407 



which shows that the equation M rf # + N d ?/ is a 

 complete differential, and may be integrated. Thus 

 integrate (a a 2 3 x y 2 ) d y, as if x were constant, and 

 add X (a function of x, or a constant), as necessary to 

 complete the integral, and we have 



ax* y ^?/ 3 + X = Z; 

 now differentiate, supposing y constant, and we have 



dz 3 , ( <ZX 



_ =( ^ 2/ _ 2/ 3 ) + __ 



(because of the criterion) = < 2axy ?/ 3 , 

 consequently -= = o, and X = C, a constant. 



CL X 



Accordingly, z = a x* y x y B 4. C ; 



and so it is, for differentiating in the ordinary way, x 

 and y being both variable, we have 



dz = Zaxydx-i r ax'*dy Sxy* dy 



which was the equation given to be integrated. 



j ~\r 



To take another instance in which -= , the differen- 



d x 



tial coefficient of the quantity added is not = o or X 

 constant. Let 



in which, by inspection, the solution is easy 



Here M = f + 3 a? 



and j = 2 y = -= 



dy d x 



