272 Mr. Challis oh the Tlieory of the 



be drawn with certainty from Lagrange's reasoning is, that if 

 a number of particles, constituting a line of fluid, are in motion, 

 the line which bounds the ordinates erected at every point, 

 proportional to the velocities at a given instant, is not necessarily 

 regular. It may consist of portions of continuous curves, con- 

 nected together at their extremities, and be expressed analytically 

 by a function, which possesses no distinctive property of dis- 

 continuity, but changes form abiuptly and in a manner always 

 given by the data of the problem to be solved. But if he 

 had limited himself to this inference, and not supposed the 

 existence, of a new order of functions, he could not have 

 determined the velocity of sound, and must have confessed that 

 the analytical theory had not succeeded in solving that problem. 

 For the demonstration he gives of it, rests altogether on the 

 existence of discontinuous functions, such as they are above 

 defined: and herein it differs entirely from Newton's solution 

 of the same problem, which requires no new projjcrty of 

 curves oi* functions, but deduces the velocity directly from the 

 constitution of the medium: — a method, which certainly at first 

 sight appears the more natural. As, however, we are sure that the 

 velocity of the propagation of sound, must be a deduction from 

 the principles on which the analytical investigation is founded, 

 if no other mode of making the deduction can be thought of, 

 we must be content to take up with discontinuous functions. 

 No person can object to them who does not suiiiily an equiva- 

 lent, provided always they be considered in the present state 

 of analytic science, not as demonstrated to exist, but as hypo- 

 thetical, and like all hypotheses, established only by the extent 

 and success of their applications. It was necessary to premise 

 so much as this about discontinuous functions, in order to give 

 a reason why any one, who treats of the vibrations of an elastic 

 medium, has a right, if he can, to leave these functions out of 



