462 Prof. Challis on the Dispersion of Light. 



we shall obtain 



rf 2 . r<r 9 9 d* . ra flrcat — r + c) 



s? — **-*-*wr> Kac = r — ' 



y_f , (fcat — r-\-c) f(/cat — r + c) 

 r r z 



the propagation being exclusively from the centre. Since in 

 the applications about to be made of these equations the func- 



tion/ will be of the form m sin -—{/cat— r + c), and the value 



A, 



of X is supposed to be extremely large compared to the values 

 of r concerned in the investigation, it is allowable to omit the 



terms containing f' } and to take V to be equal to ^-^-. This 



is, in fact, supposing, as was done in the other solution, the 

 fluid to be incompressible within the small space over which the 

 action of the small sphere is of sensible magnitude, so far at 

 least as regards the effect of that action. It should, however, 

 be noticed that the complete values of a and V prove that the 

 condensation and velocity impressed by the sphere are propagated 

 with the velocity na to an unlimited distance, the fluid being, by 

 supposition, unconfined. We are now prepared to enter upon 

 the second solution of the problem of the vibrating sphere. 



In the first place, as the sphere impresses motion only in 

 directions perpendicular to its surface, the directions of the rec- 

 tilinear axes of the component motions, being determined by 

 that circumstance alone, must be the same. That is, as we 

 have supposed the fluid to comport itself as if it were incom- 

 pressible, the instantaneous lines of motion will be in the direc- 

 tions of the prolonged radii. But it is to be observed that under 

 these circumstances the motion of a given particle is curvilinear. 

 Let T be the velocity of the vibrating sphere. Then, according 

 to the foregoing reasoning, the velocity V of the fluid at any 

 point whose polar coordinates, referred to the moving centre of 

 the sphere, are r and 6, will be wholly in the direction of r, and 



TZ> 2 

 be equal to —j- cos 6, b being the radius of the sphere. Hence 



9 9 da , W- cos dT . 

 Ka 'a¥ + -l^-'Tt= '> 



and by integration, performed necessarily along the line of 

 motion, 



9 , 6 2 cos0 dT 



r at 



No arbitrary function of t is added, because the integral must 



