308-309] MODULI OF DECAY. 567 



is aperiodic, can be investigated by the method of Arts. 293, 294, the effects of 

 inertia being disregarded. In the case of a globe returning to the spherical 

 form under the influence of gravitation, it appears that 



ao+iy+i^ 



n ga 

 a result first given by Darwin (I. c.). Cf. Art. 302 (22). 



309. Problems of periodic motion of a liquid in the space 

 between two concentric spheres require for their treatment 

 additional solutions of the equations of Art. 306, in which p is of 

 the form p- n -i , and the functions ^ n (hr) which occur in the 

 complementary functions u', v', w f are to be replaced by M/* n (hr). 



The question is simplified, when the radius of the second sphere 

 is infinite, by the condition that the fluid is at rest at infinity. It 

 was shewn in Art. 267 that the functions ^ w (?) "^n(?) are both 

 included in the form 



fd \ n A<& + Be-** 



' 



In the present applications, we have = hr, where h is defined 

 by Art. 306 (4), and we will suppose, for definiteness, that that 

 value of h is adopted which makes the real part of ih positive. 

 The condition of zero motion at infinity then requires that -4 = 0, 

 and we have to deal only with the function 



As particular cases : 



The formulae of reduction for/ n () are exactly the same as for 

 ty n () and > J r n (f), and the general solution of the equations of 

 small periodic motion of a viscous liquid, for the space external to 

 a sphere, are therefore given at once by Art. 306 (8), (10), with 

 P-n-i written for p n , smdf n (hr) for ty n (hr). 



1. The case of the rotatory oscillations of a sphere surrounded by an 

 infinite mass of liquid is included in the solutions of the First Class, with 

 n= 1. As in Art. 307, 2, we put x\ = Cz t and find 



(i), 



