D'ALEMBERT. 419 



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

 y being both variable, we have 



dz = Zaxydx + ax* dy 3xy* dy y* dx 

 = (Zaxy ?/ B ) dx + (a x 2 3 xy*) dy 



which was the equation given to be integrated. 



To take another instance in which -, , the differ- 



d x 



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

 constant. Let 



2 xy dy 

 in which, by inspection, the solution is easy 



z = xy* + x 3 + 

 Here M = * + 3 x* N = 2x 



and = y = -T- 



dy dx 



So z = x y z 4- X, and differentiating with respect to x 

 dz dX 



Hence X = x 3 4- C 



and z = xy* + x* + 0, 



the integral of the equation proposed. 



It must, however, be observed of the criterion, that an 

 equation may be integrable which does not answer the 



condition 



^M rfN 

 dy " dx 



It may be possible to separate the variables and 

 obtain X dx Y dy, as by transformation; or to find a 

 factor, which, multiplying the equation, shall render it 



* 2 E 2 



