Prof. Challis on the Dispersion of Light,. 493 



another contiguous point by substituting 6 + h6 for 6, while r 

 and t are constant. Hence the equation obtained by differen- 

 tiating (ft) with respect to 9 only will be true. We shall thus get 



V a dedr + dedt) u + \ Ka dr + dt) de 



For the disturbance of the fluid produced by the first hemi- 

 spherical surface, the relation between U and W and also that 



, . dV , dW , , Al . 



between -^- and -7-5- are known, and this equation is con- 

 sequently not required for determining that part of the motion. 

 But for the remainder of the motion, U and W, as we have 



argued above, and by consequence -j^ and —^, are related to 



each other in part by modes of fluid action ascertainable only 

 by integration. To find such relation between the latter two 

 quantities, we must assume, according to a principle already 

 applied, that 



2 2 da dV , 2 2 da dW 



Ka -Tr + iit =0 ' m * Ka -m + nr =0 - 



Also, as consequences of these equations, we shall have 

 22 d*v , d*V , 2S d*ad*W 



Ka dWr + dM = ' and Ka 7W + d0dt = °- 



The foregoing equation is therefore satisfied on this assump- 

 tion. But it is particularly to be remarked that this process 

 implies that the relation between U and W does not depend 

 wholly on the mutual action of the parts of the fluid. For if 

 that were so, the two latter equations would not be required, 

 and we should have the kind of motion which was previously 

 shown to be inconsistent with the conditions of the present 

 problem. Consequently the four equations will serve to deter- 

 mine only those parts of U and W of which the relation is not 

 otherwise assignable ; and it is clear that, since the original 

 equations are linear with constant coefficients, this portion of 

 the motion' may be determined independently of the rest. 



a a r <* U ^ V d *V i <* 2W u e*. 



Alter eliminating -=- > — — > — 7rri and —tttt, by means of the 

 D dt dt dOdt dddt J 



four equations above and the equation (7), and substituting q 



M 



for 



id the 



result is 









1 



c? 2 . qr 

 1 dt* 



d*.qr 



' dr* 



4 



d 



qr 



9-2 



