The theory of determinants in the historical order of development, by Sir Thomas Muir.

BIGRADIENTS (CAYLEY, 1844)

359

when
1 aA 1 aiA 1 MA 1 CA
A, B, C, D = 2c 6c 6 b' 2 Da
2 'ad' 6 ac' 6 ab' 2 a'
-can be expressed in the form
A 2B C. a 2b c.
A 2B C. a 2b c
B 2C D. = b 2c d. B 2C D. b 2c d
it is not as a relation between two four-line determinants that it
has been studied.
Cayley in effect says that if we wish to find substitutes
A, B, C,... for a, b, c,... so that
0,(A, B, C,...) = {O(a, b, c,..
we must (1) find a quantic u of which p(ca, b, c,...) is an invariant; (2) express p (a, b,,...) as a determinant of the same
number of lines as ut has facients; and (3) transform u into U
by a linear substitution of which the said determinant is the
modulus. The coefficients of U will then be the substitutes
required.
For example, A being an invariant of the binary cubic and
being expressible in the form
be - ad 2(c2-b bd)
2(b2-ac) be - ad
we should have to transform the said cubic by the substitution
x = (be-ad) ( + 2(c2- bd),,
y = 2(b2-ac) + (bc-ad)i
and the discriminant of the new cubic thus obtained being A
multiplied by a power of the modulus must be a power of A.
Unfortunately, in this case it would be A7, whereas in Eisenstein's
case the power-index is 3. Instead of the binary cubic, therefore, Cayley takes the binary trilinear
Caxly1z + bxlylz2 + cxy22z1 + dx1Y2z2 + eX2y1zl +fx.2yz2 + gx2Y2Zl + h2Y2z2,