;_' MR. A. B. BASSET ON THE MOTION OF 



where A is a constant, which, together with the function F (a), must be determined 

 so as to satisfy the conditions of the problem. 



14. Though I am convinced that a solution of the problem exists in the form of a 

 definite integral, I have not succeeded in obtaining it ; and therefore subjoin a 

 solution of a different character. 



Let S(r) denote the spherical function d(r~ l amr)/dr ; then a solution of (43), 

 subject to the condition of finiteness at the origin, is 



v = 2A x e- xl "S (\r) + wa, ........ (62) 



when r =a, v =o>a for all values of t, whence 



S(Xa) = 0, . ,. ,,, . ...... (63) 



and the different values of X are the roots of (63). 

 Initially v = 0, whence 



<oa = - 2A X S (Xr). 



Let X and p be different roots of (63), and let T = S(^tr), then, since S(Xr) satisfies 

 the equation 



ePS , 2dS 2S 



M + -J --- T -i- x 2 s = o, 



ar* r dr r- 

 we obtain, 



(X - /i) f STV dr + |>T f - r*S ^1 ' = 0, 64) 



Jo \_ dr drJ 







and since by (63), S and T both vanish where r = a, we obtain 



ST-*dr = 0, .......... (65) 



provided X and /* are different. To find the value of the integral where X = p., let 

 p = K -\- d\; then from (64) 



2 o ^ 



r + cH S -r- d\ = U, 

 o dr d\ dr d\j 



or, 



SV> dr = ^a 8 S' 2 (Xa), . ,,, ^ ..,,,, . . (66) 



where the accents denote differentiation with respect to Xa ; whence 



cA> /\ \ w ' f f" d s in Xr , 



' 2 (Xa) = , - dr, 



\ 3 dr r 



w . 



= - sm X. 



A, 



