Prof. Challis on the Dispersion of Light, 457 



c b 3 

 the sphere, we shall have — * = -s-j ana * thus, by substitution in 



the above equations, the values of U and W are completely de- 

 termined. Also by substituting the values of — >f x c% and — in 



the expression for a, the condensation is given to the same ap- 

 proximation by the equation 



<r=—.< sin— -{at + c)+-l-^ + 2r \ cos 6 cos —(at + c) >. 



We have thus solved the proposed problem on the principles 

 that are usually adopted. 



In order to verify the foregoing reasoning, I shall now show 

 that the problem of the simultaneous movements of a vibrating 

 sphere and the surrounding air may be solved by a like process, 

 and that the solution thus obtained is identical with that of 

 Poisson. The centre of the sphere being assumed to be always 

 on a fixed straight line, let « be its distance from a fixed point 

 of that line at any time t } and let the moving centre be the 

 origin of the coordinates x, y, z of any point of the fluid, the 

 axis of x coinciding with the fixed line. Then 



°"=/i(0 &(*-«, y> *)+/«(') &0*—*, y> *) + &c. 



7 72 



On obtaining from this equation -j- and -p-, terms will arise 



which will have f^t) and -5-, f 2 (t) and -7-, &c. as factors, and 



which will consequently be of the order of the square of the 

 velocity. These being omitted, the differential equation of which 

 <r is the principal variable will be precisely the same as in the 

 foregoing problem, and its integral will be the same ; but the 

 arbitrary quantities introduced by the integration will have to 

 be differently determined. Reverting to the expression for <r, 

 since the condition is to be fulfilled that cr = for all except very 

 small values of r, it follows that </>(r, t) = 0, and c 2 =0. Conse- 

 quently 



f\ c \ a da 2f,c x n 



c = — *V cos 6, -r = -^r- 1 cos 6. 



W = -*J- cos 0, ~^ = -Ajr sm 6. 



Now supposing the velocity of the centre of the sphere to be T, 

 the only condition that remains to be satisfied is that where 

 r — b, U = T cos 0, the fluid in contact with the sphere necessarily 



