DIFFERENTIAL AND INTEGRAL CALCULUS. 41 



du 



dz 



,- fj )/ 



In the case where a, b are independent of 2, to find -j- , 



we shall have 



j> 



t4Ai=l <f)(X) z 



J a 

 -i 



therefore Aw = / {$ (a?, z 4- As) $ (a?, 2)} efo?, 



a 



Aw f 6 (#, z + As) (oj, z) , 



hence, taking the limit, we shall have 

 da b d 



or we differentiate the quantity under the integral sign 

 with respect to z, just as if no integral sign existed. Of 

 course in exactly the same way we may integrate both 

 sides with respect to 3, when #, b are independent of z. 

 If, however, , b are related to z in any way, the complete 

 differential coefficient with respect to z will consist of three 

 parts, (1) that which we have already formed, (2) that 

 arising from the variation of (6), (3) that arising from 

 the variation of a. The first we already have, the second 



du db d , ,, .. db _ . db 



snce 



similarly -j r = & (a, z) -=- ; 



da dz ' dz J 



or the complete differential coefficient of u with respect 



to z is 



Thus 



r 4rt a ? 



V() ^ = f (8 - 1 ) = V 5 



J a. 



