136 ON THE MOTION OF SOLIDS THROUGH A LIQUID. [CHAP. V. 



a radius-vector r in the direction of the velocity (u, v, w) and 

 erect the perpendicular h from the centre to the tangent plane 

 at the extremity of r, the plane of the above couple is that of h 



A 



and r, and its magnitude is proportional to sinhr directly, and 

 to h inversely. Its tendency is to turn the body from r to h. Let 

 us suppose that A, B, C are in order of magnitude, and that the 

 direction of the velocity (u, v, w) deviates but slightly from that 

 of one of the principal axes of (4G). If this axis be that of #, 

 the tendency of the above couple is to diminish, and if that of 

 z, to increase the deviation ; whilst in the case of a slight deviation 

 from the axis of y the tendency of the couple depends on the 

 position of r relative to the principal circular sections of (46). 

 Compare Art. 118. 



Case of a Perforated Solid. 



120. If the moving solid have one or more apertures or per 

 forations, so that the space external to it is multiply-connected, 

 the fluid may have a motion independent of that of the solid, viz. 

 a cyclic motion in which the circulations in the various non-evanes- 

 cible circuits which can be drawn through the apertures may 

 have any values whatever. We will briefly indicate how the 



foregoing methods may be adapted to this case. Let x lf K Z 



be the values of the circulations in the above-mentioned circuits, 

 and let d&amp;lt;r lt cfor 2 , ... be surface-elements of the corresponding 

 barriers necessary (as explained in Art. 54) to reduce the region 

 occupied by the fluid to a simply-connected one. Further, let 

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

 the fluid at any point of the surface of the solid, or drawn on 

 the positive side at any point of a barrier. We may now write 



(/&amp;gt; = ufa + vfa + wfa + P& + q& + r% 3 + fc^ + * 2 o&amp;gt; 2 + . . . (47). 



The functions &amp;lt;/&amp;gt;, ^ are determined by the same conditions as 

 before. To determine w l we have the conditions 



(a) that it must satisfy y 2 ^ = throughout the fluid ; 



(b) that its derivatives must vanish at infinity ; 



(c) that -j- 1 = at the surface of the solid ; and 



