Ball — On So7nographic Scretv Si/stems. 435 



LXV. — On Homogeaphic Screw Systems. By Eobeet S. Ball, 



LL.D., F.E.S. 



[Read, May 9tli, 1881.] 



I HAVE lately ascertained that several of the most important parts of 

 the Theory of Screws can be embraced in a more general theory. I 

 propose in the present Paper to sketch this general theory. It will be 

 found to have points of connexion with the modern higher geometry ; 

 in particular the theory of Homographic Screws is specially connected 

 with the general theory of correspondence. I believe it will be of 

 some interest to show how these abstract geometrical theories may be 

 illustrated by dynamics. The intimate alliance which exists between 

 the higher branches of rigid dynamics and the higher branches of 

 modern geometry is perfectly natural. This will, I hope, be sufficiently 

 illustrated in the present Paper. Among the more recondite theorems 

 in rigid dynamics is that of the existence of a number of principal 

 screws of inertia equal to the number of degrees of freedom which the 

 body enjoys. Yet we shall show in this Paper that this is an instan- 

 taneous consequence of the purely geometrical theory of homographic 

 screws. 



We commence with the most general case in which the screws may 

 be regarded as existing anywhere in space. I may remind the reader 

 that a screiv in the present sense of the word denotes a right line of 

 specified situation and direction with which the linear magnitude 

 termed the piicli is associated. 



Given one screw a, it is easy to conceive that another screw yS corre- 

 sponding thereto shall be also determined. We may, for example, 

 suppose that the co-ordinates of jS (see Theory of Screws, p. 33) shall 

 be given functions of those of a, or we may imagine a geometrical 

 construction by the aid of fixed lines or curves by which, when an a is 

 given, the corresponding /3 shall be forthwith known : again, we may 

 imagine a connexion involving dynamical conceptions such as that, 

 when a is the seat of an impulsive wrench, (3 is the instantaneous 

 screw about which the body begins to twist. 



As a moves about, so will the corresponding screw /5 : we thus have 

 two corresponding screw systems generated. Eegarding the connexion 

 between the two systems from a purely analytical point of view, the 

 co-ordinates of a and jB will be connected by certain equations. It will 

 not generally happen that a single screw /? corresponds to a single 

 screw a, and that conversely a single screw a corresponds to a single 

 screw jB ; but ivlien this does happen the tivo systems of screws are said 

 to be homographic. 



A screw a in the first system has one corresponding screw jB in 

 the second system ; so also to y8 in the second system corresponds one 

 screw a' in the first system. It will generally be impossible for a and 



R.I. A. PROC, SER. II., VOL. III. — SCIENCE. 2 



