436 Proceedings of the Royal Irish Academy. 



a' to coincide, but cases may arise in which they do coincide, and these 

 will be discussed further on. 



From the fundamental property of two homographic screw systems 

 it follows that the co-ordinates of ^ must be expressed by six equations 

 of the type — 



A =/i («!' • • . ae) 

 &c., &c., 



A =/6 («i» • • • «6)- 



If these six equations be solved for ai . . . og we must have — 



ai = F^ (A, ... A) 



&c., «&c., 



ae = F, (/3i, . . . A). 



It can be easily shown, that if ai, a^ . . . a^ are to have unique values, 

 then these equations must be linear ; whence we have the following 

 important result : — 



In two homographic screw systems the co-ordinates of a screw in o?ie 

 system are linear functions with constant coefficients of the co-ordinates 

 of the corresponding screw in the other system. 



If we denote the constant coefficients by the notation (11) (22), &c., 

 then we have the following system of equations : — 



A = (11) ai + (12) a2 + (13) aj + (14) 04 + (15) aj + (16) a^, 

 yS^ = (21) ai + (22) a2 + (23) a. + (24) a^ + (25) a, + (26) ag, 



&c., 

 /^e = (61) ai -f (62) ao_ + (63) a^ + (64) a^ + (65) a^ + (66) ag. 



It is now easy to show that there are six screws which coincide 

 with their corresponding screws ; for if A = paj, A = paj, &c., we ob- 

 tain an equation of the sixth degree for the determination of p. "We 

 therefore have the following result : — 



In tivo homographic screw systems six screws can be found, each of 

 which regarded as a screw in either system coincides with its correspondent 

 in the other system. 



In two homographic rows of points we have the anharmonic ratio 

 of any four points equal to that of their correspondents. In the case 

 of two homographic screw systems we have a set of eight screws in 

 one of the systems specially related to the corresponding eight screws 

 in the other system. 



We first remark that, given seven pairs of corresponding screws 

 in the two systems, then the screw corresponding to any other 



