258 ME. C. J. T. SEWELL: EXTINCTION OF SOUND IN A VISCOUS ATMOSPHEEE 

 where <f> is any solution of the equation 



(V 8 +/i') = .......... (7). 



The complete solution of the equations of motion will be given by 



(u, v, w) = (u 1 , v', w') + grad <j> ....... (8), 



where u', v', iv' satisfy the equations 



(V 2 + P) u'=Q, (V 2 + F) v' = 0, ( V 2 + k 2 ) w' = 

 toether with 



+ 

 dx dy 3z 



/ Q \ 



+ + = 



The solution of these equations suitable to the case of waves diverging from the 

 origin is given by 



with corresponding expressions for v' and w'. Here and x are solid harmonics of 

 positive degree n, and f n (/?) is a function of A'?* given by the relation 



where ^f n () and \\> n () satisfy the relations 



ii (12) 



The general formulae for </>(), ^ (^), andy n (^), are 



1 



Z 1.3.5... 



(g) _.....- 



_ __ . . 



2(l-2n) 2;4.(l-2n)(3-2w) J 



, ,, _ < f n.(ft+l) (n-l)n(/i+l)(-H+2) 1.2.3...2n 1 , ^ 



"T^l "^^ 2.4.(iO a 2.4.6...2n(iO"J 



The functions / s (^), V n (), and /({), all satisfy recurrence formulae of the types 



these will be found useful hereafter in reductions. 



