spherical Wave of Light. jj 



Referring to (3) we see that we are considering 



4 = 



^={,:^2--^ 8/V-+^ ^^^— + etc.}sin/(/;/-r) 



{(j?i^i)(7n + 2)x"'z 1 



- y ,^V-+^ etc.|cos/(^/- .) 



Writing V1.U2.V3, etc., for the coefficients which we get 

 when we take these into account, we get 

 Vi = - 2a^x'\x + cf'+'^jr". 

 U2 = 2{?ian + Cn - d„)x"-'^{x + c)"'^'^lcr" 



- {m + i)(;// + 2)a,,x"-\x + cy"'/r'"~^ etc. 



Where we see that in such coefficient we get a part 

 independent of r, and that the coefficients of such terms in 

 order differ only from the coefficients in the corresponding 

 terms of the original expansion by a constant factor. It 

 will be noticed that these coefficients are obtained by 

 integration over the sphere merely, and are quite indepen- 

 dent of the special form of expansion which I have used in 

 the secondary wave. The use of that special form of 

 expansion has been to cause the disappearance of all terms 

 from the integral which might have appeared affected by 

 A/r, and which would have been suitable only in the 

 immediate neighbourhood of the sphere of radius c. We 

 might, in fact, take any constant coefficients in an expansion 

 of the same form, and forming the conditions for the dis- 

 appearance of the terms in X/r, etc., infer from their linear 

 character that the form of expansion must be that which I 

 have used. 



Also as we shall see later the coefficients independent 

 of c in V1.U2, etc., can be inferred from the first term of the 

 expanded expression for the original displacement, thus 

 giving additional evidence of the propriety of the form of 

 expansion used. 



Proceeding now to the consideration of the actual value 

 of the integral displacement, — if we consider first a point 

 F 



