92 
[500 
500. 
ON A THEOREM RELATING TO EIGHT POINTS ON A CONIC. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xi. (1871), 
pp. 344—346.] 
The following is a known theorem: 
“ In any octagon inscribed in a conic, the two sets of alternate sides intersect in 
the 8 points of the octagon and in 8 other points lying in a conic.” 
In fact the two sets of sides are each of them a quartic curve, hence any quartic 
curve through 13 of the 8 + 8 points passes through the remaining 3 points: but the 
original conic together with a conic through 5 of the 8 new points form together 
such a quartic curve; and hence the remaining 3 of the new points (inasmuch as 
obviously they are not situate on the original conic) must be situate on the conic 
through the 5 new points, that is the 8 new points must lie on a conic. 
We may without loss of generality take (af 2 , a 1 , 1), (a 2 2 , a 2 , 1), ... (a 8 2 , a 8 , 1), as the 
coordinates (x, y, z) of the 8 points of the octagon; and obtain hereby an a posteriori 
verification of the theorem, by finding the equation of the conic through the 8 new 
points: the result contains cyclical expressions of an interesting form. 
Calling the points of the octagon 1, 2, 3, 4, 5, 6, 7, 8, the 8 new points are 
12.45, 23.56, 34.67, 45.78, 56.81, 67.12, 78.23, 81.34, 
viz. 12.45 is the intersection of the lines 12 and 45; and so on. The 8 points lie 
on a conic, the equation of which is to be found. 
The equation of the line 12 is 
x — (a x + a 2 ) y + <x-y<x 2 z = 0, 
or as it is convenient to write it 
x — (1 + 2) y + 12 . £ = 0, 
viz. 1, 2, &c., are for shortness written in place of a 1} a 2 , &c. respectively.