1905-6.] Hessians of Certain Invariants of Binary Quantics. 529 
The Hessians of Certain Invariants of Binary Qnantics. 
By Thomas Muir, LL.D. 
(MS. received August 11, 1906. Read November 5, 1906.) 
(1) The cubinvariant J of a binary quartic being 
a b c 
ace + 2 bed - ad? - b 2 e - c 3 or 
bed 
its Hessian, H(J), is 
c d e 
e -2d c 
. - 2e 2d 2c -2b 
e 2d - 6c 2b a 
- 2d 2c 2b -2a 
c -2b a 
Performing on this the operations 
c-col 3 — e-col 5 , c-col 4 + 2<i-col 5 , 
and on the result the operations 
c-rowg - e-row 5 , c-row 4 + 2d row 5 , 
we obtain 
H(J) = - 8c 2 
- e 
cd + be 
c 2 - 2 bd 
cd + be 
- 3c 3 — ace 
be 2 + acd 
c 2 - 2 bd 
be 2 + acd, 
— ac 2 
= - 8c 3 
- ce 
cd + be 
c 2 - 2bd 
cd + be 
— 3c 2 - ae 
be + ad 
c 2 - 2 bd 
be +ad 
- ac 
The three-line determinant here is seen to contain c as a factor, 
because it manifestly vanishes when c is put equal to 0. A full 
resolution of it into factors, however, is got by multiplying it by 
the determinant form of J, an operation which curiously enough 
leads to the equation 
PROC. ROY. SOC. EDIN. — VOL. XXVI. 
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