572 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



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



(f my _ md'^ y 

 d^ ~ dx'' ' 

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

 cible, by puttfng ?^, 2 m . . . successively for v. 



Bj' 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 coeffkient of - 

 Since - -y^ 



'''■I .CO d" 



rf*" Jo dxl' 





-S: 



{-af e-'" ,1a: 



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

 following manner. 



Let ax = 6 



de 



d a =^ ■ 



X 



.^l 



and " X _ no, ( _Q\i^ —e d^ 



dxi" ~Jo V ?/ T 



= i=}^r^e-' dd 

 x^-^'-Jo 



since /^I + m) =/" d''e~^ d6 , where /~ is Legendbe's function gannna. 

 But if a: be negative, we have 



1 /"» «a- . 



— = — I e da 



X Jo 



i =- f 0^" e'^'^da 



f Jo 



dx 



Let ax = —6 



d6 



da = — 



X 



