172 
MATHEMATICS: F. MORLEY 
We now find where this curve in y meets the line 1/^. That is, we elimi- 
nate y from 
2 = 0, (iv) = 0, {y/^) ^. 0. 
We have then 
2^0 (6^ - ^2^) 
^0 ^1 b 
1/ro i/fi i/r2 
0, 
(3) 
or since {x^ = 0 
2 (ri/r2 - r2/rO (^o^ - ^i^) (^o^ - ^2^) = 0 
or if a be the join of ^ and 1/^, 
The values of ^ common to this equation and {x^) = 0 give the inter- 
sections y of px^ and 1/^. Thus when {x^ is a line of (3) then as 2 moves 
on ^ the curve px^ touches 1/^ at the point y, and re is on the envelope 
sought. 
Now the quartic (3) is two conies on the lines 1, =t 1, =t 1. And 
when 
2 -s/a^Xi = 0, (4) 
the two conies become one conic R whose equation is 
(5) 
The conic (4) occurs then twice in the envelope, the other factors being 
[xo^ + 2xiW]\ 
XqXiX^j 
and the cubic of the system with a double point at a, namely 
2 XQfii'qz 
where i) is the join of x and a. 
The conic (4) is the Feuerbach conic F, for it is on the diagonal lines 
of the four lines, and having the line equation 
2 a/^ = 0, 
it is on ^ and 1/^. 
The construction of the cubics which touch both ^ and 1 is then as 
follows. Take a point x on F, and draw from x the two tangents to the 
conic R. The diagonals of this line-pair and the pair f and 1/f give 
the points of contact of the two cubics. If the diagonals meet at J, 
