82 



Proceedings of the Royal Irish Academy. 



If the circle x^ -^y"- -\- lax + l^y + y = has double contact with 

 it, then 



{2ax + 2ySy + y)- + ax^ +ly'^ + c-\- 2gx + 2/y + <? + A.[«2 + y- + 2ax + ^jBy + y] 



= 7"2 . 



hence, as before, every minor of 



« + A + 4a2 4a)8 2ay + ^ + aA 



4/3a J + A + 4y82 2^y+/+/3A 



2ay + ^^r + Aa 2;8y +/+ A^ c + Ay + y- 



1 - 4a 



a a + \ 



(3 



A A 



4-^2 



-4y3 





 5 + A 



/ 



- (A + 2y) 



9 

 f 



A^ 

 c — - 



= 0. 



« + A g 



i + A / 

 . A^ 



= 0, or -^ + -^ 



« + A 6+ A 4 



1 -4a 



-AlB 



a « + A 







/5 



5 + A 



a « + A 







/3 



J + A 



A A 



4^2 ^ 



/ 



4a2 4^2 



= 0, or + ^ + 1 = 0; 



« + A J + A 



2aff 2y8/ A 



= 0, or + - — - = y + - . 



' a + X b +X ' 2 



i. Shows that A is determined by a quartic. 



ii. Shows that the centre of the enveloping circle moves on a fixed 

 conic, and also to each value of A we have a determinate conic, and 

 the conies are confocal. 



