PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



= -^^ + -^+0/(0) 

 {x-af x-a •' ^ ^ 



072 2X 



+ + 1 



601 



(x — af x — a 

 du 



+ 1 



d X 



d^ u du 



hence the coefficient required = ^^ .^^ = 1. 



In the same way, and with equal facility, we may find the values of the 

 other coefficients. Should we attempt to find the coefficient of such a power of h 



2 



as h^ , we shall readily find that — -|- is a finite quantity, but that f (l + ^\ 



dx^ i ^ \ 3 / 



is infinite ; and, therefore, that the coefficient = 0. 



27. In Art. 24, we assumed that 



1 



6?" 



-^^ -dr"^ where^=l + ^/-l., 



Should any difficulty be experienced respecting this assumption, it will be 

 entirely removed by means of the following proposition. 



1 



To find d- 



1 



^ ^ {a xY' 1 1 + — ^ I 



\ ax/ 



a'^x"'\ ax 1.2 a^ x^ j 



1 d" I 1 m m(m + l) 1 \ 



^ 1^ { ^~ aa''" + i "^ 1.2 a2^m + 2 ~ '^^- I 



d" 1 



T^ ■ (1 + a ^)'''' 



In + m 



= — (-!)" {- 



fm a;" + ^ 



»2 



o /»« + 



+ 1 1 m(m + V) 1 - In + m + 2 „ 1 . 



1 ■ ^n + m + i 1.2 ■ a^ /mT2..r" + '" + ^ " / 



m X 



-(-lY IT ^ |l-— . 



m (m + 1) 1 (?^ + m + 1) (^ + ??^) _1 ^ ) 



■^ 1,2 ''^' {m + \)m x" ^' ] 



w 



r 



