392 
OX THE CLASSIFICATION OF CUBIC CURVES. 
[350 
eliminating k from the second and third equations 
gz (cz + gx) — by (2gy + Svz) = 0, 
that is 
— 2 bgy 2 + cgz 2 + ggzx — 3bvyz — 0 ; 
or reducing by means of the first equation written under the form 
by' 1 + cz 1 + 4gzx = 0, 
we find 
ocgz 2 + 9ggzx — 3bvyz — 0, 
that is z = 0, which may be rejected, or else 
cgz + 3ggx — bvy = 0, 
or, as it may be written, 
g (3gx + cz) — vby = 0. 
Hence the entire series of critic centres lie on the conic 
by 1 + cz 2 + 4 gzx = 0, 
and corresponding to the satellite line gy + vz = 0, we have the two critic centres 
which are the intersections of the conic by the line 
g (3gx + cz) — vby = 0, 
the lines pass through the fixed point Sgx + cz = 0, y = 0, and form a pencil homographic 
with the satellite lines gy + vz = 0. 
113. We have a twofold centre if the line touches the conic, the condition for 
this is 
(be, — 4<g 2 , 0, 0, — 2bg, O'fySgg, — bv, eg) 2 = 0, 
that is 3bg 2 (3eg 2 + 4bv 2 ) = 0, or simply, 
3 eg 2 + 4b v 2 = 0, 
and from this and the equation gy + vz = 0, eliminating /u, and v we find 
4 by 2 + 3 cz 2 = 0, 
for the equation of the satellite lines which respectively give rise to a twofold 
centre; the lines in question are real or imaginary according as the lines by 2 + cz 2 = 0 
are real or imaginary, that is, according as the line x = 0 cuts the conic by 2 + cz 2 + 2gzx — 0 
in two real, or in two imaginary, points. 
114. Writing now 6 = 1, c=-mn, 2g = n and x + mz in the place of z, the 
equation is 
(x + mz) (y 2 + nzx) + 2kz 2 (gy + vz) = 0,