., by P, P', &c., we shall have 



F {x + h) = Fx + h. Q, 

 Q = P + h". Q', 



q = P' + h^. Q", 



Q being finite v/hen ^ = 0. Multiplying the 2d equation by h, the 3d by 

 ^ ^ +", &c., adding, and putting a for 1 + a, 5 for 1 + a + (^, &c., we have 



F{x + h) = Fx + Ph + P'lf + P"A* + h'. Q; (2) 



which differs only in the 2d term from the development assumed by Poisson. 

 The equation 



F {z + h) = Fx + Ph + h. R, 



