406 Prof. Stokes on the Communication of Vibration 



is 



. 17 n(n + l) (n — l)n(n+l)(n + 2) 1 



, u , eimr T X _ ~^ + D + (n-l)»(n + l)(n + 2) _ 1 

 + w n e ^i ^ ^ -t- 2.4(zW) 2 •••/' 



u n and w' n being evidently Laplace's Functions of the order n, 

 since that is the case with yfr n . 



It will be convenient to take next the condition which has to 

 be satisfied at a great distance from the sphere. When r is very 

 large, the series within braces may be reduced to their first terms 

 1, and we shall have 



r( j) = e im ^ at - r ^u n + e im ( ai+r ^u' n . 



The first of these terms denotes a disturbance travelling out- 

 wards from the centre, the second a disturbance travelling 

 towards the centre, the amplitude of vibration in both cases, for 

 a given phase, varying inversely as the distance from the centre. 

 In the problem before us there is no disturbance travelling 

 towards the centre, and therefore Xu' n = } which requires that 

 each function n' n should separately be equal to zero. We have 

 therefore simply 



r^ n = u n e~ imr 



f n(n + l) (n-l)...(n + 2) , 1 . 2 . 3 . . ■ 2n 1 



L 2.imr "*" 2.4(mr) 2 ^"'^ 2 A.6...2n{imr)» J ' [ } 



or, if we choose to reverse the series, 



. 1.3.5... (2» — l) 



r [imr) n 



r_ , 2n . , (2n— 2)2n ,. N2 , 2.4. 6. ..2» ,. . n 

 i 1 + rT2n mr+ 172(2^31)2^ {lMr) - + 1.2.3...2, ^ }(9). 



Putting for shortness fjr) for the multiplier of u n e~ imr in the 

 right-hand member of (8) or (9), we shall have 



It remains to satisfy the equation of condition (4). Put for 

 shortness 



so that 



Y n (r) = (\+imr)Ur)-rf,!(r), .... (10) 



