112 1U.] STEADY MOTION OF SOLID. 127 



i.e. provided the velocity of which \i, v, w are the components be 

 in the direction of one of the principal axes of the ellipsoid, 



Ax* + Bif + Cz* + 2A yz + 2B zx + ZC xy = const. 



There exist then for every body three mutually perpendicular 

 directions of permanent translation ; that is to say, if the body 

 be set in motion parallel to one of these directions, without ro 

 tation, and then left to itself, it will continue to move in this 

 manner. It will be seen that these directions are determined by 

 the ratio of the mean density of the solid to the density of the 

 surrounding fluid and by the form of the body s surface. The 

 impulse necessary to produce motion in one of these directions 

 does not in general reduce to a single force ; thus if the axes 

 of co-ordinates be chosen, for convenience, parallel to these 

 directions, so that A , B r , C 0, we have corresponding to the 

 motion u alone 



so that the impulse consists of a wrench* of pitch 



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 

 twist about a certain screwf*; and the condition that this motion 

 should be permanent is that it should not affect the configuration 

 of the impulse (which is fixed in space) relatively to the body. 

 This requires that the axes of the screw and of the corresponding 

 impulsive wrench should coincide. Since the general equations 

 of a straight line involve four independent constants, this gives 

 four relations to be satisfied by the five ratios u : v : w : p : q : r. 



* A wrench is a system of forces supposed reduced after the manner of Poinsot 

 to a force and a couple having its axis in the direction of the force. Its pitch is 

 the line which is the result of dividing the couple by the force. See Ball, Theory 

 of Screics. 



t A twist is the most general motion of a rigid body, equivalent to a translation 

 parallel to some axis combined with a rotation about that axis. Its pitch is the 

 linear magnitude which is the ratio of the translation to the rotation. Ball, 

 Theory of Screics. 



