492 Prof. Challis on the Dispersion of Light, 



But the conditions of the problem are wholly different if, as 

 I assume to be the case, the sphere vibrates in fluid of unlimited 

 extent, and impresses disturbances upon it which are propagated 

 indefinitely into space. By applying to this case the hydro- 

 dynamical principles on which our reasoning is now proceeding, 

 U and W are found to have a known relation to each other, by 

 introducing which into the equations (8) and (y) and integrating, 

 the motion of the fluid may be completely determined. I do 

 not produce the details of this reasoning here, because they are 

 contained in the former paper (pages 462 and 463), and have 

 been given in several other communications. Tt results from 

 the solution of this problem that the relative velocity of the fluid 

 in contact with the sphere is T sin 0, — T being the velocity of 

 the centre of the sphere, and the angle 6 being reckoned from 

 the axis of x on the negative side. This value, however, implies 

 that terms involving the small ratio of the radius of the sphere 

 to X have been neglected, or that the fluid comports itself as if 

 it were incompressible. 



Suppose now the velocity T to be impressed at each instant 

 oji the sphere and the fluid, so that the sphere will be reduced 

 to rest. The actual velocity of the fluid along its surface will 

 then be T sin 6. But on this supposition the fluid through 

 its whole extent moves with the velocity T, and, excepting so far 

 as condensation is produced by impact on the sphere, all its parts 

 have the same density as in the state of rest. To pass from 

 this case to that of waves of variable density impinging on the 

 sphere, it is sufficient for a first approximation, considering the 

 small ratio of the radius of the sphere to the breadth of the 

 waves, to substitute a — a l for <r in the equation which gives 

 the condensation in the first case, a } being the condensation at 

 any instant of the incident wave at the points of incidence. 

 Through the small extent of the hemispherical surface on which 

 the waves impinge, o-j may be regarded as uniform, without 

 omitting quantities more significant than those already neglected. 



But for the determination of the motion beyond the plane yz 

 passing through the centre of the sphere other considerations 

 are necessary, because, on account of the varying density of the 

 incident wave, the impressed velocity is there altered by the 

 mutual action of the parts of the fluid, and the effect of such 

 action can be ascertained only by the solution of a partial dif- 

 ferential equation. The course of the reasoning now requires 

 an investigation of the amount of this modification of the im- 

 pressed velocity. 



Since the equation (ft) is perfectly general for motion symme- 

 trical with respect to an axis, and applies to all points of the 

 fluid at all times, it will be true if we pass from one point to 



