CHAPTER XXII. 



THE GEOMETRICAL THEORY*. 



302. Preliminary. 



IT will be remembered how Poinsot advanced our knowledge of the 

 dynamics of a rigid system by a beautiful geometrical theory of the rotation 

 of a rigid body about a fixed point. We now specially refer to the geometrical 

 construction by which he determined the instantaneous axis about which 

 the body commenced to rotate when the plane of the instantaneous couple 

 was given. 



We may enunciate with a generality, increasing in successive steps, the 

 problem which, in its simplest form, Poinsot had made classical. From the 

 case of a rigid body which is constrained to rotate about a fixed point we 

 advance to the wider conception of a body which has three degrees of 

 freedom of the most general type. We can generalize this again into the 

 case in which the body, instead of having a definite number of degrees of 

 freedom has any number of such degrees. The range extends from the 

 first or lowest degree, where the only movement of which the body is 

 capable is that of twisting about a single fixed screw, up to the case in 

 which the body being perfectly free, or in other words, having six degrees of 

 freedom, is able to twist about every screw in space. It will, of course, be 

 borne in mind that only small movements are to be understood. 



In a corresponding manner we may generalize the forces applied to the 

 body. In the problem solved by Poinsot the effective forces are equivalent 

 to a couple solely. For the reaction of the fixed point is capable of reducing 

 any system of forces whatever to a couple. But in the more generalized 

 problems with which the theory of screws is concerned, we do not restrict 

 the forces to the specialized pair which form a couple. We shall assume 

 that the forces are of the most general type and represented by a wrench 

 upon a screw. Thus, by generalizing the freedom of the rigid body, as well 

 as the forces which act upon it, we may investigate the geometrical theory of 

 the motion when a rigid body of the most general type, possessing a certain 

 number of degrees of freedom of the most general type, is disturbed from a 



* Trans. Royal Irish Acad. Vol. xxi. (1897) p. 185. 



