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' J dx 

 = (Zaxy y z ] dx + (a x l 3 xy^ dy 



which was the equation given to be integrated. 



To take another instance in which j , the differ- 



a jc 



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

 constant. Let 



dz=y*dx + 3x*dx + 2 x y dy 

 in which, by inspection, the solution is easy 



z = xy* + .r 3 + C 

 Here M = /* + 3 x* N = 2x.y 



and 



dy dx 



So z x y* + X, and differentiating with respect to 

 dz dX 



Hence X = sc 3 + C 



and z = xy* + x* + C, 



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 



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 



