MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 109 
i. e. B+6A=0, 3D4-2B=0, &c. ; or putting A=l, we find B=— 6, C=4, D=4, 
E=:— 3, and the invariant is 
— Qahcd 
+4ac® 
-36 V. 
Again, there is a covariant of the order 3 and the degree 3. The coefficient of V 
or leading coefficient is 
Aa^d 
4-Ba6c 
+C6^ 
which operated upon with abi-\-2bb^-\-3cba, gives 
a“e 
ab“ 
i. e. B+3A=0, 3C+2B=0 ; or putting A=l, we have B= — 3, C=2, and the leading 
coefficient is 
a^d 
— 3abc 
+2b\ 
The coefficient of is found by operating upon this with {3b'ba-\-2cbi-\-d'd^), this 
gives 
ahd 
ac~ 
b^c 
i. e. the required coefficient of V3/ is 
3abd 
—6ac^ 
+3b^c ; 
and by operating upon this with ^ (36B„+2 cBa+c?BJ, we have for the coefficient of 
acd 
h~d 
bc~ 
i. €. the coefficient of is 
— 3acd 
+ 66V 
— 3bc\ 
Q 
+3 
-6 
' 2 
-1-^ 
^ 2 
-9 
+e 
+ 6 
-6 
— 3 
-9 
+ 12 
+ B 
+ 3A 
+ 3C 
+ 2B 
MDCCCLVI. 
