308 
ON THE PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON. 
[267 
and therefore 
these equations give 
a — L — l, 
r 2 = V +l 2 , 
R 2 =V +L 2 -, 
R 2 
k =L + l, 
and thence 
and 
V = E 2 - 
a = L — l; 
L = 
R 2 — r 2 + a 2 
~2a~ 
1 
4a 2 
(2 a 2 R 2 + 2a 2 r 2 + 2 r 2 R 2 — R 4 — r 4 — a 4 ) 
if with M. Mention we write 
r 4 
= 4a 2 v * 
1 
r 4 
(r 4 + R 4 + a 4 — 2 r 2 R 2 — 2 r 2 a 2 — 2 R 2 a 2 ) = 
v. 
The equation 
thus becomes 
(l 2 + V) 2 + 4 (Z — Z) 2 V = 0 
r 4 
r 4 4- 4a 2 . x-„ v = 0, 
4a 2 
that is, it becomes v + l =0, which is the other factor of the complete condition t'(^ + l) = 0.