spherical Wave of LigJit. 8i 



all divided by r^+\ where I have omitted the terms relating 

 to V which are identical in form with those retained. 



We may now argue that if for any reason the values of 

 7t and of the coefficients a,^ and b,^ are determinate in the 

 expressions for the displacement, we shall have the same 

 determinancy about the coefficients in the expressions for 

 the rotation which is a vector of identical form. Or we may 

 form the expressions for the secondary wave representing 

 the original rotation, which ought to agree with that just 

 written not only when the integration is performed over 

 the whole surface of the sphere, but if extended over any 

 portion of it. 



By either line of reasoning we should conclude that 



\i p be the least value of ii, we get 

 and finally 





Or 



fp+1 



0, or 2(-i)X = 0. 



From this we conclude that if 2 is the least value of n 

 h = 0, 

 as we have said before it is reasonable to expect, and of 

 which a proof will be offered in the next section. 



§ 5. 

 We have as yet considered a solution of the fundamental 

 differential equation which proceeds by powers of X/r and 

 is convergent if ?^>X, but which is not only infinite when 

 ;'=0, but is also still a function of the direction of the 

 vanishing vector. It is, however, evident that there is also 

 a solution in ascending powers of r/X, which will not be a 

 terminating series : these two series must give us no dis- 

 continuity when r=\. 



