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



128. Before leaving this part of the subject we remark that 

 the preceding theory applies, with obvious modifications, to the 

 acyclic motion of a liquid occupying a cavity in a moving solid. If 

 the origin be taken at the centre of inertia of the liquid, the 

 formula for the kinetic energy of the fluid motion is of the type 



2T = m O 2 + v* + w 2 ) 



+ Pp 2 + Qq* + Rr 2 + 2P 'qr + 2QVp + VR'pq ...... (1). 



For the kinetic energy is equal to that of the whole fluid mass 

 (m), supposed concentrated at the centre of mass and moving 

 with this point, together with the kinetic energy of the motion 

 relative to the centre of mass. The latter part of the energy is 

 easily proved by the method of Arts. 115, 118 to be a homo- 

 geneous quadratic function of p, q, r. 



Hence the fluid may be replaced by a solid of the same 

 mass, having the same centre of inertia, provided the principal 

 axes and moments of inertia be properly assigned. 



The values of the coefficients in (1), for the case of an ellipsoidal cavity, 

 may be calculated from Art. 107. Thus, if the axes of x, y, z coincide with 

 the principal axes of the ellipsoid, we find 



P'=0, Q'=0, R' = 0. 



Case of a Perforated Solid. 



129. If the moving solid have one or more apertures or per- 

 forations, so that the space external to it is multiply-connected, 

 the fluid may have a motion independent of that of the solid, viz. 

 a cyclic motion in which the circulations in the several irreducible 

 circuits which can be drawn through the apertures may have any 

 given constant values. We will briefly indicate how the foregoing 

 methods may be adapted to this case. 



Let K, K, K",... be the circulations in the various circuits, and 

 let Scr, Scr', 8cr",... be elements of the corresponding barriers, 

 drawn as in Art. 48. Further, let I, m, n denote direction-cosines 

 of the normal, drawn towards the fluid, at any point of the surface 

 of the solid, or drawn on the positive side at any point of a barrier. 

 The velocity-potential is then of the form 



