307] MOTION IN A SPHERICAL VESSEL. 568 



Again, since .ru+yv + zw = ................................. (ix), 



for r = a, we must have 



J-^ n -f^(n + l)(2w + l)^ n (Aa)0 n = .................. (x), 



where use has been made of Art. 305 (7). This determines the ratio p n : $ n . 

 In the case n = 1 , the equation ( viii) becomes 



, .v 

 (xi), 





 the lowest root of which is ha 5*764, leading to 



r= 0301-. 



v 



For the method of combining the various solutions so as to represent the 

 decay of any arbitrary initial motion we must refer to the paper cited last on 

 p. 558. 



2. We take next the case of a hollow spherical shell containing liquid, 

 and oscillating by the torsion of a suspending wire*. 



The forced oscillations of the liquid will evidently be of the First Class, with 

 n = 1 . If the axis of z coincide with the vertical diameter of the shell, we find, 

 putting xi=Ck, 



tt-Cf^A^y, v=-Cty l (hr}x, w = ................. (xii). 



If o) denote the angular velocity of the shell, the surface-condition gives 



C Vr 1 (Aa)=-6) ................................. (xiii). 



It appears that at any instant the particles situate on a spherical surface 

 of radius r concentric with the boundary are rotating together with an 

 angular velocity 



^i(Ar) , . N 



T^TT-H o&amp;gt; ................................. (xiv). 



If we assume that co = ae ( ^ +e) ................................. (xv), 



and put h?=-i&amp;lt;r/v = (I-i) 2 P* ........................... (xvi), 



where, as in Art. 297, /3 2 = o-/2j/ ................................. (xvii), 



the expression (xiv) for the angular velocity may be separated into its real 

 and imaginary parts with the help of the formula 



. ... sin cosf 



If the viscosity be so small that (3a is considerable, then, keeping only the 

 most important term, we have, for points near the surface, 



* This was first treated, in a different manner, by Helmholtz, 1. c. ante p. 513. 



362 



