ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL 
EQUATIONS. 
[From the Cambridge and Dublin Mathematical Journal, vol. vm. (1853), pp. 97—101.] 
Suppose 
x + y — 0, x 2 = a, y 2 = b\ 
then if we multiply the first equation by 1, xy, and reduce by the two others, we have 
x+ y = 0, 
bx + ay = 0, 
from which, eliminating x, y, 
1, 1 
b, a 
= 0; 
which is the equation between a and b; or, considering x, y as quadratic radicals, 
the rational equation between x, y. So if the original equation be multiplied by x, y, 
we have 
a + xy = 0, 
b + xy = 0 ; 
or, eliminating 1, xy, 
a, 1 
b. 1 
= 0, 
which may be in like manner considered as the rational equation between x, y. 
The preceding results are of course self-evident, but by applying the same process 
to the equations 
x + y + z = 0, x 2 = a, y 2 = b, z 2 = c,