267-268] SOLUTION IN SURFACE HARMONICS. - 487 



and by repeated applications of this result it appears that (12) is satisfied by 



where R is the solution of 



that is R, = - .............................. (ii)*. 



268. A simple application of the foregoing analysis is to the 

 vibrations of air contained in a spherical envelope. 



1. Let us first consider the free vibrations when the envelope is rigid. 

 Since the motion is finite at the origin, we have, by (9), 



<t> = A+ n (kr}r*S n .e i(rt ................................. (i), 



with the boundary-condition 



kaW(ka)+nty n (kd) = Q .............................. (ii), 



a being the radius. This determines the admissible values of k and thence of 



<r( = kc}. 



It is evident from Art. 267 (11) that this equation reduces always to the 

 form 



tan &*=/(&) ................................. (iii), 



where f(ka) is a rational algebraic function. The roots can then be calculated 

 without difficulty by Fourier's method, referred to in Art. 266. 



In the case w = l, if we take the axis of x coincident with that of the 

 harmonic S lt and write ,zr = rcos 6, we have 



in kr 



and the equation (ii) becomes 



9^/TT 



(v). 



The zero root is irrelevant. The next root gives, for the ratio of the 

 diameter to the wave-length, 



and the higher values of this ratio approximate to the successive integers 

 2, 3, 4.... In the case of the lowest root, we have, inverting, 



X/2a= 1-509. 



* The above analysis, which has a wide application in mathematical physics, 

 has been given, in one form or another, by various writers, from Poisson (TMorie 

 matliematique de la Chaleur, Paris, 1835) downwards. For references to the history 

 of the matter, considered as a problem in Differential Equations, see Glaisher, "On 

 Kiccati's Equation and its Transformations," PhiL Trans., 1881. 



