496 Prof. Challis on the Dispersion of Light. 



Similarly for the second half of the surface, 



__ — -r— — — - - = — sin 0(1 + a cos 6) +p—r^ sin cos 0. 

 cdd at at dr 



IT 



Hence, by integrating from # = -o to#=7r, 



«V=«V 1+p .^^cos0+-cos^+- 2 - ^ _■ 



and the whole pressure, resolved as before in the direction of inci- 

 dence, is 



ft 2 r flV, c dT/1 a\ £c d 2r n 



8/ 8* 2 '^ 2 J- 



Hence, if the mass of the sphere be A x its volume, A being an 

 unknown constant, and if the sum of the two pressures above 

 obtained be divided by this mass, the acceleration which the 

 waves tend to produce in the direction of their propagation is 



J_ r^T/ 1 _3a\ 3/3 d?T-\ 

 k 2 A ' \dt\ 16/ " 16 ' dt* J ' 



Passing now to the case of a moveable sphere, it is allowable, 

 on the principle of the coexistence of small vibrations, to deter- 

 mine the dynamical effect of a given series of waves by consi- 

 dering, apart from any other motion the sphere may have, that 

 which the waves would produce by themselves. But it is evi- 

 dent that the waves are effective only in proportion to the rela- 

 tive velocity of the fluid and the sphere; that is, x being the 

 coordinate of the centre of the sphere, and regarded positive in 

 the direction of propagation, the effect is proportional to 



„, dx . 277- , dx 



T j-3 or m sin — (bt + c) j- > 



dt X v dt 



the term — x in the expression for T being omitted, because the 



excursions of the sphere are supposed to be extremely small 



about a mean position, which may be taken for the origin of x. 



dx 

 Now -=- f since it depends only on T, will be a periodic function 



dx 

 having the same period as T, and consequently T — j- will be 



similarly periodic. Hence, calling this quantity T', if we sub- 

 stitute T ; for T in the foregoing expression for the accelerative 

 force the sphere being fixed, we shall have the acceleration of the 

 sphere in motion, so far as it is due to the action of the waves. 



Again, as I have argued in the article " On Double Refrac- 

 tion " in the Philosophical Magazine for December 1863 (p. 472), 



