264 THE THEORY OF SCREWS. [246- 



If these six equations be solved for Oj, ... otg we must have 



a, =., .... 



As a single a is to correspond to a single /3, and vice versa, these equations 

 must be linear : whence we have the following important result : 



In two homographic screw systems the co-ordinates of a screw in one system 

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

 sponding 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 : 



13, = (11) a, + (12) a, + (13) o 3 + (14) 4 + (15) a 5 + (16) 6 , 

 & = (21) a + (22) as + (23) a 3 + (24) 4 + (25) a 5 + (26) a,, 



& = (61) a, + (62) a, + (63) a 3 + (64) a 4 + (65) a 5 + (66) cv 



247. The Double Screws. 



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

 with their corresponding screws; for if ^ 1 = p%i, j3 2 = pa. 2 , &c., we obtain an 

 equation of the sixth degree for the determination of p. We therefore 

 have the following result : 



In two homographic screw systems six screws can in general be found, each 

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

 in the other system. 



248. The Seven Pairs. 



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 given screw is deter 

 mined. For from the six equations just written by substitution of known 

 values of ail, ... 6 and /3 1; ... /3 6 , we can deduce six equations between (11), 

 (12), &c. As, however, the co-ordinates are homogeneous and their ratios are 

 alone involved, we can use only the ratios of the equations so that each pair 

 of screws gives five relations between the 36 quantities (11), (12), &c. The 



