D'ALEMBERT. 415 



integration that they appeared to set at defiance the 

 utmost resources of the calculus. When a close and 

 rigorous inspection shewed no daylight, when expe- 

 riments of substitution and transformation 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 inte- 

 grable either entirely, or by circular arches, or by loga- 

 rithms, or by series. D'Alernbert, in all probability, drew 

 his new method of treating the subject from the considera- 

 tion that, in the process of differentiation we successively 

 assume one quantity only to be variable and the rest con- 

 stant, and we differentiate with reference to that one vari- 

 able ; so that x d y + y d x is the differential of x y, a 

 rectangle, and xydz + xzdy + yzdx the differential of 

 xyz,& parallelepiped, and so of second differences, d 2 z 

 being (when z = x m ) = (m*-m) x m -* dx* + m x~ 1 d* x. 

 He probably conceived 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 constant, he might succeed in going a certain 

 length, and then discover the residue by supposing an un- 

 known function of the variable which had been assumed 

 constant, 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 neces- 

 sary part of it consisted of the equations of con- 



D'Alembert's solution led to an equation of partial differences of 

 this form f^L?\ = a 2 f^Ci in which t is the time of the vibra- 



tion, x and y the co-ordinates of the curve formed by the vibration. 



