572 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



If there be any doubt about the correctness of the assumption that 



d'^ my _ md'^y 

 ~d^ ~ dxi' ' 



it may be removed by means of equation (1), from which this equation is dedu- 

 cible, by puttfng u,2u . .. successively for v. 



B)^ means of this fundamental formula, or definition (as it may be termed) 

 of the calculus, we are enabled to obtain the differential coefficients of different 

 functions of x. 



4. To find the differential coefficient of - ■ 

 Since - ■= 1'^ e—^' da 



X Jo 



d:>^ Jo dxT 



= n {-aY e-'-'^da: 



a form which is readily put into a numerical shape, when x is positive, in the 

 following manner. 



Let ax = 6 



d6 



da=^ 



X 



rf'.l 



and * X p«, / 6\ f^ —( dd 



e 



dxf" ~Jo \ x) X 



^{-^r-e\-' ds 



since /(l + /u) =/* 6'^e—^d6, where /" is Legendre's function gamma. 

 But if X be negative, we have 



— = — / e d a 



X Jo 



rf''! 



= — # oT e a a, 



dx^ o/^ 



Let ax = —6 



de 



da= 



