131-132] MOTION OF A RING. 197 



The ring then rotates about an axis in the plane yz parallel to that of z, at a 

 distance u/r from it. 



For further investigations on the motion of a ring we refer to papers by 

 Basset*, who has discussed in detail various cases where the axis moves in 

 one plane, and Miss Fawcettf. 



Equations of Motion in Generalized Coordinates. 



132. When we have more than one moving solid, or when the 

 fluid is bounded, wholly or in part, by fixed walls, we may have 

 recourse to Lagrange's method of ' generalized coordinates.' This 

 was first applied to hydrodynamical problems by Thomson and 

 Tait. 



In any dynamical system whatever, if f, ij, % be the Cartesian 

 coordinates at time t of any particle m, and X, Y, Z be the com- 

 ponents of the total force acting on it, we have of course 



raf = X, mij=Y, m% = Z (1). 



Now let f + Af, rj + A?;, ?4-Af be the coordinates of the same 

 particle in any arbitrary motion of the system differing infinitely 

 little from the actual motion, and let us form the equation 



2m (f Af + 7/Aij + A?) = 2 (ZAf + FAT? + ZAf) (2), 



where the summation 5) embraces all the particles of the system. 

 This follows at once from the equations (1), and includes these, on 

 account of the arbitrary character of the variations A, A??, Af. 

 Its chief advantages, however, consist in the extensive elimination 

 of internal forces which, by imposing suitable restrictions on the 

 values of Af, A?;, Af we are able to effect, and in the facilities 

 which it affords for transformation of coordinates. 



If we multiply (2) by Bt and integrate between the limits t 

 and ti, then since 



* "On the Motion of a Ring in an Infinite Liquid," Proc. Gamb. Phil. Soc., 

 t. vi. (1887). 



t 1. c. ante p. 191. 



Natural Philosophy (1st ed.), Oxford, 1867, Art. 331. 



