PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 291 



Hence, if we equate the coefficients of a" in the two equivalent expressions 



a 

 1 J / • 1 r+A \ 



l-a(l + A; Vn_„)2_^ (l_a)2-^>' 



^ ^ 1 + A ^ ' 1 + A 



the result will be 



r W2-P a2 ^ 1 



-»* 1 1+ 17273 rT^ + ^°- j IT^ "- 

 f M-'-l-' 1 



24. Let us apply these formulte to examples. 

 Ex. 1. Let M^=e"^ then 



d 



A>, = (e ■"■-I)" e»'"= (e«-l)" e"'^ (by A.) 



Ex. 2. Let «^=ap, n^i, then, formula (2), 



-i(l + |+Y^ + &c.) 



=V^3i^(l-l)^-'^7-^(l-l)-* 



= 00 



Ex.3. A"a-=(-l)"(^-»(.r + l) + "-^^=^(^ + 2)-&c. ) 



= (-1)" ^ (!-!)"-» (-l)"(l-l)"-i 



which is zero when n is greater than 1, finite only when »=!, in which case it is 

 1 ; and infinite when n is less than 1 . 



It is evident that this introduction of oo may indicate simply that the form 

 of the expansion is incorrect : for Acox=ao(i- + V)-ccx=x + const, is the analytical 

 result of the equation ax^{x + 1)-z^0x . + const., by dividing both sides by the 

 symbol 0. 



