408 D'ALEMBERT. 



So z = x 2/ 2 -|- X, and differentiating with respect to x 



d x d x 



Hence X = x 3 4. C 



and z = # ?/ 2 + z? + 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 



d y d x 



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 only true in theory : 

 we cannot resolve the general equation by any such 

 means ; for that gives us 



M d N\ ^ d F , f d F 

 y d x ) ' d x d y 



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 quite unaccountable* that 

 Clairaut should, in reference to his equation, which is 

 substantially the same with the above, describe it as 

 " d'une grande utilite, pour trouver p " (that is F). 



* Mem. del'Acad. 1740, p. 299. I find my surprise shared by a very 

 learned mathematician to whom I had mentioned it. 



