CHAPTER XIX. 



HOMOGRAPHIC SCREW SYSTEMS*. 



243. Introduction. 



Several of the most important parts of the Theory of Screws can be 

 embraced in a more general theory. I propose in the present chapter 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. 



244. On Plane Homographic Systems. 



It may be convenient first to recite the leading principle of the purely 

 geometrical theory of homography. We have already had to mention a 

 special case in the Introduction. 



Let a be any point in a plane, and let ft be a corresponding point. Let 

 us further suppose that the correspondence is of the one-to-one type, so that 

 when one a is given then one ft is known, when one ft is given then it is the 

 correspondent of a single a. The relation is not generally interchangeable. 

 Only in very special circumstances will it be true that ft, regarded as in the 

 first system, will correspond to a in the second system. 



The general relation between the points a and ft can be expressed by the 

 following equations, where a 1} a. 2 , 3 are the ordinary trilinear co-ordinates of 

 a, and J3 1} /3 a , fi 3 , the co-ordinates of ft, 



ft = (11)0! +(12)0, + (13) a,, 



ft = (31) 0^(32) a, + (33) 03. 



In these expressions (11), (12), &c., are the constants defining the particular 

 character of the homographic system. 



* Proc. Hoy. Irish Acad. Ser. n. Vol. HI. p. 435 (1881). 



