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



It will be necessary to modify, to some extent, our previous 

 notation. Let us now denote by %, %', %", ... the total fluxes 

 relative to the several barriers of the region, which we shall as 

 before regard as ideal surfaces fixed relatively to the solid surfaces 

 on which they abut ; and let ^, %', %", ... be the time-integrals 

 of these fluxes, reckoned each from an arbitrary epoch. We 

 shall shew, in the first place, that the Lagrangian equations 

 (Art. 133 (17)) will still hold in the case of cyclic motion, provided 

 these quantities ^, ^', ^", ... are treated as additional generalized 

 coordinates of the system. 



Let q lt ^ 2 , ... be, as before, the system of generalized coordi- 

 nates which specify the positions of the moving solids. The 

 motion of the fluid at any instant is completely determined by 

 the values of the velocities q^ q. 2 , ..., and of the fluxes %,%',, as 

 above defined. For if there were two types of irrotational motion 

 consistent with these values, then, in the motion which is the 

 difference of these, the bounding surfaces, and therefore also the 

 barriers, would be at rest, and the flux across each barrier would 

 be zero. The formula (5) of Art. 55 shews that the kinetic energy 

 of such a motion would be zero, and the velocity therefore every- 

 where null. 



It follows that the velocity-potential (4>, say) of the fluid 

 motion can be expressed in the form 



+ ............ (1), 



where <f> 1} for example, is the velocity-potential of the motion 

 corresponding to 



& = 1, #> = 0, g s = 0,..., 



x = o, %=o, jp-o,..., 



which we have just seen to be determinate. 



The kinetic energy of the fluid is given by the expression 



Substituting the value of <E> from (1), and adding the energy of 

 motion of the solids, we see that the total kinetic energy of the 

 system (T, say) is a homogeneous quadratic function of the quan- 

 tities q lt q 2 , ..., %, %', ..., with coefficients which are functions of 



