78 Mr. Gwyther on a 



outside the sphere then p ><r, and we need only consider 

 the first coefficient and the wave motion will be reproduced 



provided 



S(-i)X-0. 



. X 

 / = -. 



4 



If we consider a point within the sphere, so long as p is 

 a quantity of the same order of magnitude as c, we have no 

 backward wave since 2(— i)'X = ^- 



If we consider a point near the origin, so that p is small 

 compared with c, but bears a finite ratio to A, so that we 

 may neglect x-\-c in comparison with x, but retain it 

 ( = pcos0) when compared with X, we thus get 



^1= -(-I)"2^„0'" + 1COS"' + 10 



//2 = ( - \Y\m + \){m + 2)^„p"'cos'"0. 



v,= - (- ir ^-^'^«p"^-^cos--ia 



But the conditions which we have previously obtained 

 ensure that each of these will vanish. 



The sole condition which we find then to ensure that 

 the system of secondary waves, of which we have previously 

 obtained the form, should reproduce a spherical wave 

 is 2(— i)X = 0, provided that \lc should be a very small 

 quantity, which is a condition essential to the very con- 

 sideration of the matter. 



Here it will be noticed that the coefficients of S(- i)X» 

 at a point near the origin are the coefficients of the cor- 

 responding terms in the originally used expansion. These 

 now appear as deduced from the first term of that expan- 

 sion, and are independent of n and do not depend upon the 

 method by which the expansions were formed, but appear 

 solely as a result of integration. 



If the wave were travelling inwards towards the centre, 

 wc should find the solution reproduced in its entirety as 

 j> approaches the value X. 



