CALCULUS OF PAETIAL DIFFERENCES. 47 



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 

 isF). 



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

 apparently, first of all the geometricians, given the criterion 

 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 - 



d X 



stand for the differential coefficient of y when x only varies, 



and - for the same differential coefficient when y only 

 dy 



varies. Hence he takes ~ x dx, not, as in the ordinary 

 d x 



notation it would be, = city, the complete differential of (f> ; 

 whereas that differential would, in this solution, be 



dq> dtp 



- 1 - x dx + -/- X dy. 



dx dy 



Thus, if = xy*, its complete dif. d(j> = 2yxdy + y*dx, but 



df 



= y . 

 dx 



It is quite clear, therefore, that Fontaine gave the notation of 

 this calculus. 



But D'Alembert had been anticipated in the method itself, 

 as well as in the notation or algorithm ; for Euler, in a paper 

 entitled ' Investigatio functionum ex data differential ium con- 

 ditione,' dated 1734,t integrated an equation of partial differ- 



* Mem. de 1'Acad. 1740, p. 299. I find my surprise shared by a very 

 learned mathematician to whom I had mentioned it, Prof. Heaviside. 



f Petersburg!! Memoirs,' vol. vii. That Euler, in the Memoir pub- 

 lished in 1734, solved an equation of Partial Differences is quite incon- 



