CALCULUS OF PARTIAL DIFFERENCES. 43 



by mathematicians.* In all these inquiries the differential 

 equations which resulted from a geometrical examination of 

 the conditions of any problem, proved to be of so difficult 

 integration that they appeared to set at defiance the utmost 

 resources of the calculus. When a close and rigorous inspec- 

 tion showed no daylight, when experiments 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 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 differentiation 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 dy + y d x is the differential of x y, 

 a rectangle, and x y d z -\- xzdy + yzdx the differential of 

 x y z, a parallelepiped, and so of second differences, d? z being 

 (when z = x m ) = (m 2 in) x m ~ s dx* + m x m ~ l d' 2 x. He pro- 

 bably 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 unknown function of the variable 

 which had been assumed constant, to be added, and after- 



* Taylor (' Methodus Incrementorum ') had solved the problem of the 

 vibrating cord's movement, but upon three assumptions that it departs 

 very little from the axis or from a straight line, that all its points come to 

 the axis at the same moment, and that it is of a uniform thickness in ite 

 whole length. D'Alembert's solution only requires the last and the first 

 supposition, rejecting the second. The first, indeed, is near the truth, 

 and it is absolutely necessary to render the problem soluble at all. The 

 third has been rejected by both Euler and Daniel Bernoulli, in several 

 cases investigated by them. D'Alembert's solution led to an equation of 



partial differences of this form (-^-f ) = o 2 (-=-?) in which t is the time 



\dt 2 / \dx 2 / 



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

 vibration. 



