of a Free or Constrained Rigid Body. 275 



less constrained, the problems possess a high degree of geo- 

 metrical interest. It is also the occasion of some surprise that, 

 notwithstanding the infinite variety of conceivable constraints 

 (such as fixed points, axes, contact with fixed surfaces, ar- 

 rangements of link-work and the like) by which the move- 

 ments of the rigid body may be hampered, their geometrical 

 classification is of the utmost simplicity. It will be shown 

 that there are six fundamental descriptions of constraint, 

 which include every conceivable arrangement, from leaving the 

 body absolutely free on the one hand, or absolutely immovable 

 on the other. With the investigation of these fundamental 

 forms of constraint we may fitly commence our inquiry. 



In the first place, it is to be observed that, as the constraints 

 limit the movements of a body, we may adequately describe 

 the nature of those constraints either by pointing out all the 

 movements of the rigid body which are prohibited, or, on the 

 other hand, by ascertaining all the movements which are per- 

 mitted. It is obviously more to the purpose to adopt the latter 

 method of viewing the subject ; and therefore we shall proceed 

 to indicate the method by which a complete inventory may be 

 made of the possible movements which a rigid body can 

 execute. It is also to be continually borne in mind that we 

 are only considering the initial movements of the body, and that, 

 consequently, it is only necessary to consider movements which 

 are of indefinitely small magnitude. 



Let it therefore be supposed that the rigid body is submitted 

 to our examination when it occupies a definite position A. We 

 are not now going to apply the impulsive forces to it ; we are 

 at present merely making a preliminary trial of a purely geo- 

 metrical or kinematical character of its capability for displace- 

 ment. It is at once perceived that the body, not being fixed, 

 can be moved into many closely adjacent positions. Take any 

 one of these positions and call it B. 



By a celebrated theorem of Chasles it is known that the 

 displacement of the body from A to B can be produced by 

 translating the body parallel to a certain line, and at the same 

 time rotating the body about the same line. We may, for con- 

 venience, speak of this motion as a twist ; and we may term 

 the angle of rotation the amplitude of the twist. The distance 

 of the translation is proportional to the amplitude of the twist, 

 and may be taken to be equal to the product of the amplitude 

 of the twist and a certain linear magnitude called the pitch. 

 The axis about which the rotation is made, associated with the 

 linear magnitude termed the pitch, constitute what is called a 

 screw. We therefore say that the displacement of the rigid 

 body from A to B is effected by a twist about a screw. 



T2 



