PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 

 '^"^ +0/(0) 



+ -^— + 1 



601 



{x — af x — a 



{x-df 



du 

 dx 



+ 1 



rf' M du 



hence the coefficient requu-ed = '^ '^ z= i. 



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 



as ifi , we shaU readily find that ^ is a finite quantity, but that / (l +4- ) 



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

 27. In Art. 24, we assumed that 



d" ;^^^- 



(l + V-lx) _ d" 1 



daf 



= ^--.— where w=l + A/-la;. 

 dy" y ^ 



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

 entirely removed by means of the following proposition. 



1 



To find 



rf». 



(1 + a x)" 



d" 



^1 + "^)'" iaxr(l^l-Y 

 \ ax/ 



_ 1 r.. m m(m + l) 1 ^ 



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



} 



1 d' 



m m(m + l) 



dx" ' {l + axy a"" da" \ ^™ ax'" + ^ 



O*" ^ M /^ X" + 



1 



1.2 



&c. 



m fn + m + 1 



a /m + 1 



m (m + 1) 1 . /n + m + 2 „ 



^+™ + i 1.2 ' a^ /W+2.a^ + '" + ^ 



m n + ni 1 



-r-ir ^^ ^ (i-— . 



m{m + V) 1 (n + m + l){n + m) 1 



1.2 



(w + l)i 



&c. I 



-^~^' 1^ ' a'"x" + "'' \ «^ 



