116] 
91 
116. 
CORRECTION OF TWO THEOREMS RELATING TO THE PORISM 
OF THE IN-AND-CIRCUMSCRIBED POLYGON. 
[From the Philosophical Magazine, vol. vi. (1853), pp. 376—377 ] 
The two theorems in my “Note on the Porism of the in-and-circumscribed Polygon” 
(see August Number), [115], are erroneous, the mistake arising from my having in 
advertently assumed a wrong formulae for the addition of elliptic integrals. The first 
of the two theorems (which, in fact, includes the other as a particular case) should be 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 £ of the square root of the discriminant of £Ï7-1- F ; viz. if 
this square root be 
A + BÇ + C£ 2 + D£ 3 + E? + F? + G? + H? + ..., 
then for n = 3, 5, 7, &c. respectively, the conditions are 
1(7 1=0, 
C, 
JD 
= 0, 
77, 
D, 
E 
= 0, &c. ; 
f 
D, 
E 
P, 
E, 
F 
E, 
F, 
G 
and for n = 4, 6, 8, &c. respectively, the conditions are 
I D I =0, 
= 0, &c. 
The examples require no correction ; since for the triangle and the quadrilateral 
respectively, the conditions are (as in the erroneous theorem) 77—0, D — 0. 
1 2— 2 
D, 
E 
= 0, 
D, 
E, 
F 
E, 
F 
D, 
F, 
G 
F, 
G, 
H