3AL EQUATIONS. [109 
= 0, 
re obtained, but may be 
s the two forms are not 
interchanging lines and 
two forms cannot be so 
above process applies to 
¡St case, suppose 
109] ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS, 
or, eliminating, 
43 
a 1 
b . 1 
. 1 1 
0; 
where it is to be remarked that the second and third forms are not essentially distinct, 
since the one may be derived from the other by the interchange of lines and columns. 
Applying the preceding process to the system 
x + y + z = 0, a? = a, y s = b, z 2 = c ; 
multiply first by 1, xyz, x 2 y 2 z 2 , x 2 z, y 2 x, z 2 y, x 2 y, y 2 z, z 2 x, reduce and eliminate the 
quantities in the outside row, 
the result is 
the result is 
X, y, Z, 
y 2 z 2 , x 2 yz, y 2 zx, 
z 2 xy, z 2 x 2 , x 2 y 2 
1 1 1 
111 
. 
a b c 
a 
1 . . 
. 1 . 
b . . 
. 1 . 
. . 1 
c 
. . 1 
1 . . 
. a 
1 . . 
. . 1 
. . b 
. 1 . 
1 . . 
G 
. . 1 
. 1 . 
z, y 2 z 2 , Z 2 X 2 , 
ow, 
x 2 y 2 , x 2 yz, 
y 2 zx, z 2 xy, 
x 2 , y 2 , z 2 , 
yz, zx, xy,xy 2 z 2 ,yz 2 x 2 , zx 2 y- 
1 . . 
. 1 1 
. 
. 1 . 
1 . 1 
. 
. . 1 
1 1 . 
. . 
c b 
• 
1 . . 
c . a 
. 
. 1 . 
b a 
. . 
. . 1 
a 
. 1 1 
• • 
. b . 
1 . 1 
• • • 
c 
1 1 . 
0; 
= 0; 
6—2