and 



PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 573 



dx" J" \ ^) V ^ / 



^ {-irr(x+ix) 



the same form as before. 



5. To find the differential coefficient of — • 



If a; be positive, f^ e-"'' a'-'^da. becomes, by the substitution of a for ax. 



X X" 



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



In the first case, if we differentiate with respect to w, to the index ju we get 



1 Pa , x« ~a.x n — 1 



dxi^ Injo 



In Jo 



d c 



<a- n + a — l • 

 a (la 



In 



I {n + jX) 



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



rf''.— 



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

 ties ; consequently when either w+M 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 



