238 
ME. A. CAYLEY ON THE POEISM OE 
conics. But for a figure of an even number, 2w, of sides, then starting from a point of 
intersection of the two conics, the side will terminate at a second point of intersec- 
tion, and then the same series of sides will be repeated in the reverse order, so that the 
sides will coincide in pairs, first and last, second and last but one, ?^th and (7i-|-l)th. 
For a figure of an odd number of sides, the relation involves only a single point of 
intersection, but for a figure of an even number of sides, it involves two points of inter- 
section. 
Now in the case of two circles, for a polygon of an odd number of sides, the same 
relation is obtained, whether we take as the point of intersection one of the actual points 
of intersection, or a circular point at infinity, and the relation [2% + l]=:0 does not break 
up into factors. And so for a polygon of an even number of sides, then taking for the 
two points of intersection, the two actual points of intersection, or the two circular points 
at infinity, we have one form of resu.lt; but taking for them an actual point of intersec- 
tion and a circular point at infinity, we have a different form of result ; and the equation 
[2??] = 0 does break up into factors. 
This is verified very simply in the case of the quadrangle. Taking for the two points 
of intersection the circular points at infinity, the line joining them is the line infinity, 
and its pole, with respect to the inscribed circle, is the centre of this circle ; the relation 
therefore is that the centre of the inscribed circle lies on the circumscribed circle. But 
when this is the case, it is easy to see that the pole (with respect to the inscribed circle) 
of the radical axis, lies also on the circumscribed circle ; this pole and the centre of the 
insciibed circle are in fact the extremities of a diameter of the circumscribed circle. 
The condition thus obtained is — a^=0 (which is M. Mention’s condition f=0). We 
have next to find the analytical relation when the pole (with respect to the inscribed) 
circle, of the line joining one of the actual points of intersection with a circular point at 
infinity is a point on the circumscribed circle. This I effect as follows : — Taking 2:=0 as 
the equation of line infinity, if the origin be taken on the middle point of the radical 
axis, and if ^=0 be the radical axis, then the equations of the two circles may be taken 
to be 
Inscribed circle, 2Z sz—Vz^=0, 
Circumscribed circle, — 2L.r2 — V2®=0. 
A circular point at infinity is 
a^:^:z=l:i:0, (^■=^— 1). 
An actual point of intersection is 
: y : z=0 : \/v : 1. 
The line joining these is 
xi—y-\-z\/ V=0. 
Its pole, with respect to the inscribed circle, is 
X'.y. z= — V • 'V V—il : 1. 
