458 Prof. Challis on the Dispersion of Light. 



moving in the normal direction with the same velocity as the 

 points of the sphere with which it is in contact. Hence 



-^cosflj/^Tcostf. 



By means of this equation it will readily be found that 



TT Tb* ' w Tb 3 . A 9 dT & a 

 U = _cos<9, W=^sin^ aV=-^-.^cos<9. 



As these results agree exactly with those of Poisson's solution, 

 they may be considered to prove that the principles of the two 

 solutions are the same, although the mathematical processes are 

 considerably different. It may also be concluded that the case 

 of waves impinging on a sphere at rest is treated above in exact 

 accordance with the reasoning usually adopted in hydrodynamical 

 questions. Waiving for the present the statement of the diffi- 

 culties presented by these results, I shall now only direct atten- 

 tion to two inferences which may be drawn from them. First, 

 the solution of the first problem might be deduced from that of 

 the other, so far at least as regards the velocity of the fluid, by 

 conceiving a velocity equal to that of the vibrating sphere to be 

 impressed at each instant on the sphere and on the fluid in the 

 contrary direction, so as to reduce the sphere to rest. Secondly, 

 the velocity of the fluid in contact with the fixed sphere, where 

 it passes the plane through the centre of the sphere perpendicular 

 to the direction of the incidence of the waves, is to the velocity 

 in the undisturbed wave as 3 to 2. Hence it will be found, by 

 calculating according to the above law of the variation of W in- 

 versely as r 3 , that the whole quantity of fluid which passes that 

 plane is just as much as would have passed it if the waves had 

 been undisturbed, and that thus the mean quantity which at each 

 instant is diverted in the contrary direction is zero. So also the 

 mean quantity of the fluid which the vibrating sphere pushes 

 or draws at each instant in the direction of its motion is zero. 

 I proceed now to apply to the same two problems the hydrody- 

 namical principles which I long since enunciated in this Journal. 



After expressing the law that the lines of motion are normals 

 to a continuous surface by the equation 



u v "W 



(d&A = — dx + - dy + — dz, 

 A. A. A 



an equation necessary for determining the unknown function X 

 was obtained on the principle that that law of continuity holds 

 good for any given particle in successive instants, or, that the 

 motion conforms to the law at all points and at all times. In 

 like manner the equation is obtained which expresses that the 



