516 
A FOURTH MEMOIR UPON QUANTICS. 
[155 
65. The law of reciprocity for the number of the invariants of a binary quantic 1 , 
leads at once to the theorems in regard to the number of the quadrinvariants, cubin- 
variants and quartinvariants of a binary quantic of a given degree, first obtained by 
the method in the second part of my original memoir 2 . Thus a quadric has only a 
single invariant, which is of the degree 2 ; hence, by the law of reciprocity, the number 
of quadrinvariants of a quantic of the order m is equal to the number of ways in which 
m can be made up with the part 2, which is of course unity or zero, according as 
m is even or odd. And we conclude that 
The quadrinvariant exists only for quantics of an even order, and for each such 
quantic there is one, and only one, quadrinvariant. 
66. Again, a cubic has only one invariant, which is of the degree 4, and the 
number of cubinvariants of a quantic of the degree m is equal to the number of 
ways in which m can be made up with the part 4. Hence 
A cubinvariant only exists for quantics of an evenly even order, and for each 
such quantic there is one, and only one, cubinvariant. 
67. But a quartic has two invariants, which are of the degrees 2 and 3 respectively, 
and the number of quartinvariants of a quantic of the degree m is equal to the number 
of ways in which m can be made up with the parts 2 and 3. When m is even, 
there is of course a quar tin variant which is the square of the quadrinvariant, and which, 
if we attend only to the irreducible quartinvariants, must be excluded from consideration. 
The preceding number must therefore, when m is even, be diminished by unity. The 
result is easily found to be 
Quartinvariants exist for a quantic of any order, even or odd, whatever, the quadric 
and the quartic alone excepted ; and according as the order of the quantic is 
6g, 6g + 1, Qg + 2, 6g + 3, 6g + 4, 6g + 5, 
the number of quartinvariants is 
g, g , g > g +1, g , g+1- 
In particular, for the orders 
2, 3, 4, 5; 6, 7, 8, 9, 10, 11; 12, &c., 
the numbers are 
0, 1, 0, 1; 1, 1, 1, 2, 1, 2; 2, &c. 
Thus the ninthic is the lowest quantic which has more than one quartinvariant. 
68. But the whole theory of the invariants or covariants of the degrees 2, 3, 4 is 
most easily treated by the method above alluded to, contained in the second part of my 
original memoir; and indeed the method appears to be the appropriate one for the 
1 Introductory Memoir, [139], No. 20. 
2 Ibid. Nos. 10—17.