396] 
TRIANGLE INSCRIBED IN A CIRCLE. 
79 
It is interesting to verify this; I write Z = — X — F, the cubic function then 
assumes the form 
(«, /3, ry, 8$X, F) 3 , 
where (a, /3, 7, 8) have the values presently given. 
The Hessian is 
(2a 7 - 2/3-) X* + (aS - /3 7 ). 2XY + (2/38 - 2 7 2 ) F 2 , 
or writing 2X Y = Z 2 — X- — F 2 , this is 
= (2ay - 2 / 8 2 - «8 + £7) X 2 + (2/38 - 2 7 2 - a8 - £7) F 2 + («8 - £7) P. 
We find after some easy reductions, 
¿a = X' + X, = — 3 (a + c) (ac + M), 
/3 = — H' + 2K' +1' + F', = 3a(a + c)(b +c ), 
— 7 = — K' + 2H' + J' + G', = 3b (a + c)(b + c ), 
— %8=H'+J' } = — 3 (b 4- c) (be + M), 
and heiice 
a8 — /3<y = — 81 (a + c) (b + c) {{ac + M) (be + M) — ab (a + c) (b + c)}, 
where the expression in { } is 
= (ab + 2ac + be) (ab + ac + 2be) — ab (ab + ac + bc + c 2 ), 
= c {ab (3a + 3b) + c(b+ 2a) (a + 2b) —ab(a + b + c)j, 
= 2c (a + b) (be + ca + ab), 
and therefore 
aS — £7 = — 162 (b + c)(c + a) (a + b) Me ; 
the other coefficients may be similarly calculated, and omitting the merely numerical 
factor, we have 
Hessian = (b + c) (c + a) (a + b) M (aX- + bY- + cZ 2 ), 
which is right. 
1 write next 
HU+3MU-*rU= X‘ 2 (-a^x + I'Y -F'Z) 
+ Y- (- Vby + J'Z - GX) 
+ Z 2 (-a*z+KX-H'Y),