AND ON ANOTHER ALGEBRAICAL SYSTEM. 
467 
79] 
Hence, observing the values of yz, zx, xy, we find 
x 2 = ^ {abc —fgh + /V(23(S7) — g\J{^^!) — 
t=^ {<*&> -fgh -M&®)+ 9 m®) - 
22 = fc ^ abc ~f gbj ~ — 
Hence, forming the value of any one of the functions by 2 + cz 2 + 2fyz, cz 2 + ax 2 + 2gxy, 
ax 2 + by 2 + 2hxy, we obtain s = Ô 2 ; or we have 
' æ2z= Ta i abG ~f gh + ~ 
y 2 = ^ {a&c -fgh -/V(33<2D) + (<2D&) - W(838)}, 
- ^ ^ {a&c -fgh -/V(230T) - 
yz=\6 2 {JF + V(38®)}, 
^2/=i^ 2 {?§+V№3)}. 
It may be remarked that the equations 
b y 2 + c z 2 + 2f yz = L , 
c' z 2 + a! x? + 2g yz — M, 
a"x 2 + b"y 2 + 2b!'xy = N, 
in which the coefficients are supposed to be such that the functions 
M (a'x 2 + b"y 2 A 2b!'xy) — N (o' z 2 + a' x 2 + 2 g'y z), 
N (b y 2 + c z 2 + 2 f yz) — L (ia”x 2 + b"y 2 + 2b!'y z), 
L (c f z 2 + a'x 2 + 2g’yz) — M (b y 2 + c z 2 + 2 f yz), 
are each of them decomposable into linear factors, may always be reduced to a system 
of equations similar to those which have just been solved. 
Suppose 
f — 9 — h — tj 2 
abc 
a + b + c 
= r, 
and write fX, aJY, *JZ instead of x, y, z. The equations to be solved become 
bY + cZ + 2r\J(YZ) = (b + c) r, 
cZ + aX + 2r\J(ZX ) = (c + a) r, 
aX + bY + 2 rs/(XY) = (a + b)r, 
59—2