[810 
810] 
that is, 
NOTE ON A SYSTEM OF EQUATIONS. 
49 
ex 2 — (a — d) xy — by 2 = 0, 
ex 2 — (c —f) xy — dy n - = 0, 
and eliminating the (x, y) from these equations we have an equation il = 0 as the 
condition that the original three equations may have a single common root; the 
b a — d 
Q 
-, are the conditions in order that the three 
before-mentioned equations 
d c —/ e 
equations may have two common roots, that is, that there may be two systems of (x, y) 
satisfying the three equations. 
We have, moreover, y(x — d) = cx, x(y - c) = dy, and substituting these values, say 
be , de 
that is, 
x- — (a + d) x + ad — be = 0, 
y 2 - (c +/) y + ef - de = 0, 
•] 
which are quadric equations for x and y respectively; it is easy to express the second 
equation (like the first) in terms of (a, b, c, d), and the first equation (like the second) 
in terms of (c, d, e, f), but the forms are less simple. 
Suppose (a, b, c, d) = (— 1, -1, 1, 0), then we have (e, f) = (0, 1), the two equations 
in x : y become x 2 + xy + y 2 = 0, and 0=0 respectively; those in x and y become 
a? + x + 1 = 0, y 2 — 2y + 1 = 0 respectively; this is right, for the three equations are 
x 2 =-x-y, xy = x, y 2 = y; 
viz. from the third equation we have y = 1, a value satisfying the second equation, 
and then the first equation becomes x 2 + x + 1 = 0; or, if we please, x 2 + xy + y 2 = 0, 
the values in fact being x=o), an imaginary cube root of unity, and y = 1. 
uations ; viz. 
In the general case, the values (x, y) may be regarded as units in a complex 
numerical theory, viz. if (a, b, c, d, e, f) are integers, and p, q, p', q', P, Q are also 
integers, then the product of the two complex integers px + qy and p'x + q'y will be 
a complex integer Px + Qy. 
C. XII. 
7