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



fulfils the following conditions : (1) the value of - d^/dn, where Sn 

 denotes as usual an element of the normal at any point of the 

 surface of the solid, drawn on the side of the fluid, must be equal 

 to the velocity of the surface at that point normal to itself, and 

 (2) the differential coefficients d(f>/dx, dfyfdy, dfyjdz must vanish at 

 an infinite distance, in every direction, from the solid. The latter 

 condition is rendered necessary by the consideration that a finite 

 velocity at infinity would imply an infinite kinetic energy, which 

 could not be generated by finite forces acting for a finite time on 

 the solid. It is also the condition to which we are led by supposing 

 the fluid to be enclosed within a fixed vessel infinitely large and 

 infinitely distant, all round, from the moving body. For on this 

 supposition the space occupied by the fluid may be conceived as 

 made up of tubes of flow which begin and end on the surface of 

 the solid, so that the total flux across any area, finite or infinite, 

 drawn in the fluid must be finite, and therefore the velocity at 

 infinity zero. 



It has been shewn in Arts. 40, 41, that under the above con- 

 ditions the motion of the fluid is determinate. 



115. In the further study of the problem it is convenient to 

 follow the method introduced by Euler in the dynamics of rigid 

 bodies, and to adopt a system of rectangular axes Ox, Oy, Oz fixed 

 in the body, and moving with it. If the motion of the body at 

 any instant be defined by the angular velocities p, q, r about, and 

 the translational velocities u, v, w of the origin parallel to, the 

 instantaneous positions of these axes, we may write, after 

 Kirchhoff, 



............ (2), 



where, as will appear immediately, fa, fa, fa, % 15 % 2 > % 3 are certain 

 functions of x, y, z determined solely by the configuration of 

 the surface of the solid, relative to the coordinate axes. In fact, 

 if I, m, n denote the direction- cosines of the normal, drawn 

 towards the fluid, at any point of this surface, the kinematical 

 surface-condition is 



-^ = I (u + qz ry} + m (v + rx pz) + n (w -\-py qx), 



