348 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. [349 
71. I remark in passing that the equation of the one-with-twofold centre locus 
may also be obtained by means of the equations 
3q(2/i --gQ y 1 Oi-^i) ^i(^i-yi) =0 
x y z ’ 
(- «1 + Vi + z-i) X + (x 1 -y 1 + z 1 )y + (x 1 + y 1 -z 1 )z = 0, 
which determine the remaining two critic centres corresponding to a given critic centre 
(x 1} y l , Zj); in fact, in order that the centre (x 1} y 1} z x ) may be accompanied by a 
twofold centre the line must touch the conic; and the analytical condition, substituting 
therein (x, y, z) in the place of (x ly y 1} zj, is found to be 
xyz I* 3 + y 3 + z 3 — (yz 2 + y~z + zot? + z-x + xy 2 + x 2 y) + Sxyzj = 0, 
the three lines xyz = 0 are not properly part of the locus, but their appearance may 
be accounted for without difficulty. 
72. Assume that the line \x + /j,y + vz = 0 passes successively through the points 
(x = 0, y z — 0) (y — 0, z-x = 0), (z = 0, x- y = 0), 
or, what is the same thing, the points (0, 1, 1), (1, 0, 1), (1, 1, 0): then {ante, No. 56) 
the critic centres are in all these cases respectively on the conics. 
or, as these may be written, 
11 4 
—I — Q 
y Z X + y + z 
11 4 
—I — — 0, 
Z X X + y + z 
11 4 
1 ; ;—- = 0 ; 
X y X + y + z 
(y - z) 2 + x(y + z) = 0, 
(z -x) 2 + y(z + x) = 0, 
(x -y) 2 + z (x + y)= 0, 
the first of which is a conic touching the lines x = 0, y + z = 0 at the points of inter 
section with the line y — z = 0; and similarly for the other two conics. 
73. Suppose that the line \x + /xy + vz = 0 passes through the point (4, —1, — 1), 
nr let 4\ — fi — v = 0; we have (a, /3, 7) = (4, —1, —1); and the critic centres lie on 
the curve 
xyz x+y+z 
4 (y + z) y + z 
x(x+y + z) yz 
that is