[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