296 Mr. Challis on the Theory of the 



instance, to the action of a solid on the fluid. It must be 

 remembered, that the proper proof of the discontinuity of the 

 motion consists in the circumstance, that the motion is to be 

 determined by the integration of a partial differential equation, 

 the peculiar property of which is, that it may be satisfied by 

 giving a series of values, linked together by no law of con- 

 tinuity, to that variable which is considered a function of the 

 other two. The demonstration of this property must be con- 

 ducted by pure analysis. It has been given by Lagrange ; for, 

 as I believe, the reasoning on this point, in the second volume 

 of the Misc. Taur. may be abstracted from the physical question 

 with which it is involved. 



11. Lastly, we have to consider what happens when the 

 diaphragm goes on moving uniformly with the velocity with 

 which it was made to commence. Take ac (Fig. 7.) to repre- 

 sent the impressed velocity ; af, the distance over which its 

 influence is felt at the first instant; and join cf. The ordinates 

 of this line, as has been sliewn, will represent the velocities 

 initially impressed on the particles included in af. Bisect «c in 

 e, take ae in ca produced equal to ae, and join ef, ef. The 

 motion in the first instant will be the same as if two waves, 

 designated by ef, ef, the former condensed, the latter rarefied, 

 were moving simultaneously in opposite directions, ef will be 

 moving in the direction «/, e/ in the contrary direction, and the 

 velocities of the particles at the same distance from a, will be 

 the same and in the same direction in both, so that the line 

 cf will represent the .compound velocities. As the motion pro- 

 ceeds, the wave e/ will be reflected, and ef will go on in its 

 original direction. Also the diaphragm will generate conden- 

 sations proportional to its velocity. Hence the motion after a 

 given time may be represented in the following manner. Let t 

 be the given time. Take aa' (Fig. 8.) the .space over which the 



