214 MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. VI 



Equations of this type present themselves in various problems 

 of ordinary Dynamics, e.g. in questions relating to gyrostats, where 

 the coordinates %,% , , whose absolute values do not affect the 

 kinetic and potential energies of the system, are the angular 

 coordinates of the gyrostats relative to their frames. The general 

 theory of systems of this kind has been treated independently by 

 Routh* and by Thomson and Tait-)-. It may be put, briefly, as 

 follows. 



We obtain from (1), by integration, 



(2), 



where, in the language of the general theory, C, C , ... are the 

 constant momenta corresponding to the coordinates %, % , ____ 



In the hydrodynamical problem, they are equal to p, prc , . . . , as 

 will be shewn later, but we retain for the present the more 

 general notation. 

 Let us write 



S = T-C x -ff x - ..................... (3). 



The equations (2) when written in full, determine %, % ,. as 

 linear functions of C, C , ... and &amp;lt;j,, q. 2) ... , and by substitution in 

 (3) we can express as a quadratic function of q 1} q 2 , ...,C,G ..... 



On this supposition we have, performing the arbitrary variation A 

 on both sides of (3), and omitting terms which cancel, by (2), 

 d d d 



where, for brevity, only one term of each kind is exhibited. 



Hence 



d = dT d = dT_ 



dq\ dq dq. 2 dq. 2 



d dT d dT 



.(5). 



d = . d = ., 

 dO~ X dC f X&amp;gt; 



* On the Stability of a given State of Motion (Adams Prize Essay), London, 1877. 

 t Natural Philosophy, 2nd edition, Art. 319 (1879). 



See also von Helmholtz, &quot;Principien der Statik monocyclischer Systeme,&quot; Crelle, 

 t. xcvii. (1884). 



