142 
DEVELOPMENTS ON THE PORISM OF THE 
[128 
We have, therefore, 
A = l, 
2jB = M + 2, 
- 8(7 = M 3 - 4if, 
16D = M 3 -2M 2 , 
- 128E = oif 4 - 8if 3 , 
&c. 
1024 (CE - D-) = M i (M 2 -\2M+lO), 
Szc. 
Hence for the triangle, quadrangle and pentagon, the conditions are— 
I. For the triangle, 
M+2 = 0. 
II. For the quadrangle, 
if-4 = 0. 
III. For the pentagon, 
and so on. 
if 2 - 12if + 16 = 0 ; 
It is worth noticing, that, in the case of two conics having a 4-point contact, 
we have F = 0, and consequently if = 1. The discriminant is therefore (1 + £) 3 , and 
as this does not contain any variable parameter, the conics cannot be determined so 
that there may be for a given value of n (nor, indeed, for any value whatever of 
n) an infinity of w-gons inscribed in the one conic, and circumscribed about the 
other conic. 
The geometrical properties of a triangle, &c. inscribed in a conic and circum 
scribed about another conic; these two conics having double contact with each other, 
are at once obtained from those of the system in which the two conics are replaced