[109 
41 
ALGEBRAICAL 
m. (1853), pp. 97—101.] 
the two others, we have 
y as quadratic radicals, 
in be multiplied by x, y, 
ion between x, y. 
pplying the same process 
109] ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. 
we have results of some elegance. Multiply the equation first by 1, yz, zx, xy, reduce 
and eliminate the quantities x, y, z, xyz, we have the rational equation 
1 1 
1 . c 
1 c . 
1 b a 
and again, multiply the equation by x, y, z, xyz, reduce and eliminate the quantities 
1, yz, zx, xy, the result is 
a b 
a . 1 
b 1 . 
c 1 1 
c 
1 
1 
= 0, 
which is of course equivalent to the preceding one (the two determinants are in fact 
identical in value), but the form is essentially different. The former of the two forms 
is that given in my paper “On a theorem in the Geometry of Position” {Journal, 
vol. H. [1841] p. 270 [1]): it was only very recently that I perceived that a similar 
process led to the latter of the two forms. 
Similarly, if we have the equations 
x + y + z + w = 0, x 2 = a, y 2 = b, z 2 — c, w 2 = d, 
then multiplying by 1, yz, zx, xy, xw, yw, zw, xyzw, reducing and eliminating the 
quantities in the outside row, 
we have the result 
X, y, z, 
W, 
yzw, zwx, wxy 
xyz 
1 1 1 
1 
c b 
1 . . 
] 
c . a 
. 1 . 
1 
b a 
. . 1 
1 
d . . 
a 
. 1 1 
. d . 
b 
1 . 1 
. . d 
c 
1 1 . 
a b c 
d 
so if we multiply the equations by x, y, z, w, yzw, zwx, wxy, and xyz, reduce and 
eliminate the quantities in the outside row, 
C. II. 
6