466 Prof. Challis on the Dispersion of Light. 



Consequently, by substituting in the foregoing equation, 



/ 2 2 da du\ u , ( o <2 da dv\v / da t dw\ w 



As this equation has been obtained by perfectly general consi- 

 derations, it is applicable to any instance of small motions. In 

 the case of symmetrical motion relative to an axis, produced by a 

 vibrating sphere, as also in that of -the like motion due to the 

 reaction of the first hemispherical surface of the fixed sphere, it 

 was found that the motion was wholly in lines normal to the sur- 

 face of the sphere, and the dynamical equation employed is that 

 which the above equation becomes when the condition of central 

 motion is satisfied. But in the present question there is no other 

 condition than that the motion is symmetrical about an axis ; in 

 which case a, u } v, w are functions of the polar coordinates r and 

 0. Let u and co be respectively the velocities parallel and per- 

 pendicular to the axis, so that co 2 = v 2 + w' 2 , and - = — . Taking 



these equations into account, and transforming into polar coor- 

 dinates, the last equation becomes 



V U" dd r ) + dtj V 



} 22 (da . a da cos 0\ dco\ co 



Since no general relation exists between u and co other than that 

 which results from the mutual action of the parts of the fluid, 

 and since to find this relation is precisely what the analysis is 

 required to do, we must equate separately to zero the multipliers 

 of u and co in brackets. By combining the two equations thus 

 obtained with the equation of constancy of mass, which, expressed 

 in the same coordinates, is 



da du n du sin 6 dco . n dco cos co 



— ■ + -J-COS0— -jz h^-sm^+^^ 1- . ^ =0, 



dt dr dd r dr dd r rsmd ' 



there results 



1 d 2 .ar d 2 .ar 1 (d 2 .ar d.ar \ 



+ -* \ — 77^ H T7T COt U I . 



kV dt 2 dr* ' r 2 V d6 2 ' dO 



This equation, after substituting — n 2 a for -j-^-, differs in no 



respect from the one used in the first solution of the problem, 

 excepting in having K 2 a 2 in the place of a 2 . It applies, however, 



