349] on 
then observing that 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
347 
d + A d + ya 0 + v 
A /a v 
the first part is 
Sfr-,)(«+£)( 
.(d + A)(d + ya) (0 + X)(0 + v) 
which is 
- (0 + x)(0 + /*)(0+z/) S ^ ( d + 2 t) {"( 0 + ^( d + ^ + X + ^ + y )} > 
1 
£ (ya + Z>) ^d + (Ad 2 — ’hfiv), 
(0 + A) (d + ya) (0 + r*) 
and observing that the sum is 
the first part is 
2\/jiv (/a — v){v — A) (A — ya) 
0 {0 + A) (0 + fi) (0 + v) 
The second part is 
2\/iv 
£ (ya — v) (A — 20) (A + 0), 
0 (0 + A) (A + fi) (0 + v) 
in which the sum is 
— £ (ya — v) (A 2 — Ad — 2d 2 ) = £ A 2 (ya — v) = — (ya — v) (v — A) (A — ya), 
so that the second part is 
2\fjbv (/a — v) {v — A) (A — ya) 
d (d + A) (d + ya) (d + v) 
and the sum of the two parts is = 0, which proves the theorem. 
70. Let x 1} y 1 , z 1 be the coordinates of a critic centre, then the equation of the 
polar in regard to the twofold centre conic is 
(- x i + y l + z x ) x + Oi - y 1 + z^y + + y 1 -z 1 )z = 0, 
and the equation of the conic through the five points is 
«1 (yi -*i) 
Vi (zi-m,) 
y 
z 1 (a?i — yi) _ q 
x 
and these equations together determine the remaining two critic centres. 
44—2