392 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[146 
and combining this with the foregoing equation, 
X (x 2 + 2 lyz) + Y (y 2 + 2 Izx) + Z(z 2 + 2 Ixy) = 0 
of the pair of lines through F, viz. multiplying the two equations by 
X 2 X' + PF + Z 2 Z', - (XX' 2 + YY' 2 + ZZ' 2 ), 
and adding, then if as before 
a : b : c = yy — /3% : a^ — yt; : /3£ — 007, 
we find as the equation of a conic passing through the points A, B, C, D, the equation 
a (.x 2 + 2lyz) + b(y 2 + 2Izx) + c(z 2 + 2Ixy) — 0. 
But putting, as before, 
a' : b' : c' = a% : /3y : yÇ, 
then a', b', c' are the coordinates of the point 0, and the equations 
aa! +1 (be' + b'c) = 0, 
bb' + l (ca'+ c'a) = 0, 
cc' + l (ab 1 + a'b) = 0, 
show that the conic in question is in fact the pair of lines through the point 0. 
16. To find the coordinates of the point T, which is the harmonic of G with 
respect to the points E, F. 
The coordinates of the point in question are 
uX-vX', uY-vY', uZ-vZ', 
where u, v have the values given in No. 10, viz. 
u = - 31 2 (XX' 2 + YY' 2 + ZZ' 2 ) + (1 + 2P) ( Y'Z'X + Z'X' Y + X' Y'Z), 
v = SI 2 (X 2 X' +Y 2 Y' + Z 2 Z') - (1 + 21 s ) ( YZX' +ZXY' + XYZ') ; 
these values give 
uX -vX' = - 31 2 {2X 2 X' 2 + (IF + X'Y) YY' + (XZ' + X'Z) ZZ'} 
+ (1 + 21 s ) {(IF + X'Y) (XZ' + X'Z) + XX' (YZ' + Y'Z}; 
and therefore 
and consequently, omitting the constant factor, the coordinates of T may be taken to be 
— la 2 + /3y, —1/3 2 + 7 a, — ly 2 + a/3. 
uX -vX' = - 31 2 j2a* - j £7j + (1 + 21 s ) ji ßy