99 100.] KINEMATICAL INVESTIGATIONS. Ill 



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. 



In the case of a fluid occupying a cavity in a moving solid, 

 the condition (6) does not apply; the surface of the cavity is 

 then the complete boundary of the fluid, and the condition (a) 

 is therefore sufficient. 



We have seen in Arts. 49, 52 that under either of the above 

 sets of conditions the motion of the fluid is determinate. Our 

 problem then divides itself into two distinct parts ; the first or 

 kinematical part consisting in the determination of the motion of 

 the fluid at any instant in terms of that of the solid, and the 

 second, or dynamical part, in the calculation of the effect of the 

 fluid pressures on the surface of the latter. 



Kinematica I Investigations. 



100. Let us take a system of rectangular axes Ox, Oy&amp;gt; Oz 

 fixed in the body, and let the motion of the latter at any time t 

 be defined by the instantaneous 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 then write, 

 as Kirchhoff does, 



&amp;lt;/&amp;gt; = w^ + ^ + wk + ^Xi + ZXa + rXs (2), 



where, as will immediately appear, &amp;lt;f&amp;gt; l , &c., ^, &c., are certain 

 functions of x, y y z depending only on the shape and size of the 

 solid. In fact, if I, m, n denote the direction-cosines of the normal 

 (drawn on the side of the fluid) at any point (x, y, z) of the sur 

 face of the solid, the condition (a) of Art. 99 may be written 



-~ = (u + qz - ry] I + (v + rx pz] m + (w +&amp;gt;?/ -qx)n; 

 and equating this to the value of f- obtained from (2) we find 



~^ l &amp;gt; ~~r^ m &amp;gt; ~r~ 



an an an j , Q v 



-W 



_-- = 7i u mz, -7 = c* /to/, -5 *- 



c??i * c??z a;i 



