292] DEVELOPMENTS OF THE DYNAMICAL THEORY. 309 



In general, when two pairs of screws are given on two cylindroids, the 

 chiastic homography between the surfaces is determined. If, however, it 

 were possible to determine two chiastic homographies having two pairs in 

 common, then every homography is chiastic, and the cylindroids are normal. 



Let a, 77 and ft, be the two pairs of correspondents, and let 7 have the 

 correspondents and , then we have 



whence 



&quot; 



i.e. the two cylindroids are normal. 



292. General Conditions of Chiastic Homography. 



We shall now discuss the relations of chiastic homography between two 

 systems of screws in the same w-system. The first point to be demonstrated 

 is, that in such a case every pair of the double screws are reciprocal. 



Take a and (3 as two of the double screws, and 77 and will coincide 

 with them ; whence the general condition, 



becomes 



= 0. 



One or other of these factors must be zero. We have to show that in general 

 it is impossible for 







to vanish. 



For, take 7 reciprocal to a but not to ft, then r av = ; but OT^ Y is not 

 zero, and therefore -sr a f would have to be zero ; in other words f must be 

 reciprocal to a. But this cannot generally be the case, and hence the other 

 factor must vanish, that is 



UTaft = 0. 



In like manner it can be shown that every pair of the double screws must 

 be reciprocal. 



Conversely it can be shown that if the double screws of two homo- 

 graphic systems are co-reciprocal, then the homography is chiastic. 



Let the w-double screws of the two systems be taken as the screws of 

 reference ; then if one screw in one system be denoted by the co-ordinates 



!, ... On, 



