260 ME. C. J. T. SEWELL : EXTINCTION OF SOUND IN A VISCOUS ATMOSPHERE 

 Expanding in terms of the functions /, we obtain 



....... (3). 



The scattered waves will be symmetrical about the axis of x, and so we may assume 

 for them 



where 



Now, by means of the recurrence formulae, it may be proved without difficulty that 



At the surface of the spherical obstacle r a, we must have 

 WO + M! = 0, fo + t'i = 0, W + w, = 0. 

 Hence, when r = a, we have 



grad to, + M + J o {(+ l)/-i (*r) gd ^-nW^+%^ (A;r) grad ^) J = 0. 



Introducing the expressions given above for < > <i, , we obtain for all values of n, 

 when ? = a, 



grad {(2+ 1) t "/A/, n (^-) ^P B } + grad {A,,/, (7 t r) r"PJ 



4- (w+ !)/-! (Ar) B. grad (^P n ) - /dV +3 / M+1 (Jr) B n grad = 0. 



Hence, by means of the identity (7), we find that the following expression 



+1 (Aa) 



n +1 



and two other similar expressions must vanish, when r = a, for all values of n. 

 These three conditions will be satisfied if 



A / 1 (/ia)= -^(ha) ......... (8), 



