456 Prof. Challis on the Dispersion of Light. 



following considerations. Let the condensation be <r l at a point 

 for which the value of r is so large as to make the term -A in- 

 sensible, and where, consequently, the effect of the disturbance 

 of the incident waves vanishes. Then 



o-j = <f>(r } t) — fiCegr cos 0. 

 But by the given conditions of the problem, 



Y = aar l = m sin-— (at — x + c), 

 A 



the factor k being unity in the process of investigation now em- 

 ployed, and the coordinate x being reckoned positive in the direc- 

 tion of propagation. Hence, taking the origin of x to be the 

 centre of the sphere, we shall have — #=rcos#; and by what 

 is said above, this coordinate may always be regarded as very 

 small compared to \. Thus, very nearly, 



m . 2-7T . J . Iirm „ 2ir . , . 



0"! = — sin — (at + c)+ — — r cos cr cos— - (at + c). 



a A \CL A 



Comparing this value of o-j with that above, it follows that 



. . A m . 2ir . J N , e lirm 2tt . M . 



(p(r, t) = — sin — (at + c), and — /iC 2 =-r — cos — (at + c), 



a a Ka a 



Hence also 



By substituting the values of -^ and -=- that have been thus 

 obtained, we have 



The integration of these equations gives, regard being had to 

 the preceding value of f x c^ 



U = ( — \— 1 \m cos sin-— (at + c), 



W = (-\ + 1) m sin sin ^ (at + c), 



no arbitrary function of coordinates being added, because U and 

 W are by hypothesis periodic functions of the time. The only 

 remaining condition to be satisfied is that U=0 at every point 

 of the surface of the sphere. If, therefore, h be the radius of 



