51*" Prof. Challis oti some Points relating to 



consequently be called the equation of continuity, while the 

 equation usually so named may with more propriety be called 

 the equation of constancy of mass, with reference to the prin- 

 ciple on which it is based. 



To the equations (1.), (2.), (3.), and (4.) are to be added 

 the two following : - 



the fluid being supposed to be acted upon by no impressed 

 forces. When the relation between the pressure p and density 

 p is given, these six equations serve to determine the six un- 

 known quantities, \|/, X, ?/, t;, w and p. 



The equation (5.) is equivalent to the following : 



dp dp dx dp dy dp dz /du ,dv dwX _ 

 dt dx' dt dy' dt ds' dt '^Xdx dy dz/~ 



Hence, i^u^ + v^ + ^'^-V^ and ds=\dt, 



dp TT dp (du dv d'm\ 



But by what is proved in the Cambridge Philosophical Trans- 

 actions (vol. vii. part 3. p. 385, 386), where, however, it is 

 proper to remark, the use of equation (4.) is not absolutely 

 necessary, we have 



du dv dw _ dV ,y{^ -4- M 

 dx dy dz ~ ds \r r^J* 



r and r^ being the principal radii of curvature at the point 

 xyz, of the surface which cuts the directions of motion at 

 right angles. Hence, by substitution, 



This equation has also been derived from elementary consi- 

 derations in the memoir above cited (p. 387), By whatever 

 process it be obtained, it involves the principle expressed ana- 

 lytically by the equation (4.), viz. that the directions of motion 

 are normals to surfaces of continued curvature. It cannot, 

 therefore, be identical with any equation which does not in- 

 volve the same principle, or does not contain exiMcitly the 

 radii of curvature r and r^. An application of the equation 



