CALCULUS OF PAETIAL DIFFERENCES. 45 



to the proposition, and from which he says that Clairaut 

 derived his equation of condition to differentials involving 

 three variables. It is possible ; but as this never was men- 

 tioned in Clairaut's lifetime, although there existed a sharp 

 controversy between these two great men on other matters, 

 and especially as the equation of conditions respecting two 

 variables might very easily have led to the train of reasoning 

 by which this extension of the criterion was found ont, 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, 



dz =. (2axy y^dx + (ax* 3xy*)dy, 

 where M = 2axy y a , N = ax 9 3xy*. 



For the criterion -^- = 2 ax 3 if. 

 dy 



dx 



which shows that the equation Mdor + Ndyis a complete 

 differential, and may be integrated. Thus integrate (a x z 3 x ?/ 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 xy 3 + X = Z ; 



now differentiate, supposing y constant, and we have 



dz dX 



- = (2a^-,3 ) + _ 



(because of the criterion) = laxy y 3 , 



J -*T 



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

 Accordingly, z = ax^y - xy 3 + C ; 



