350 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
we have 
y : z : x + y : x + z 
2 
0 + y ' 0 + v 0 0 + v ' 0 0 + ia ’ 
1 1 0 + 2v 0 + 2y 
and consequently 
6 + y 6 + v 6 (6 + v) 6 (0 + y) ’ 
y : x + z=0 \ 0 + 2y\ z\ x+y—0\ 0+ 2v, 
the foregoing equation 
gives 
that is 
№ _ (0 + 2y) (0 + 2v) = 0 
oyz ~{x + z){x + y) = 0, 
x {x + y + z) — fyz = 0 ; 
or the critic centres corresponding to the two values of 0 lie on the conic. The 
line joining them is the polar of the point —1, 1^ in regard to the twofold 
centre conic; the equation therefore is 
(y — v) x — (3/a + v)y + (y + 3v) z= 0. 
77. Starting with a critic centre on the line y + z-0, the other two critic centres 
lie on the conic x (x + y + z) — tyz = 0, and they are the intersections of the conic by 
the polar of the first centre in regard to the twofold centre conic. 
78. Starting with a critic centre on the conic x{x + y + z) — 4>yz = 0, the other- 
two critic centres lie one on the conic, and the other on the line y + z— 0; viz. the 
polar of the first centre in regard to the twofold centre conic meets the line in one 
point, and the conic in two points; of these one is the harmonic of the point on 
the line in regard to the twofold centre conic; this point on the conic, and the 
point on the line, are the other two centres. 
79. The point (4, —1, —1) is of course one of a system of three points; viz. 
these are (4, — 1, — 1) (- 1, 4, —1), (—1, —1, 4); and the corresponding loci of the 
critic centres are 
(;y + z) |x {x + y + z) — 4xy^ = 0, 
(z + x) jr/ (x + y + z) - 4^| = 0, 
(x + y)\z{x + y + z)- 4«?/J = 0, 
the three points in question are {ante, No. 24) shown to be nodes of the twofold centre 
envelope.