ON THE CLASSIFICATION OF CUBIC CURVES. 
374 
[350 
Memoir on Involution, viz. the critic centres corresponding to the satellite lines through 
the point 
(-4, 1, 1) or (2, -J, -!) 
lie one of them on the line y + z — 0, and the other two on the conic x (x + y + z) — \yz — 0 ; 
reducing by the condition x+y+z= 1, these equations become respectively x=l, and 
(y + i)(* + i)-&=0. 
66. The foregoing positions of the satellite line, and the critic centres, as exhibited 
in the figure, were selected partly for facility of delineation; I wished however to 
examine the effect of the passage of the satellite line through a node of the envelope; 
and it appears that such passage does not give rise to any marked peculiarity in 
regard to the critic centres. The - selected positions are sufficient to indicate the 
circumstances of the critic centres as the satellite line passes from the position A at 
infinity continuously to the position A' at infinity; in particular we see that as the 
line passes from A to G, or from C’ to A', there are three real centres; but that as 
the line passes from G to G' there is only one real centre. 
67. The case of the satellite line parallel to the asymptote x — 0, is included (as 
already mentioned) as a limiting case in the foregoing one; the harmonic conic is 
here the pair of lines x (y — z) = 0; and we have two critic centres on the line 
y — z= 0, (the perpendicular), and the third (not properly a critic centre) at infinity on 
the asymptote x = 0; in fact, starting with a critic centre on the line y — z — 0, the 
polar of the centre in regard to the twofold centre conic or circle is a line parallel 
to the asymptote x — 0, and which therefore meets the harmonic conic x {y — z) = 0 in 
a second centre on the line y — z= 0, and in the point at infinity on the line x — 0. 
But the analytical theory of the case is peculiar and may be specially considered. 
68. Writing y = v, the equation of the satellite line is Xx + y (y + z) = 0, or 
putting x+y+z= 0 this is x — ——The equation in 9 (see Memoir on Involution, 
y — A 
No. 20) becomes 
(9 + y) {9°- — 9y — 2Xy) = 0; 
or, disregarding the factor 9 + y = 0, which corresponds to the centre at infinity, the 
equation is 
9 2 — 9y — 2 \y = 0, 
which is a quadratic equation, giving therefore two values of 9, and the corresponding 
critic centres lie on the perpendicular y — z = 0, the x coordinate being given by the 
equation 
1 / 1 2 \ 
9 + \ ' \6 + \ + 9 + y) 
1 
9 + X 
Q — 2 0 ■=" (0 + ^)- 
We have therefore conversely