491 Prof. Challis on the Dispersion of Light. 



which, as might have been anticipated, is the equation (8) dif- 

 ferentiated with respect to 0. By a like process we might have 

 obtained the equation resulting from the differentiation of (8) 

 with respect to r ; but the above equation suffices for the present 

 purpose. I proceed now to obtain a particular and exact 

 solution of it. 



Let it be assumed that qr = (j> sin 6 cos 6, and that <fi is a 

 function of r and t only. Then, by substituting this value of qr 

 in the equation, the result is 



which accords with the assumption that (j) is a function of r and t. 

 This differential equation, which admits of exact integration, 

 gives for <f> an expression containing arbitrary functions of 

 r — /cat and r + feat. Excluding the function of r-\-fcat, which is 

 inapplicable to the question, it will be found that 



do- /f(r—fcat) f(r—/cat) , f"(r — fcat)\ . a n 



But if W, be the part of W which depends only on the mutual 



action of the fluid elements, from what is said above the left- 



1 dW 

 hand side of this equation is equal to ^ • —rr* Hence, by 



K ft CLZ 



integration, 



no arbitrary function of r and 6 being added, because by the 

 conditions of the problem W x is a periodic function of the time. 

 The preceding investigation would suffice for finding the corre- 

 sponding part of U, and the condensation at any point due to the 

 mutual action of the parts of the fluid. But as the sequel of 

 the reasoning requires only the knowledge of the velocity and 

 condensation at the surface of the sphere, this inquiry may be 

 omitted. It may also be here remarked that as the form of the 

 function / is determined by the given expression for the velocity 

 T of the incident waves, and as the conditions that W 1 = where 



0— 2 and = 7r are satisfied, all the conditions of the problem 



are definitely satisfied by the foregoing value of W v which may 

 therefore be considered as the only solution of the equation (e) 

 appropriate to the question. 



By supposing r in the general expression for W, to be equal 

 to c the radius of the sphere, we obtain the velocity along the 



