424 
ME. A. CAYLEY’S MEMOIE 02s CLEYES OF THE THIED OPuDEE. 
Let one of the lines be 
then the other is 
and we find 
—W -\ — ?/+ -2=0; 
-Xv^+21YvX-Zk^ =0 
-X/A^--Y?i^+2/Zp=0, 
any tsvo of which determine the ratios "k, [Jj, v. 
The elimination of X, Y, Z gives 
which is equivalent to 
or omitting a factor, to 
2l[/jV , — V^, — 
— , 21va, — 
—[j/, —X% 2IX[/j 
= 0 , 
X[x>v{ — + ( — \-\-^l^)X^v} =0 ; 
which shows that the line in question is a tangent of the Pippian. 
15. To find the equation of the pair of lines through O. 
The equation of the pair of lines through E is in hke manner 
li!{x^+2lyz)-^Y\y^-\-2lzx)-\-7J{z^-^2lxy)^^-, 
and combining this with the foregoing equation, 
^{x‘^-\-2lyz) -\-Y{y‘^-\-2lzx) + Z{z^-{-2Jxy) = 0 
of the pair of lines through F, viz. multiplying the two equations by 
X^X'+Y^Y'+Z^Z', -(XX'=^+YY'^+ZZ'^), 
and adding, then if as before 
a:h: c=yyi—^l : al—yl : f3^—af!, 
we find as the equation of a conic passing through the points A, B, C, D, the equation 
a(x^-i-2lyz)-{-d(y^-i-2 Izx ) + + 2 Ixy) = 0 . 
But putting, as before. 
a' : y ; c' = cc^ : (5}] : 
then a!, h\ c' are the coordinates of the point O, and the equations 
aa! -{-h' c) =0, 
hV +Z(m' +<?'«) = (), 
cc' a'b) = 0 
show that the conic in question is in fact the pair of lines through tlie point (). 
