110 
MR. A. CAYLEY’S SECOND xMEMOIR UPON QUANTICS. 
Finally, operating upon this with ^ we have for the coefficient of 
a(F 
bed 
,.3 
i. e. the coefficient of is 
-\-3bcd 
— 2c\ 
and the covariant is 
I have worked out the example in detail as a specimen of the most convenient method 
for the actual calculation of more complicated covariants *. 
35. The number of terms of the degree 6 and of the weight q is obviously equal to 
the number of ways in which q can be made up as a sum of B terms with the elements 
(0, 1,2, ...m), a number which is equal to the coefficient of in the development of 
1 . 
and the number of the asyzygetic covariants of any particular degree for the quantic 
* Note added Feb. 7, 1856. — The following method for the calculation of an invariant or of the leading 
coefficient of a covariant, is easily seen to be identical in principle with that given in the text. Write down 
all the terms of the weight next inferior to that of the invariant or leading coefficient, and operate on each of 
these separately with the symbol 
h . c . b' 
ind. S • - + 2 ind. c>i" • +m— 1 ind. 5'*— » 
a b a 
where we are first to multiply by the fraction, rejecting negative powers, and then by the index of the proper 
letter in the term so obtained. Equating tbe results to zero, we obtain equations between the terms of the 
invariant or leading coefficient, and replacing in these equations each term by its numerical coefficient in the 
invariant or leading coefficient, we have the equations of connexion of these numerical coefficients. Thus, for 
the discriminant of a cubic, the terms of the next inferior weight are a-cd, abrd, aber, b^c, and operating on each 
of these separately with the symbol 
ind. A • - + 2 ind. c • - + 3 ind. c? • - > 
a b c 
we find 
abed 
+ 6 a~d~ 
3 ¥d 
+ 2 abed 
2 ¥(? 
+ 6 ae^ 
+ 3 abed 
+ 4 b‘(? 
+ 3 bH 
and equating the horizontal lines to zero, and assuming a‘d‘=\, we have a~d“=l, abcd=—6, ac®=4, ¥d—4, 
h^c-=—S, or the value of the discriminant is that given in the text. 
a~d 
Zabd 
—Zaed 
— a(P 
— Zabe 
— 6ac" 
+ 66V 
-\~Zbed 
+ 36* 
+ 36"c 
-Zbe^ 
-2c* 
-1 
-3 
+ 8 
-2 
-2 
