64 Proceedings of (he Royal Irish A cademy. 



If we now seek the locus of the intersection of tangents at corresponding 

 points, we obtain, without difficulty, the following expressions for the 

 coordinates S. Y. Z of a point on this locus : — 



X = 3(2BJ-A*K)x - SAL. I 

 I* = 3 ( 1B-T - A-K | j - :U/.' ; L (6) 



Z = 4[8©/ + AJ' -3BK] I 

 where ./' = J( Q, }: . 



w, as the coordinates of this curve are thus expressed rationally in 

 terms of a parameter xij or 9, and involving that parameter in the seventh 

 degree, the locus we seek is a uni-cursal curve of the seventh degree. 



We now pass on and seek to determine the parameters of the two points 

 in which the tangent at a point, whose parameti igain meets the curve. 



This is effected by eliminating Z l>etween the equation of t! _ it, as 



:i in <4>. and that of the curve in 1 1. 



The eliminant is clearly a homogeneous elation of the fourth degree in 

 1 }"; and if we put A' = 9'Y and divide by V. we obtain 



^V -By-UBA* [A -.i.m'- ii. ' -:•.!•. I'/ = 0, 



whe A' " - " md 



A = 3A(B 



A*K- IBJ . 



- A | ft - J 



_ I: 



ear that this equation must contain th< " - 0)\ and 



of which it must -1 before we can obtain the quadratic equation in 



6', the roots of which determine the parameters "f the points 



The work now 1 ■ xceedingly complicated and intricate, and I only 



: the method which I have adopted in clearing this equation 

 of the factor <0' - 9v. 



If we call V - 9 = h, we may write the equation given in (7) in the 

 form 



.: . • V .[a.i 2 A'B-AB (B-t v 2) 



-0. (8) 

 •hat the coefficient of 9A : in the above equation 

 is divisible by h, and I find that 



- ![-6 r.-yi:)- a- a i\ (9 



where 



h*' = A'B- J//. , 



- V-d 



