20 
PORISM OF THE IN-AND-CIRCUMSCRIBED POLYGON, 
the new coefficients are 
(a , b , c v)\ (a , b , c v\p, p), (a , b , c ^p, p) 2 , 
(a', b', c'$>, z/) 2 , (a', 6', c'][A, v^p, p), (a', b', c'l[p, p) 2 , 
(a", b", c"#\, v) 2 , (a", b", c"%\ v^p, p), (a", b", c"\p, p) 2 : 
assume now 
ci'X + (b' — 9) v — ap — bp = 0, 
(b' + 9) \ + c'v — bp — cp = 0, 
a"\ + b"v- a'p-(b' + 9)p = 0, 
b"\ + c"v — (b' — 6) p— cp = 0, 
then it is easy to show that 
(a, b, c v][p, p) = (a', b', c' v) 2 , 
(a', b’, c'^p, p) 2 = (a", b", c''-$X, v\p, p), 
(a, b, c\p, p) 2 = (a", b", c"J\, v) 2 
[=(a', b', c'$>, v\p, p) + 9 (\p - pv)\ 
and the equations give 
a , b' — 9, a , b =0, 
b' + 9, c' , b , c 
a" , b" , a , b' + 9 
b" , c" , V - 9, c' 
that is 
(a'c' - b' 2 + 9 2 ) 2 + (a'c" - b" 2 ) (ac - b 2 ) 
— (a'b" — a"b' + a"9) (be' — b'c + c9) 
+ (a'c" - b"b' + b"9) (bb' - a'c + b9) 
+ (b'b" - a"c' + b"9) (ac' - b'b + b9) 
- (b'c" - b"c + c"9) (ab' - a'b + a9) = 0, 
which is 
(a'c' - b' 2 ) 2 + (a"c" - b" 2 ) (ac - b 2 ) - 29 
+ a' 2 (— cc") 
+ b' 2 (- ac" - 2bb" - a"c) 
+ c' 2 (— aa") 
+ 2b'd (ab" + a"b) + 9 2 (2 (a'd - b' 2 ) - ac" + 2bb" - 
+ 2cV (- bb") 
+ 2 a'b' (be" + b"c) 
+ 9 4 = 0.