46 DYNAMICAL PBINCTPLE. 



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

 being both variable, we have 



dz = 2axydx + ax*dy 3xy*dy y 3 dx 



= (2axy y a ~)dx + (ax* 3xy^dy ; 

 which was the equation given to be integrated. 



To take another instance in which - , the differential co- 



dx 



efficient of the quantity added is not = o or X constant. Let 



dz = y* dx -\- 3 x* dx + 2 xy dy, 

 in which, by inspection, the solution is easy 



z = xy* -f- x a -|- C. 

 Here M = y + 3 x\ N = 2 xy, 



and = 2y = - . 



ay dx 



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



dz c?X 



= f + = w 2 + 3a; 8 . 



^;r rfa; 



Hence X = x 3 + C, 



and 2 = a??/ 8 + x 3 + C, 



the integral of the equation proposed. 



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

 tion may be integrable which does not answer the condition 



dy dx ' 



It may be possible to separate the variables and obtain 

 X d x = Y d y, as by transformation ; or to find a factor, 

 which, multiplying the equation, shall render it integrable, 

 by bringing it within that condition. The latter process is 

 the most hopeful ; and it is generally affirmed that such a 

 factor, F, may always be found for every equation of the first 

 order involving only two variables. However, this is onl}' 

 true in theory : we cannot resolve the general equation by 

 any such means ; for that gives us 



F.f ^W.1I_M^, 

 \dy dx J dx dy 



