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383] 
PROBLEMS AND SOLUTIONS. 
581 
above- 
convex 
situate 
of the 
c curve, 
lea cum 
branch 
nt cases 
Teoria 
ere are 
ie three 
, inter- 
fh each 
e cubic 
an find 
ie fifth 
a curve 
, which 
points; 
of the 
it, and 
*7=0; 
lave a 
ing the 
I assume that the three points are given by the equations (x = 0, f = 0), (y = 0, rj = 0), 
(z = 0, £ = 0), respectively. This being so, we may write 
Q — + zxr\V + xyt/8" + xyze = 0, — P = yzgX + zxrj Y + xy£Z + xyz% = 0, 
for the equations of the cubic curve and the quartic curve respectively. We have 
of course M = xyz = 0 for the equation of the three sides of the triangle, and the 
identity to be satisfied is xyzTJ — AP 2 + BPQ + CQ 2 . 
I was led to the values of A, B, G by considerations founded on the theory of 
curves in space. We have 
A = 88'8", B = (S'a" + 8"a) 8x + (8"/3 + 8/3") 8'y + (8y' + 8'y) 8"z, 
G = a'a?8a? + £"/38'y* + yy'8"z 2 + (y/3''8' + y'/38") yz + (a'y8" + *"y'8) zx + (/3"a'S + /3a"8') xy ; 
and with these values it is easy to show that the function AP 2 + BPQ+ CQ 2 contains 
the factor xyz\ for substituting the values of P, Q, all the terms of AP 2 + BPQ + CQ 2 
contain explicitly the factor xyz, except the terms 
A (y 2 z 2 £ 2 X 2 + z 2 x 2 rfY 2 + x 2 y% 2 Z-) — B (y' 2 z 2 ^~X8 + z 2 x 2 rj 2 Y8' + x 2 y‘ 2 £ 2 Z8') 
+ G (y 2 z 2 g 2 8 2 + z 2 x 2 rf8' 2 + x 2 y 2 £ 2 8" 2 ); 
and these terms will contain the factor xyz, if only the expressions AX 2 — BX8 + C8 2 , 
AY 2 — BY8'+ G8' 2 , AZ 2 — BZ8" + G8" 2 contain respectively the factors x, y, z. But 
AX 2 — BX8 + G8 2 will contain the factor x, if only the expression vanishes for x = 0; 
and for « = 0 we have 
AX 2 - BX8 + C8 2 = 0 = 
88'S" (ßy + yz) 2 — [8'8" (ßy + yz) + 8 (ß"8'y + y'8"z)\ 8 (ßy + yz) + (ßy + yz) (ß"8'y + y'8"z) S 2 ; 
that is, AX 2 — BX8 + C8 2 contains the factor x; and by symmetry the other two 
expressions contain the factors y and z respectively. The excepted terms contain therefore 
the factor xyz; and there exists therefore a quintic function U = (AP 2 + BPQ + GQ 2 ) xyz; 
which proves the theorem. 
The values of A, B, G were obtained by considering the surface w = 
as is at once seen, contains upon itself the three lines 
P 
Q ’ 
which, 
(y = 0’ w = ~s') ’ 
z — 0, w = — 
8" 
or as these equations may be written 
(x — 0, . ¡3y + yz + 8w = 0), 
(y = 0, a'x . + y z + 8'w = 0), 
(z = 0, a"x + f3"y . + 8"vj = 0); 
le, and