18 
PORISM OF THE IN - AND-CIRCUMSCRIBED POLYGON, 
[489 
or, as it may be written, 
x 2 y 2 (46 2 — 6a0) 
4- (x 2 y + xy 2 ) (— 4ad + 12be — 1260) 
+ (x 2 + y 2 ) (— ae + 9c 2 — 18c# + 9# 2 ) 
+ xy (— 2ae — 86# 4- 18c 2 — 18# 2 ) 
4- (x 4- y) (— 46e 4- 12c# - 12dd) 
4- 4# 2 — 6c# = 0. 
Comparing this with the original integral equation V= 0, and the form of differential 
equation deduced therefrom, we ought to have identically 
[(46 2 - 6a#) x 2 4- (- 2ad 4- 66c - 66#) x + (- ae + 9c 2 - 18c# 4- 9# 2 )] 
x [(— ae 4- 9c 2 — 18c# 4- 9# 2 ) x 2 4- (— 26c 4- 6c# — 6##) x + (4# 2 — 6e#)] 
— [(— 2ad 4- 66c — 66#) x 2 4- (— ae — 46cZ 4- 9c 2 — 9# 2 ) x 4- (— 26c 4- 6c# — 6##)] 2 
= multiple of X, 
= {(— 4a# 2 — 46 2 e 4- 246c#) 4- (6ae — 246# — 54c 2 ) # 4- 108c# 2 — 54# 3 } (a, 6, c, #, e\x, l) 4 , 
by comparing the coefficients of x 1 . 
I obtain this otherwise : 
Write 
V=aU+6fiH, 
then, forming the Hessian of V, we have 
HV = (a 2 - SI/3 2 ) H 4- (/a/3 4- 9 J/3 2 ) U, 
= (a *~ 6 3J/3 ’' ) ( V - a V) + (Ia/3 + 9 J/3 1 ) U, 
= V + h ( ~ 0? + + 54W U ’ 
that is 
djVd/V-(dd,Vf - (MJV+ ZcydJ„r+ y’djV) = ^ (- or> + №/3 ! + 54,J0>) U, 
or writing 
this is 
K = - 
2 (a 2 - SI/3 2 ) 
P 
24 
(d x 2 V + Ky 2 ) (d y 2 V + Kx 2 ) - (d x d y V — Kxy) 2 = j (- a 3 4- 9/a/3 2 4- 54J/3 2 ) 67, 
so that the components are 
#* 2 F 4- Ky 2 , d x d y V — Kxy, d y 2 V 4- Kx 2 , 
V — ocU + 6 (311 = 
a (a, 6, c, #, c$æ, l) 4 4-6/3 (ac — 6 2 , 2a# — 26c, ac 4-26# — 3c 2 , 26c —2c#, cc — # 2? $æ, l) 4 ,