[500 
500] 
ON A THEOREM RELATING TO EIGHT POINTS ON A CONIC. 
93 
Î A CONIC. 
>1. xi. (1871), 
sides intersect in 
lence any quartic 
points : but the 
:s form together 
nts (inasmuch as 
te on the conic 
> 8 2 , a 8 , 1), as the 
•y an à posteriori 
Dugh the 8 new 
points are 
H . 34, 
The 8 points lie 
7- 
The coordinates of the 12.45 are consequently proportional to the terms of 
1, -(1 + 2), 12, 
1, - (4 + 5), 45, 
or say they are as 
12 (4 + 5) - 45 (1 + 2) : 12-45 : 1 + 2 - (4 + 5). 
The equation of the line (12.45) (23.56) which joins the points 12.45 and 
23.56 thus is 
12 (4+ 5)-45 (1 + 2), 
23 (5 + 6) - 56 (2 + 3), 
V > 
12 + 45, 
23 - 56, 
£ 
= 0, 
(1 + 2) — (4 + 5) 
(2 + 3) — (5 + 6) 
where the determinant vanishes identically if 2 - 5 = 0 (a, - a 5 = 0); it in fact thereby 
becomes 
x , y , z 
2 2 (l-4), 2(1-4), (1-4) 
2 2 (3 — 6), 3(3-6), (3-6) 
which is = 0 ; the determinant divides therefore by 2 — 5 ; the coefficient of x is easily 
found to be 
= (2 - 5) (12 - 23 + 34 - 45 + 56 - 61), 
and so for the other terms ; and omitting the factor 2 — 5 the equation is 
x {12 - 23 + 34 - 45 + 56 - 61} 
-y{12(4 + 5)- 23 (5 + 6) + 34 (6 + 1) - 45 (1 + 2) + 56 (2 + 3) - 61 (3 + 4)} 
+ z {1234 - 2345 + 3456 - 4561 + 5612 - 6123} = 0. 
There is now not much difficulty in forming the equation of the required conic ; 
viz. this is 
(2-8) {«-(6 + 7)y+67s} x 
' æ [12 - 23 + 34 - 45 + 56 - 61] 
- y [12 (4 + 5) - 23 (5 + 6) + 34 (6 + 1) - 45 (1 + 2) + 56 (2 + 3) - 61 (3 + 4)] - 
[ + z [1234 - 2345 + 3456 - 4561 + 5612 - 6123] 
+ (6 —8) {« — (1 + 2) y + 12s} x 
' «[23-34 + 45-56 + 67-72] 
- y [23 (5 + 6) - 34 (6 + 7) + 45 (7 + 2) - 56(2 + 3) + 67 (3 + 4) - 72 (4 + 5)] 
+ z [2345 - 3456 + 4567 - 5672 + 6723 - 7234] 
= 0. 
7