48 
[810 
810. 
NOTE ON A SYSTEM OF EQUATIONS. 
[From the Messenger of Mathematics, vol. xn. (1883), pp. 191, 192.] 
The equations are 
where 
x- 2 = ax + by, xy = cx 4- dy, y 1 = ex + fy, 
b a — d c 
d c —f e ’ 
or, what is the same thing, if a, b, c, d are given, then 
cd - d(a — d) 
e =T’ f =c b— ; 
and this being so, the equations are equivalent to two independent equations; viz 
starting from the first and the second equations, we have 
that is, 
and thence 
dx 2 — bxy = (ad — be) x, 
dx —by = (ad — be) : 
dxy — by 2 = (ad — be) y, 
or 
d (ex + dy) — by 2 = (ad — be) y; 
which, attending to the values of e and f is the third equation 
We have 
y 2 = ex +fy. 
x _ ax + by y 
y ex + dy' x 
ex ±fy. 
cx + dy ’