137] CYCLIC MOTION. 209 



and (*, *)=-p /T-j- dv, 



.............. (9). 



dot' ' 



It is evident that K is the energy of the cyclic motion which 

 remains when the solids are maintained at rest in the configuration 

 (?i, &, ..-) We note that, by (3), (7), and (9), 



dK dK 



+ (11). 



If we add to (5) the kinetic energy of the solids themselves, we 

 obtain for the total kinetic energy of the system an expression of 

 the form 



where 2^ = A n q-f + A^+ ... -\-^A^q^ + (13), 



the coefficients being in general functions of q lf q 2 , 



To obtain the equations of motion we have recourse as before 

 to the formula 



tl {AT+ 2 (XAf + FAr; + Af)} dt 



The only new feature is in the treatment of the expression on 

 the right-hand side. By the usual method of partial integration 

 we find 



= p 



+ pK I |(/Af + roA); + nAf ) dir+pie' | |(/Af + mA?? + nAf ) da +,.. 



............... (15), 



where /, m, n are the direction-cosines of the normal to any 



element $S of a bounding surface, drawn towards the fluid, or 



L. H 



