PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 597 



which may, in certain cases, be reduced by the addition of the quantities 



(l-^ITla;)" and (l + ^/"3l a;)" (_!)"-! 



or by the subtraction of {—l-s/'^xy from (l-y/^-ixf . 



\in is negative, or positive and less than 1, the above reasoning does not 

 apply, except in as far as to shew that the result is the expansion of one of the 

 complete results, if there are many. 



Section IV. — Expansion of Functions. 



25. Our object in this section is the general expansion of a function of ,/+/-!, 

 by a process analogous to that which constitutes Taylor's theorem. 

 Let u be any function of sc ; 



u' the same function of x + h; 

 then, if u' be expanded in terms of h, the result will be of the form 



2AA» 

 where A is a function of x. 



d" u' d" u' 



2 — — . A" = 2 A . ---— 

 dx" dh" 



= 2A/{n + l).h' 



adopting the notation of art. 9. 



Now, the only term on the left hand side of this equation which contains 

 h", is that which originally contained this power of h: 



„ . d" u 



Call it u, or 



dx" 



d" H 







or A 



u' = 1 



d" u„ h" 



Cor. 1. If the expansion contain no negative powers of Z^, u„ coincides with 

 u; for if we put /i=0, we obtain u as one side, and Uo as the other side of the 

 equation. 



Cor. 2. Since each coefficient is determined from — -^- independently of its 



