36 Proceedings of fhe Royal Irish Academy. 



the triangle PQR ; in the latter, the twist velocities must be pro- 

 portional to the sides of the triangle P'Q'R'. If, therefore, <? be a 

 constant, we have 



rP'Q' = dPQ, 



qP'P' = dPP, 



pQ'P' = dQP. 



We have similarly the three other groups of equations 

 rQ'S' = aQS, qP'S' = cPS, rP'S' = IPS, 



qR'S' = aRS, pQ'S' = cQS, pR'S' = bRS, 



sR' Q' ^aRQ. s QP' = c QP. sRP' = bRP. 



"Whence we easily deduce 



ap = bq = er = ds = hpqrs ; 



whence J3"is a new constant. We hence obtain from the first equa- 

 tion 



P'Q' = hPQpq. 



As this is absolutely independent of R and S, it follows that h must 

 be independent of the special points chosen, and that consequently for 

 any two points on the circle P and Q, with their corresponding points 

 P' and Q', we must have 



PQ' 



In the limit we allow P and Q to coalesce, in which case, of course, 

 P' and Q' coalesce, and ^j and q become coincident ; but obviously wo 

 have then 



PQ : ML : : PX : LX, 



P'Q' : ML :: PY: LY; 



whence 



1 





P'Q' PY LX 





PQ " PX~ LY' 



and as 



PY-Jc^^^^ and PXoc 



we have finally 





LX 



LY' 



