Prof. Challis on Hydrodynamics. . 217 



at which the fluid flows into the wave, exceeds the rate 



— ^ of flowing out of it by just the quantity that this 



change of the law requires. This result is evidently a conse- 

 quence of the principle of constancy of in ass, which was taken 

 account of at the beginning of the reasoning. Since also for all 

 positions outside the wave/'(r — at o )—0, it follows that for the 

 same positions f{r—at ) is constant, and consequently that the 

 velocity where there is no condensation varies inversely as the 

 square of the distance ; which, again, is consistent with the prin- 

 ciple of constancy of mass. But if f(r— at ) be constant, r — at 

 must also be constant, although r is variable; which is an ab- 

 surdity. We are thus brought to a conclusion which indicates 

 that there is fault or defect in the premises of the reasoning, 

 assuming the l'easoning to be good. This contradiction is of 

 like character with that which is met with in the solution, on the 

 same principles, of the problem of motion perpendicular to a 

 plane, which leads to the conclusion that at the same point of 

 space there may be maximum velocity and no velocity at the 

 same time. Defective premises would account for both con- 

 tradictions. 



I shall now explain in what manner the problem of central 

 motion is solved after the reasoning is supplemented by employ- 

 ing the third general equation. It will be unnecessary to repeat 

 here the argument, indicated in art. 37, by which the expressions 

 for the velocity and condensation obtained after taking that equa- 

 tion into account are shown to be in no respect different from 

 those discussed above, except in having ica in the place of a. Now 

 if the new expressions be such, it might reasonably be supposed 

 that they would conduct to the same contradictions as the others. 

 The sequel of the argument will show why that is not the case. 

 From the principle of constancy of mass, combined with that 

 which is the foundation of the third general equation, namely, 

 that the directions of motion in each element are always normals 

 to a continuous surface, the following equation, cited in art. 10, 

 is deduced : — 



***£+"G*S)-» ■■■•;■« 



Here r and r 1 are the principle radii of curvature of the surface 

 of displacement of an element whose velocity is V and density p 

 at the time t; and, as was shown in the investigation of the 

 equation, the focal lines through which the radii of curvature pass 

 may either be fixed in space, or vary their positions with the 

 time. Let us now introduce into this equation the law of recti- 

 linear axes of motion, and the law of uniform propagation along 



