404 D'ALEMBERT. 



to be of so difficult integration that they appeared to 

 set at defiance the utmost resources of the calculus. 

 When a close and rigorous inspection showed no day- 

 light, when experiments of substitution and transfor- 

 mation failed, the only resource which seemed to remain 

 was finding factors which might, by multiplying each 

 side of the equation, complete the differential, and so 

 make it integrable either entirely, or by circular arches, 

 or by logarithms, or by series. D'Alembert, in all pro- 

 bability, drew his new method of treating the subject 

 from the consideration that, in the process of differen- 

 tiation we successively assume one quantity only to be 

 variable and the rest constant, and we differentiate with 

 reference to that one variable ; so that x d y -f y d xis 

 the differential of x y, a rectangle, and xydz-j-xzdy 

 \-yzdx the differential of x y z, a parallelopiped, and 

 so of second differences, d 2 z being (when z = # m ) = 

 (m 2 m) # m ~ 2 dx 2 -f m x m ~ l d 2 x. He probably con- 

 ceived from hence that by reversing the operation and 

 partially integrating, that is, integrating as if one only 

 of the variables were such, and the others were con- 

 stant, he might succeed in going a certain length, and 

 then discover the residue by supposing an unknown 

 function of the variable which had been assumed con- 

 stant, to be added, and afterwards ascertaining that 

 function by attending to the other conditions of the 

 question. This method is called that of partial dif- 

 ferences. Lacroix justly observes that it would be more 

 correct to say partial differentials; and a necessary part 

 of it consisted of the equations of conditions, which other 

 geometricians unfolded more fully than the inventor 

 of the calculus himself; that is to say, statements of 

 the relation which must subsist between the variables 

 or rather the differentials of these variables, in order 

 that there may be a possibility of finding the integral 

 by the method of partial differences. It appears that 

 Fontaine,* a geometrician of the greatest genius, gave 

 * Euler had so high an opinion of Fontaine, that in 1751 he told La- 



