349] 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
343 
Consider any three lines x, y, z, a line S, and the line T\ then the harmoconics 
being all as to the same line T, we have the theorem 
Harmoconic of intersection of x, S in regard to pair of lines y, z, 
Ditto 
Ditto 
of y, S 
of 2, S 
z, x, 
all pass through the same three points. 
And taking x — 0, y = 0, z = 0 for the equations of the lines x, y, z; Xx + /zy + vz = 0 
for the equation of the line S; and x + y + z = 0 for the equation of the line T, the 
harmoconics just spoken of are the above-mentioned three conics respectively. 
59. In fact, considering the harmoconic of intersection of x, S in regard to the 
pair y, 0; and taking x', y', z' as the coordinates of a point P of the harmoconic, 
then the equation of the line AP is 
x, y, z = 0, 
x', y', z' 
0, v, -fi 
which is the line through the last-mentioned point and the point (y = 0, z = 0). 
The line from the point A to the point (y — 0, z = 0) is 
y Z ' - Z y' = 0. 
60. By the definition of the harmoconic, the last-mentioned two lines are harmonics 
in regard to the lines y = 0, 0 = 0; that is, we have for the equation of the harmoconic 
in question 
— y'(fix' + /zy' + vz) + z (vx + /zy' + vz') = 0 ; 
this equation may also be written 
(vz' - /iy) O' + y’ + z') - 2 (v - fi) y'z' = 0, 
or, what is the same thing, 
X + y' + z' 
whence writing x, y, z in place of x, y', z\ we see that this harmoconic is in fact the 
first of the above-mentioned three conics.