152 Dr. T. J. Fa. Bromwich 



on 



If now Y n -i is a solution of (3*42) with (n — 1) replacing n, 

 and if we write 



Vdr r) 



it will be seen that Gr satisfies (3*42) as it stands. We can 

 therefore take T? n = — G. 



The equation (3*44) can be written 



r n+i - r fr y r n j-y r ^ r ) y rn -i y y° ^) 



and so we arrive at the formula 



£.-(-&)'(?) <™> 



where F satisfies the simple wave-equation 

 d 2 F 1 d 2 F 



a*-* d 2 a** 



(3-47) 



given by writing n = in (3*42). As is well known, the 

 solution of (3* 47) is given by 



¥ =f{c l t r -'r)-Vg[c 1 t + r), . . (3'48) 



where the functions /, g are arbitrary. 

 Thus finally we may write 



In problems involving divergent waves only, the function g 

 will be zero ; and if the origin r = falls within the space 

 considered, we must take 



9(S) = -f(S) (3-491) 



in order that F» may be continuous at r = 0. 



It will be convenient sometimes to have a formula for F n , 

 expressed explicitly in terms ol the differential coefficients 

 of the arbitrary functions ; such a formula is 



n(n + l) , , (n-l>(w+l)(n + 2) - 

 J« — /»i ^— / ?J _ii 2 . 4 . r 2 ^ n ~ 2 



M)...(n + 3) / 1.2...2w \ 1 



+ — 2 .4 . 6 . f*— r *-* + ~ ' + \2 . 4 . . .2nJ ^ /o ' (3 492) 



We have taken g = 0, and we write /^, # /„_i, . . . , f x to 

 denote the nth., (w — l)th ... first differential coefficients 



