350] 
ON THE CLASSIFICATION OF CUBIC CURVES. 
389 
and thence 
/j, (a, b, c§x, y) 2 — A (b, c, (T§x, y) 2 = 0, 
or as it may also be written 
(/ia — \b, fib — Ac, fic — A<T§x, y) 2 = 0, 
and also 
2 (Xx + fiy) — 3vz = 0, 
which two equations determine the critic centres for a given position of the satellite 
line; the first of them gives a pair of lines through the asymptote point; the latter 
is a line parallel to the satellite line : there are thus two critic centres. 
104. The condition for a twofold centre is 
(ac — b 2 , be — ad, bd — c 2 \A, fif = 0, 
so that there are a pair of twofold centres which will be real if 
imaginary if 
(be — ad) 2 — 4 (ac — b 2 ) (bd — c 2 ) = +, 
(be — ad)' 2 — 4 (ac — b 2 ) (bd — c 2 ) = —, 
that is, the twofold centres will be real or imaginary, according as the equation 
(a, b, c, d~§x, yf = 0 
has its roots one real and two imaginary, or all three real; viz. the twofold centres 
are real for the Hyperbolas © Defective; imaginary for the Hyperbolas © Redundant. 
And we see also that for the Hyperbolas © Redundant the critic centres are always 
real; but that for the Hyperbolas © Defective, they may be both real, or both 
imaginary, or may coincide together, giving a twofold centre. But the two cases are 
best studied by assuming different special forms for the equation. 
The Hyperbolas © Redundant. Article Nos. 105 to 107. 
105. The equation may be taken to be 
xy (x — y) + kz 2 (Ax + py+ vz) = 0, 
or writing z= 1, then, the equation is 
xy (x — y) + k (A* 1 + fiy + vz) = 0. 
We may, to fix the ideas, consider the case where the three asymptotes are 
parallel to the sides of an equilateral triangle; x, y, and x — y will then denote the 
perpendicular distances of the point from the three asymptotes respectively.