558 VISCOSITY. [CHAP, xi 



out any space, the values of u 1 , v', w' are rendered by (2) completely 

 determinate. For if there were two sets of values, say u, v, w 1 

 and u", v", w", both satisfying the prescribed conditions, then, 

 writing 



u^ = u ' u", V-L = v' - v", w^ w' ?(/', 



we should have 



anti + yVi -I- zw^ = 0; 



.(18). 



== 

 dx dy dz 



Regarding u^ , v 1} w^ as the component velocities of a liquid, the 

 first of these shews that the lines of flow are closed curves lying 

 on a system of concentric spherical surfaces. Hence the ' circula- 

 tion' (Art. 32) in any such line has a finite value. On the 

 other hand, the second equation shews, by Art. 33, that the 

 circulation in any circuit drawn on one of the above spherical 

 surfaces is zero. These conclusions are irreconcileable unless 

 MI, v lf w l are all zero. 



Hence, in the present problem, whenever the functions <f> n and 

 % n have been determined by (9) and (12), the values of u', v', w' 

 follow uniquely as in (3) and (6). 



When the region contemplated is bounded internally by a 

 spherical surface, the condition of finiteness when r = is no longer 

 imposed, and we have an additional system of solutions in which 

 the functions ty n (K) ar e replaced by ^(f), in accordance with 

 Art. 267* 



* Advantage is here taken of an improvement introduced by Love, "The Free 

 and Forced Vibrations of an Elastic Spherical Shell containing a given Mass of 

 Liquid," Proc. Lond. Math. Soc., t. xix., p. 170 (1888). 



The foregoing investigation is taken, with slight changes of notation, from the 

 following papers : 



" On the Oscillations of a Viscous Spheroid," Proc. Lond. Math. Soc., t. xiii., 

 p. 51 (1881) ; 



"On the Vibrations of an Elastic Sphere," Proc. Lond. Math. Soc., t. xiii., 

 p. 189 (1882) ; 



"On the Motion of a Viscous Fluid contained in a Spherical Vessel," Proc. 

 Lond. Math. Soc., t. xvi., p. 27 (1884). 



