PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 573 



and X roi / 6\f^ ^g / dd' 



1^ " Jo \ x) \ X ) 





the same form as before. 



5. To find the differential coefficient of — • 



If ic be positive, f^ e~'^'' a~^da. becomes, by the substitution of d for ««, 



Jo \x) X «« ' 



if X be negative, f^ e"'"-'^ a~^da. becomes 



In the first case, if we differentiate with respect to x, to the index }x we get 



d^".^ 



I' Injo ^ 



In Jo 



\u, — kx n — 1 J 

 — a) e a, a a 



CO — aX n+u—l , 

 a da 



I n X '^ 



In the second case, if we differentiate with respect to x as before, we get 

 d'' .\ 



X 1 /'oo m + M— 1 ax , 

 = __ # a e «« 



^;jr'" (—1)" /»«y<' 



_ (_!)'' + ^ /~(W + /X) 



= ^):/>^^Us before. 



Now, Legendre considered the function r as restricted to positive quanti- 

 ties ; consequently when either w+// is negative, or n negative, the above expres- 

 sion appears to fail, and others quite different have been shewn to apply to these 

 cases. If we have no means of remedying this defect, the system is utterly use- 

 less as a branch of analysis, and we should do well to attempt to establish another 

 in its place ; but, fortunately, there is no occasion for this, as we shall shew that 



