158 EXPANSION OF F {x + h). 



must contain a term A {x — r), and M a term A (x — rY, and it is only ne- 

 cessary to give this term its proper place in the series in order to get the same 

 result which a = 2 will furnish. If a < 2, the 1st term will be infinite, and 

 cannot be cancelled by any of the terms antecedent to M", since they all con- 

 tain powers of X — r diiferent from the first; nor by any term in M", since 

 that term would then contain the same power of a; — r with the first, which 

 is contrary to the arrangement of the series. We must have, then, a = 2. 

 Therefore a {a — 1) = 1 . 2, — all the terms after the first vanish, since M" 

 cannot remain finite for the same reason as in the preceding case, — and we 

 have 



^dx'^ ^ ^ l.2^dx^' 



In the same way we can show that 



1 /d^ w 



p" = 





As r is any value of x between the supposed limits, the results obtained are 

 evidently general, and will give all the terms of the series until we reach a co- 

 efficient which becomes infinite for x = r, and then the remainder of the ex- 

 pansion must be supplied in some other way. Using the notation of Lagrange, 

 we have generally, therefore, 



■ Fx==Fr +F'r{x — r) +F"r. ^^ "J"^' + {x — ry. q; 



which, when r = 0, becomes 



Fx = F (0) +x.F'{ 

 Equation (2) also becomes 



F {x-\.h) = Fx + h. F'x + ^ F"x + h\'q. 



The limits within which the value of the last term is found may be deter- 

 mined by the method of Lagrange, and the development will be complete. 

 The reader will find a summary view of this method in the subjoined note, 

 furnishing itself an exact though indirect solution of the problem. 



Fx = F{Q)+x.F'{0)+-f-F"{0)+ x\ q' 



