340 Mr. G. H. Bryan on the Equations of 



is found to be X. where 



X=-| P jj^^rfS + i P 2«j|^rf<r=S 1 + K. . (3) 



Here Xi is a quadratic function of the velocity-components of 

 the solid, and is the kinetic energy when the motion is acyclic, 

 and K is a quadratic function of the circulations. 



If the axes were fixed in space, the pressure equation 

 (supposing no forces to act on the liquid) would be 



(where p 1 = pressure, q x = resultant velocity of liquid). 

 Owing to the motion of the axes, however, 'd^/^t must be 

 replaced by the rate of change of <j> at a fixed point, that is by 



whence the pressure equation becomes 



^ + ^-{u-yr + zq)^-^-zp + xr)^-{ic-xq+yp)^ 



+ \q^=. const. ... . (4) 



The Mutual Reactions between the Solid and Liquid. 



4. Let X l7 Y lf Z x , Li, M. v Nj be the component forces and 

 couples which the solid exerts on the liquid ; then we have 

 evidently 



X 1 = jj/^> 1 rfS, L x = \\ (ny — mz)p 1 d$. . . (5) 



To reduce these expressions to the required form, we shall 

 have to resort to repeated applications of Green's formula. 

 Since the velocity-potential <£ is a multiple- valued function, 

 it follows that in transforming volume integrals involving <£ 

 we shall obtain surface integrals over the barriers <r 1} c 2 , . . . a lH 

 as well as over S the surface of the solid. On the other hand, 

 the pressure p x and the velocity-components ~d(f>l~d%, d</>/dy, 

 'dty/'dz are single-valued and do not contribute barrier terms 

 to the surface integrals. Moreover, since the circulations k 

 are independent of the time, 



-^1 = #« + v 4> v + w $w +P<I> P + # 2 + r$ r ; 



and ~d<j)fdt is therefore a single- valued function of the velocity- 

 components of the solid satisfying Laplace's equation. 



