Equations of Motion of a Perforated Solid, 339 



function ' ; obtained by "ignoring" the velocity-components 

 corresponding to the constant momenta or circulations. Either 

 of these expressions involves constants which cannot be deter- 

 mined from the ordinary expression for the energy alone, and 

 to determine them in the usual way it is necessary to resort 

 to arguments based on a consideration of the "impulse" by 

 which the motion might be set up from rest. 



In the following investigation the equations of motion are 

 deduced from purely hydrodynamical considerations, and from 

 them the modified function is found. In §§ 12-16 the equa- 

 tions of motion are interpreted for the case in which the solid 

 is a light rigid framework and the inertia is entirely due to 

 the circulation of the liquid, and the results are applied to 

 interpret the effective forces of the cyclic motion for a per- 

 forated solid in general. 



General Hydrodynamical Equations. 



2. Let a perforated solid bounded by the surface S be 

 moving through an infinite mass of liquid (density p) with 

 translational and rotational velocity-components u, v,w,p, q, r, 

 referred to axes fixed in the solid, and let /e,, k 2 , k 3 . . . K m be 

 the circulations in circuits drawn through the various aper- 

 tures. Then we know that </> the velocity-potential of the 

 fluid motion may be expressed as a linear function of the 

 velocities and circulations in the form 



$ = ii$ u + v$ v + w<$> w +p$ p + q$ q + r$ r + %K$ K , . (1) 



where evidently cf> u ="d(j)/'du &c, and the coefficients <f> u . . . 

 depend only on the form of the solid. 



If dv denotes the element of the normal to S measured 

 from the solid into the liquid, (I, m, n) its direction-cosines, 

 then, in the usual way, we have 



-^- z=l(ii—ry + qz) + m(v —pz + rx) + n (iv — qx +py) . . (2) 



The six coefficients (j> u . . . <j) r are single-valued functions of 

 the coordinates, while the coefficients </> K which determine the 

 part of the velocity-potential due to the circulations are cyclic 

 functions making b^) K /B^ = at the surface of the solid; these 

 coefficients are supposed known for each form of solid, although 

 their determination in any given case is generally beyond the 

 range of mathematical analysis. 



3. Let <r ly <r 2 , . . . <j m be barriers drawn across the perfora- 

 tions ; then, in the usual way, the kinetic energy of the liquid 



