420 D'ALEMBERT. 



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 only true in theory: we 

 cannot resolve the general equation by any such means; 

 for that gives us 



r M 



= N. - M - 



-j = . -y - 

 ax/ ax ay 



an expression as impossible to disentangle, it may safely 

 be asserted, as any for the resolution of which its aid 

 might be wanted. It is only in a few instances of the 

 values of these functions (M and N) that we can succeed 

 in finding F. 



It is here to be observed, that not only Fontaine had, 

 apparently, first of all the geometricians, given the crite- 

 rion of integrability, but he had also given the notation 

 which was afterwards adopted for the calculus of Partial 

 Differences. </> being a function of two variables, x and 



y, he makes 2 stand for the differential coefficient of 

 dx 



6 when x only varies, and ^ for the same differential 



y d6 



coefficient when y only varies. Hence he takes - x d x, 



dx 



not, as in the ordinary notation it would be, d^ the 

 complete differential of <; whereas that differential 

 would, in this solution, be 



d 6 7 , d cf> 



__r x dx + L x dy 



dx dy 



