342 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
57. If we take for (a, b, c) the coordinates of the point of intersection of the 
line \x + /xy + vz = 0 with any one of the lines a? = 0, y = 0, z = 0, x + y + z = 0, then in 
each case the cubic breaks up into the same line and a conic, viz. we have the conics 
- _ £ _ 2 JT ~ _ o 
y Z X + y + Z ’ 
A v 2 (A — v) _ 
2 x x+y+z ’ 
+ _ ^ _ 2 (> z29 - o 
x y x+y+z ’ 
ix — v v — A A — it 
+ +^-U = 0; 
xyz 
and it is to be noticed that in each case the critic centres all of them lie on the 
conic. In fact, since the point (0, v, — fx) is an arbitrary point on the line x = 0, a 
line \x + /xy + vz = 0 passing through the point in question is an absolutely arbitrary 
line, and the corresponding critic centres therefore do not lie on the line x = 0; that 
is, they lie on the conic 
v _ P _ 2 0-/*) = q . 
y z x+y+z ’ 
and it may also be remarked that the elimination of A, 6, from the system 
6 + \ : 6 + fx : 6 + v : 0 = - : 2 , 
x y z x+y+z 
or, what is the same thing, the elimination of 6 from the system 
11 9 
0 + fx : 0 + v : 0 = — , 
y z x+y+z 
gives the last-mentioned equation, unencumbered by the factor x — 0. 
We have thus four conics, each of them passing through the three critic centres 
which correspond to the line \x + /xy + vz = 0; as to the signification of the first three 
of these conics, I remark as follows. 
58. The ‘ harmoconic ’ of a point A as to the line T in respect of the conic ®, 
may be defined as follows; viz. considering the pencil of lines through A, the locus 
of the fourth harmonic of the point in which a line of the pencil meets T, in 
regard to the two points in which the same line meets the conic ©, is a conic 
which is the harmoconic in question. (In particular, if the line T pass through the 
point A the harmoconic breaks up into the line T and into the polar of A.) The 
conic © may of course be a pair of lines.