MancJicster Memoirs, Vol. Ivi. {]g\2), No. 4. 13 



This form of relation between ?- and r' proves the 

 theorem.'* 



If <p is the angle which the tangent plane at P{r, O, O) 

 makes with that at O it has been shown by Chasles 

 that y = (i»tan^, w being the pitch of the screw by means 

 of which one generator is moved into the position of the 



next, i.e. («» = -, where 00' = d and 6 is the angle between 

 n ° 



G and G' . If this relation (which is not difficult to prove) 



is assLMTied we find 



(y-/ + m) tan 4- (;• - /-')w = O. 



If we write w= — i it is easily seen that w = -j , so that 



this relation is identical with (x). 



If ^ = _ we get rr' -\- w- = O , so that the points form 

 2 



a system in involution. This is a known theorem.'' 



^ Russell's " Pure Geometry," p. 1 14. 



" A. J. C. Allen, Mess, of Math., vol. xii., p. 26. 



