121124.] CIRCULAR RING. 141 



right-hand side of (48) should be negative ; and the time of a small 

 oscillation, in the case of disturbed stable motion, is 



123. The general equations of motion of the ring are also 

 satisfied by f, 17, f, X, //, = 0, and v constant. We have then 



u , r = const. 



The motion of the ring is then one of uniform rotation about an 

 axis in the plane yz parallel to that of y, and at a distance - 

 from it. 



Case of two or more moving solids. 



124. The foregoing methods fail when we have two or more 

 moving solids, or when the fluid does not extend in all directions 

 to infinity, being bounded externally by fixed rigid walls. In such 

 cases we may suppose the position at the time t of each moving 

 solid to be defined by means of six co-ordinates, in the manner 

 explained in treatises on Kinematics. It is easy to see that &amp;lt;f&amp;gt; 

 must be a linear function of the rates of variation of these co 

 ordinates (in other words, of the generalized velocity-components 

 of the system), and thence that the kinetic energy of the system 

 is, as in Art. 110. a homogeneous quadratic function of these 

 generalized velocities, with however the important change that the 

 coefficients in this function are not constants, but themselves func 

 tions of the co-ordinates of the system. The equations of motion 

 are then most conveniently formed by Lagrange s method*, the 

 applicability of which to systems of the peculiar kind here con 

 sidered requires however to be in the first place established f. 



The accompanying references will be of service to the reader 

 who wishes to pursue the study of the general problem in the 

 manner indicated. We content ourselves here with the discussion 



* See Thomson and Tait, Art. 329. 



t See Thomson, Phil. Nag. May, 1873, and Kirchhoff, Vorlesungcn fiber 

 Math. Physik. Mechanik, c. 19, 1. 



