430 Proceedings of the Royal Irish Academy. 



whence each of these equations reduces to 



-v/fJ^ - i^y « + V^Py - Po.y + y Pa. - Pp S 



+ "/Pp - Py V Py - Pa ^ Pa ' Pfi =0. 



Through each point in this plane a line can be drawn, whose direction 

 cosines are proportional to 



^yp^-Pv "^Py-Pa, V fa- 



p^> 



It is remarkable that this line, besides lying in the plane, is also normal 

 thereto, and with anyp)itch ivhatever this line will be a screw of the system. 



Each of these planes of screws will correspond to one of the four 

 remarkable points through which all the pitch conies pass. The plane 

 representation of the screws in the three-system is thus seen to be more 

 complete than the family of pitch quadrics which, until supplemented 

 by the four imaginary planes, is not an adequate locus for all the screws 

 of given pitch. 



It will be convenient for our immediate purpose to designate the 

 conies by the pitches to which they correspond. Thus we have the 

 %ero-pitch conic, the infinite-pitch conic, and their points of intersection 

 we may speak of as th.e fundamental poijits. 



Let 6 and <^ be two screws of the system, if they are reciprocal 

 then {Screws, p. 35) 



Pa Ol ^1 + Pp O2 ^2 +Py O3 S^3 = 0, 



whence we deduce the result that 



If two screws are reciprocal, then their correspondiny points are conju- 

 gate with respect to the zero-pitch conic. 



If the two screws were at right angles, then we would have the 

 following relation between their co-ordinates : 



di 4>i + $2 (J32 ,+ ^3 ^3 = ; 

 whence we find 



If two screws are reciprocal, then their correspondiny points are conju- 

 gate with respect to the infinite-pitch conic. 



It will also be easy to show that 



The anyle between two screws is proportional to the loyarithm of the 

 anharmonic ratio in lohich the line joininy their correspondiny points is 

 divided by the infinite-pitch conic} 



^ If we regard tlie infinite-pitch conic as the absolute, then the angle hetween 

 two screws is nothing else than the "distance" in the non-Euclidian sense between 

 their two corresponding points. 



