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



these permanent translations does not in general reduce to a single 

 force ; thus if the axes of coordinates be chosen, for simplicity, 

 parallel to the three directions in question, so that A , B , C = 0, 

 we have, corresponding to the motion u alone, 



f = 4a, i) = 0, =0, 



\ = Lu, /JL L u, v = L&quot;u y 

 so that the impulse consists of a wrench of pitch L/A. 



With the same choice of axes, the components of the couple 

 which is the equivalent of the fluid pressures on the solid, in the 

 case of a uniform translation (u, v, w), are 



L = (B - C) vw, M = (C - A) wu, N = (A - B) uv. . .(6). 

 Hence if in the ellipsoid 



A 2 + B?/ 2 + Cz 2 = const (7), 



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

 and erect the perpendicular h from the centre on the tangent 

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

 of h and r, its magnitude is proportional to sin (h, r)/h, and its 

 tendency is to turn the solid in the direction from h to r. Thus if 

 the direction of (u, v, w) differs but slightly from that of the axis of 

 x, the tendency of the couple is to diminish the deviation when A 

 is the greatest, and to increase it when A is the least, of the 

 three quantities A, B, C, whilst if A is intermediate to B and C 

 the tendency depends on the position of r relative to the circular 

 sections of the above ellipsoid. It appears then that of the three 

 permanent translations one only is thoroughly stable, viz. that 

 corresponding to the greatest of the three coefficients A, B, C. 

 For example, the only stable direction of motion of an ellipsoid 

 is that of its least axis; see Art. 118*. 



122. The above, although the simplest, are not the only 

 steady motions of which the body is capable, under the action 

 of no external forces. The instantaneous motion of the body at 

 any instant consists, by a well-known theorem of Kinematics, of a 



* The physical cause of this tendency of a flat-shaped body to set itself 

 broadside-on to the relative motion is clearly indicated in the diagram on p. 94. 

 A number of interesting practical illustrations are given by Thomson and Tait, 

 Art. 325. 



