42 
ON THE RATIONALISATION OF CERTAIN ALGEBRAICAL EQUATIONS. [109 
we have the result 
1, 
yz, zx, xy, 
xw, yto, zw, 
xyzw 
a 
. 1 1 
1 . . 
b 
1 . 1 
. 1 . 
c 
1 1 . 
. . 1 
d 
1 1 1 
d . . 
c b 
1 
. d . 
c . a 
1 
. . d 
b a 
1 
a b c 
1 
which however is not essentially distinct from the form before obtained, but may be 
derived from it by an interchange of lines and columns. 
And in general for any even number of quadratic radicals the two forms are not 
essentially distinct, but may be derived from each other by interchanging lines and 
columns, while for an odd number of quadratic radicals the two forms cannot be so 
derived from each other, but are essentially distinct. 
I was indebted to Mr Sylvester for the remark that the above process applies to 
radicals of a higher order than the second. To take the simplest case, suppose 
x + y = 0, x 3 = a, y z = b\ 
and multiply first by 1, x*y, xy 2 -, this gives 
or, eliminating. 
x + y . = 0 
. ay + x 2 y 2 = 0 
1 
b 
+ x 2 y 2 = 0 ; 
1 . = 0 ; 
a 1 
. 1 
next multiply by x, y, x 2 y 2 \ this gives 
x 2 . + xy = 0 
y 2 + xy = 0 
bx 2 4- ay 2 . — 0 ; 
or, eliminating, 
1 
1 
= 0 ; 
b a 
and lastly, multiply by x 2 , y 2 , xy, this gives 
a + x 2 y . = 0 
b . + xy 2 = 0 
x 2 y + xy 2 = 0 ;