397 
1909-10.] Dr Muir on the Theory of Bigradients. 
we must (1) find a quantic u of which <p(a, b, c, . . .) is an invariant ; (2) 
express cp(a, b, c, . . .) as a determinant of the same number of lines as u 
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 (e 2 — bd) 
2 (b 2 — ac ) be - ad » 
we should have to transform the said cubic by the substitution 
x = (be - ad)£ + 2(c 2 - bd)rj ) 
y = 2 (b 2 - ac)£ + (be - ad)rj j ’ 
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 A 7 , whereas in Eisenstein’s case the power-index is 3. Instead of 
the binary cubic, therefore, Cayley takes the binary trilinear 
ax^j^ + bx-^^ + cx 1 y 2 z 1 + dx^j.fc + ex 2 y 1 z 1 +fx 2 y 1 z 2 + gx 2 y 2 z 1 + hx 2 y $ 2 , 
of which a generalisation of A, namely, 
a 2 h 2 + b 2 g 2 + c 2 / 2 '+ d 2 e 2 + 4 adfg + 4 bceh 
- 2ahbg - 2ahcf - 2ahde - 2 bgcf - 2bgde - 2 efde 
is an invariant ; expresses Q as a two-line determinant 
ah - bg - cf + de - 2 [eh - fg) 
-2 (ad -be) ah-bg-cf+de j 
makes the substitution 
x 1 = (ah-bg - cf+de)£ 1 -2(eh-fg)£ 2 
y 1 = - 2 (ad - bc)£ 1 + (ah - bg - cf+de)£ 2 
and, as the multiplier connecting the new Q and the old is now the second 
power of the modulus, he obtains what was wanted.* 
The substitutes found turn out to be 
or Q say, 
0Q 
‘da 
l^Q i^Q 
2 db’ 2 dc ’ 
but no explanation of this is vouchsafed. 
Eisenstein’s case is the degeneration made by putting 
a, b, c, d, e, f t g, h 
= a , b, b, c , b , e, c , d . 
* This short paper of Cayley’s teems with misprints, both in the original and in the 
Collected Math. Papers. 
