[ 225 ] 
XI. On the Po7'ism of the in~and-circumscribed Polygon. 
By Arthur Cayley, Esq., F.B.S. 
Eeceived February 20, — Read March 7, 1861. 
The Porism referred to is as follows, viz. that two conics may be so related to each other, 
that a polygon may be inscribed in the one, and circumscribed about the other conic, 
in such manner that any point whatever of the circumscribing conic may be taken for 
a vertex of the polygon. I gave in the year 1853, in the Philosophical Magazine*, a 
general formula for the relation between the two conics, viz. if U=0 is the equation of 
the inscribed conic, V = 0 that of the circumscribed conic, and if disct. (U+|V), where 
^ is an arbitrary multiplier, denotes the discriminant of U+|V in regard to the coordi- 
nates {x, y, z) (such discriminant being of course a cubic function in regard to and 
also in regard to the coefficients of the two conics U, V, jointly), then if we write 
v/disct. (U+|V) A + + Cr H- Df + Er + Ff + Gf + &c. , 
the relations for the cases of the triangle, pentagon, heptagon, &c. are 
c= 
0, 
C, 
D 
= 0, 
C, 
D, 
E 
D, 
E 
D, 
E, 
F 
E, 
F, 
G 
respectively, while those in 
the cases of the quadr 
angle, h 
D= 
0, 
D, 
E 
= 0, 
E>, 
E, 
F 
E, 
F 
E, 
F, 
G 
F, 
G, 
H 
=0, &c. 
= 0, &c. 
respectively. The demonstration of this fundamental theorem is for greater complete- 
ness here reproduced ; but the chief object of the memoir is to direct attention to a 
curious analytical theorem which is an easy d piiori consequence of the Porism, and to 
obtain the relations for the several polygons up to the enneagon, in a new and simple 
form which puts in evidence d qjosterion for these cases the analytical theorem just 
referred to. The analytical theorem rests upon the following considerations : — the 
relation for a hexagon ought to include that for a triangle ; in fact a triangle with its 
• See the papers — “On the Geometrical Representation of the Integral ^dx-i-\/ {x+a)(x + h){x + c),” 
Phil. Mag. April 1853. 
“Note on the Porism of the in-and-circumscribed Polygon,” Phil. Mag. August 1853. 
“ Correction of two Theorems relating to the Porism of the in-and-circumscribed Polygon,” Phil. Mag. 
November 1853. 
“Developments of the Porism of the in-and-circumscribed Polygon,” Phil. Mag. May 1854. 
MDCCCLXI. 2 I 
