324 TIDAL WAVES. [CHAP. VIII 



ponent of extraneous force corresponding to the coordinate q t . 



Also, since 



dx . dx . 



X = -j (ft + j (7 2 + . ., 



efyi dq^ 

 dy . dy . 



*-S* + i*^- 



we have 



^ /. dy . dx\ ^ (d(x,y) . d(x,y) . } 



Sm (x -/- -y-r- = 2mJ j/ \gi + j/ xga + -r- 



V % &amp;lt;y %/ (&amp;lt;* (&&amp;gt;?) d(q 2 ,q s )* j 



We will write, for shortness, 



Finally, we put 



r f in*Sm(Mvy*) ..................... (6), 



viz. T denotes the energy of the system when rotating with the 

 solid, without relative motion, in the configuration (q 1} q. 2) ...). 



With these notations, the typical equation (2) takes the form 



and it is to be particularly noticed that the coefficients [r, s~\ are 

 subject to the relations 



[r,s] = -|&amp;gt;,r], [s,s] = .................. (8). 



The conditions for relative equilibrium, in the absence of ex 

 traneous forces, are found by putting ^ = 0, g 2 = 0, ... in (7), or more 

 simply from (2). In either way we obtain 



which shews that the equilibrium value of the expression V- T is 

 stationary. 



196. We will now suppose the coordinates q s to be chosen 

 so as to vanish in the undisturbed state. In the case of a small 

 disturbance, we may then write 



. + 2a l2 q,q 2 + ...... (1), 



.+ 2c 1 ,q l q 2 + ...... (2), 



* Of. Thomson and Tait, Natural Philosophy (2nd ed.), Part i. p. 319. It 

 should be remarked that these equations are a particular case of Art. 139 (14), 

 obtained, with the help of the relations (7) of Art. 141, by supposing the rotating 

 solid to be free, but to have an infinite moment of inertia. 



