226 SIR W. THOMSON ON VORTEX MOTION. 



22. In Poinsot's system of the statics of a rigid body we may pass from the 

 resultant force and couple along and round the central axis to an equal resultant 

 force along the parallel line through any point, and a greater couple the resultant 

 of the former (or minimum) couple, and a couple in the plane of the two parallels, 

 having its moment equal to the product of their distance intp the resultant force. 

 So we may pass from the force-resultant and rotational moment of the impulse 

 along and round its axis, to an equal force-resultant and greater moment of im- 

 pulse, by transferring the former to any point, Q, not in the axis (§ 6) of the 

 motion. This greater moment is (§ 18) equal to the moment of momentum round 

 the point Q, of the motion within any spherical surface described from Q as 

 centre, which encloses all the vortices or moving solids. 



23. Hence a group of solids or vortices which always keep within a spherical 

 surface of finite radius, or a single body, moving in an infinite liquid, can have 

 no permanent average motion of translation in any direction oblique to the direc- 

 tion of the force-resultant of the impulse, if there is a finite force-resultant. For 

 the matter within a finite spherical surface enclosing the moving bodies or body, 

 cannot have moment of momentum round the centre increasing to infinity. 



24. But there may be motion of translation when the force-resultant of the 

 impulse vanishes ; and there will be, for example, in the case of a solid, shaped 

 like the screw-propeller of a steamer, immersed in an infinite homogeneous liquid, 

 and set in motion by a couple in a plane perpendicular to the axis of the screw. 



25. And when the force-resultant of the impulse does not vanish, there may be 

 no motion of translation, or there may even be translation in the direction opposite 

 to it. Thus, for example, a rigid ring, with cyclic motion, established (§ 63) through 

 it, will, if left at rest, remain at rest. And if at any time urged by an impulse 

 in either direction in the line of the force-resultant of the impulse of the cyclic 

 motion, it will commence and continue moving with an average motion of trans- 

 lation in that direction ; a motion which will be uniform, and the same as if there 

 were no cyclic motion, when the ring is symmetrical. If the translatory impulse 

 is contrary to the cyclic impulse, but less in magnitude, the translation will be 

 contrary to the whole force-resultant impulse. 



If the translatory impulse is equal and opposite to the cyclic impulse, 

 there will be translation with zero force-resultant impulse — another example of 

 what is asserted in § 24. In this case, if the ring is plane and symmetrical, or 

 of any other shape such that the cyclic motion (which, to fix ideas, we have sup- 

 posed given first, with the ring at rest,) must have had only a force-resultant, 

 and no rotational moment, we have a solid moving with a uniform motion of 

 translation through a fluid, and both force and couple resultant of the whole 

 impulse zero. 



26. From §§21 and 4, we see that, however long the time of application of 

 the impressed forces may be — provided only that, during the whole of it, the 



