422 SURFACE WAVES. [CHAP. IX 



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

 steady motion, with q lt q 2 , ... constant, are 



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

 values of the coordinates q l} 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 C, C ,... corresponding to the ignored coordinates 

 2, 2 7 , ..,, and of the palpable coordinates q l} q z , It may how 

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

 the coordinates q lt # 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 2) ... is that the corresponding value of 

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



Further, if in the equations of Art. 139 we write dV/dq s + Q 8 

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

 find, on multiplying by q lt q. 2) ... in order, and adding, 



Q 1 &amp;lt;z 1 +Q 2 j 2 + ............... (4), 



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

 q lt fa, .... 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 q z , ..., 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- 



