Prof. Challis on the Dispersion of Light. 465 



sider the following argument to be a step in advance of the re- 

 searches that I have hitherto published upon it, and at least to 

 suffice for the application I have in view at present. If the usual 

 treatment of hydrodynamical questions be legitimate, the problem 

 now under consideration presents no more difficulty than that of 

 the vibrating sphere, the analysis to the first order of small quan- 

 tities being precisely the same for the one as for the other. But 

 that this cannot be true is evident from the consideration that 

 the action on the fluid of that hemispherical surface of the fixed 

 sphere on which the waves are incident differs essentially from 

 that of the other, inasmuch as the latter does not directly impress 

 any velocity on the fluid compelled to move in contact with it. 

 The only arbitrary conditions which the part of the motion now 

 under consideration is required to satisfy, are the compulsory 

 movement along the surface of the sphere, and the state of the 

 fluid as to velocity and condensation in the transverse plane 

 passing through the centre of the sphere. By the results ob- 

 tained above relative to the action of the first hemispherical sur- 

 face the velocity is T, and the condensation a l} at every point of 

 that plane at all times. In other respects the motion is deter- 

 mined by the mutual action of the parts of the fluid. 



Let us now conceive the whole motion to consist of motions in 

 an unlimited number of very slender tubular spaces, varying in 

 position and transverse section, but always so that the axis of the 

 tube coincides with the direction of the motion of the particles 

 through which it passes. Let V be the velocity and a the con- 

 densation at any point of one of these tubes at any time t. Then, 

 from what has been already shown, 



Ka ' ds + dt U > 



the line s being reckoned along the axis of the tubular space, 

 and da in brackets signifying that the variation is from point to 

 point of that line. Now since a must be regarded as a function 

 of x, y, z, and /, we have 



(da) _ da dx da dxj da dz 



ds dx ds dy ds dz ds 

 Also 



dx _ u dy _ v dz __ w 



ds~~T' ds~~T Js~Y' 



and since V 2 =--w 2 + v 2 + mA 



dY _u du v dv u dw 



T dt~y'di Jr Y'lt + Y'lti' 



Phil Mag. S. 4. Vol. 27. No. 184. June 1864. 2 H 



