Motion of a Perforated Solid in Liquid. 347 



" But by § 2, "d(j> K fdy = all over the surface S of the solid. 

 Hence, equating the terms independent of the six velocities 

 (t* . . .,p . . .) on the two sides of (29), we have 



2K^2K P {\^ v (tKcf> K )da- (30) 



But by (16) (17), 



= -S^jJ|;( W c/> M +...+^+...)^ 



-\-'%Kp\\{lu< + mv + 7iiv-{- (ny — mz)p + . . .}dcr. 

 Hence, by (28) and (30), 



2^x=2^Jj ^ («*. + . ■ ■ +M, + • • ■ + 2«k)<fcr 



— 2/epff{Z(w— yr + zq) + m(v—zp + xr) + n(w — xq+yp)}d<r; 

 and therefore 



%m= U^-^(^-i/>' + ^) — (two similar) Jt/cr w . . (31) 



Now c)(£/dv is the velocity of the liquid resolved along the 

 normal to the barrier <r m ', and 



l(u —yr + zq) + m(v—zp + at) + n{w — xq +yp) 



is the velocity of the barrier <r m resolved normally to itself at 

 the point #, y, z, supposing the barrier to be fixed relatively 

 to the solid. Their difference, therefore, represents the nor- 

 mal relative velocity of the liquid with respect to the barrier. 



Hence x m represents the total rate of flow of the liquid 

 across the barrier <r m relative to the solid; in other words, the 

 generalized velocity corresponding to the ignored momentum 

 p/c m is the volume of liquid per unit time flowing through the 

 aperture relatively to the solid. 



This property is proved in a different way by Basset in his 

 ' Hydrodynamics,' vol. i. page 176. 



The Form of the Modified Function. 



11. It may be interesting to examine a little more closely 

 the effect of the circulations on the motion of a solid. 



When the motion of the liquid is acyclic, the kinetic 

 energy is a homogeneous quadratic function of (u, v, w,p, q, r) . 

 In general it therefore involves 21 constants, but by a suitable 



