138 
[128 
128. 
DEVELOPMENTS ON THE PORISM OF THE IN-AND-CIRCUM- 
SCRIBED POLYGON. 
[From the Philosophical Magazine, vol. vii. (1854), pp. 339—345.] 
I propose to develope some particular cases of the theorems given in my 
paper, “ Correction of two Theorems relating to the Porism of the in-and-circumscribed 
Polygon” (Phil. Mag. vol. VI. (1853), [116]). The two theorems are as follows: 
Theorem. The condition that there may be inscribed in the conic U = 0 an 
infinity of w-gons circumscribed about the conic V= 0, depends upon the development 
in ascending powers of f of the square root of the discriminant of i;U+V; viz. if 
this square root be 
A+B£ + C? + D? + E? + F? + G? + H? + ..., 
then for w = 3, 5, 7, &c. respectively, the conditions are 
G, 
D =0, 
G, 
D, 
E 
D, 
E \ 
D, 
E, 
F 
E, 
F, 
G 
and for n = 4, 6, 8, &c. 
respectively, 
the conditions 
are 
\E 
1=0, D, 
E =0, 
D, 
E, 
F 
E, 
F | 
E, 
F, 
G 
F, 
G, 
H 
= 0, &c.