II }.' : 
! 
NOTE ON THE PORISM OE THE IN-AND-CIRCUMSCRIBED POLYGON. 
(I now use the term discriminant in the same sense in which determinant is sometimes 
used, viz. the discriminant of a quadratic function ax 2 + by- + cz 1 + 2fyz + 2gzx + 2hxy 
or (a, b, c, f g, h) (x, y, z) 2 , is the determinant k = abc — af 2 — bg 2 — ch? + 2fgh), and if 
n ^ = f 
J or 
then the following theorem is true, viz. 
Jot’ 
“ If (6, oo), (O', oo) are the parameters of the points P, P' in which the conic 
U=0 is intersected by the tangent, the parameter of which is p, of the conic 
kU + V—0, then the equations 
U6 = Up- nk, 
\W= Ylp + Uk, 
determine the parameters 0, 6' of the points in question.” And again,— 
“ If the variable parameters 0, 0' are connected by the equation 
U0' -U0=2Uk, 
then the line PP' will be a tangent to the conic kU +V = 0.” Whence, also,— 
“ If the sides of a triangle inscribed in the conic U = 0 touch the conics 
k U+ F = 0, 
k' U+ V=0, 
k"U+ V=0, 
then the equation 
TLk+Wc' + Uk" = 0 
must hold good between the parameters k, k', k"P 
And, conversely, when this equation holds good, there are an infinite number of 
triangles inscribed in the conic TJ = 0, and the sides of which touch the three conics; 
and similarly for a polygon of any number of sides. 
The algebraical equivalent of the transcendental equation last written down is 
1, k , = 0; 
1, k', VqF 
1, k", VqF 
let it be required to find what this becomes when k — k' = k'' = 0, we have 
VOk=A+Bk+Ck 2 + ... ,