418 D'ALEMBERT. 



great men on other matters, and especially as the equa- 

 tion of conditions respecting two variables might very 

 easily have led to the train of reasoning by which this 

 extension of the criterion was found out, the probability 

 is, that Clairaut's discovery was in all respects his own. 



The extreme importance of this criterion to the 

 method of partial differences, only invented, or at least 

 applied, some years later, is obvious. Take a simple 

 case in a differential equation of the first order, 



d z = (Zaxy y*) dx + (ax* 3xy*) dy 

 where M = 2 axy y\ 'N=ax* 



For the criterion -j- = 2 a x 3 



dM rfN 



gives us -j = -j > 



dy dx 



which shows that the equation M.dx + Ndy is a 

 complete differential, and may be integrated. Thus inte- 

 grate (ax 1 3 xy*) dy, as if x were constant, and add 

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

 plete the integral, and we have 



ax*y xy* + X = Z; 

 now differentiate, supposing y constant, and we have 



dz dX 



-j x =(2axy-y) + ^ 



(because of the criterion) = Zaxy y s , 



dX 

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



Accordingly, z = a x 2 y x y* + C ; 



