422 SURFACE WAVES. [CHAP. IX 



where V is the potential energy, it appears that the conditions for 

 steady motion, with q lt q Z) ... constant, are 



where K is the energy of the motion corresponding to any given 

 values of the coordinates q lt q 2 , ..., when these are prevented from 

 varying by the application of suitable extraneous forces. 



This energy is here supposed expressed in terms of the constant 

 momenta G, C',... corresponding to the ignored coordinates 

 %,%',, and of the palpable coordinates q l} q 2 , .... It may how- 

 ever also be expressed in terms of the velocities ^, ;', ... and 

 the coordinates q lt q 2 , ...; in this form we denote it by T . It 

 may be shewn, exactly as in Art. 141, that dT /dq r = dK/dq r) so 

 that the conditions (2) are equivalent to 



Hence the condition for free steady motion with any assigned 

 constant values of q lt q z >... is that the corresponding value of 

 V + K, or of V-T 9 , should be stationary. Cf. Art. 195. 



Further, if in the equations of Art. 139 we write dV/dq s + Q s 

 for Q 8 , so that Q s now denotes a component of extraneous force, we 

 find, on multiplying by q lf q. 2} ... in order, and adding, 



Q 1 q 1 +Q&+ ............... (4), 



where & is the part of the energy which involves the velocities 

 q 1} q< 2 , .... It follows, by the same argument as in Art. 197, that 

 the condition for 'secular' stability, when there are dissipative 

 forces affecting the coordinates q lt <7 2 , ..., but not the ignored 

 coordinates ^, %', ..., is that V+K should be a minimum. 



In the application to the problem of stationary waves, it will 

 tend to clearness if we eliminate all infinities from the question 

 by imagining that the fluid circulates in a ring-shaped canal of 

 uniform rectangular section (the sides being horizontal and 

 vertical), of very large radius. The generalized velocity % corre- 



