Prof. Challis on the Vibrations of an Elastic Fluid. 361 



maintain, that to obtain true and consistent results in hydro- 

 dynamics another general equation, distinct from those com- 

 monly recognised, is absolutely necessary. In the foregoing 

 investigation no reference has been made to that equation, and 

 the reasoning is on that account defective. Before stating the 

 modification which the reasoning must undergo, I will briefly 

 advert to the evidence for the new equation which I have 

 given in a communication to the Philosophical Magazine for 

 May 1845 (p. 4-25). I have there obtained, by elementary 

 considerations which admit of no dispute, the following equa- 

 tion, 



l*^+*(£tffh* • • • (L) 



in which p is the density and V the velocity at the time /, at 

 a point where the principal radii of curvature of the surface 

 cutting at right angles the directions of motion are R and II'; 

 and ds is the increment of a line drawn always in the direction 

 of the motion of the particles through which it passes. This 

 equation being true and general^ is also necessary in the solu- 

 tion of every hydrodynamical problem. It is not identical 

 with the recognised equation, 



dp d.pu d.pv d.pta 



Tt+ ~dJ- + ~df + -dT~ ~°> ' ' ' <"•> 



because it rests on two principles, that of constancy of mass, 

 and the principle that the directions of motion are normals to 

 a surface of continued curvature; whilst equation (II.) rests 

 only on the principle of the constancy of mass. As however 

 equation (II.) is true, there must exist another equation, which, 

 combined with it, conducts to (I.). That additional equation 

 is the one I contend for. I derive it from the single principle, 

 that the directions of motion in a given element are at each 

 instant normals to a surface of continued curvature, — a con- 

 dition necessary for the continuity of the motion, — and I have 

 shown how, by its combination with (II.), the equation (I.) 

 is arrived at (see Camb. Phil. Trans., vol. vii. part 3, p. 385 ; 

 and Phil. Mag., S. 3, vol. xx. April 1842). Lastly, in a par- 

 ticular instance I have obtained an absurd result by employing 

 legitimately only the commonly received equations (Phil. 

 Mag. for May 1845, p. 429). The evidence for the necessity 

 of the new equation seems therefore to be in every respect 

 complete. I am not, however, at present aware of any other 

 use required to be made of it than that of deducing equation 

 (I.). That deduction being once made, it follows that in every 

 hydrodynamical problem equation (I.) must supersede equa- 

 tion (II.). 



Phil. Mas. S. 3. Vol. 33. No. 223. Nov. 1848. 2 B 



