MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
115 
And these covariants are connected by a single syzygy of the degree 5 and of the 
order 1 1 ; in fact, the table shows that there are only two asyzygetic covariants of 
this degree and order ; but we may, with the above-mentioned irreducible covariants 
of the degrees 1, 2, 3 and 4, form three covariants of the degree 5 and the order 11 ; 
there is therefore a syzygy of this degree and order. 
40. I represent the number of ways in which q can be made up as a sum of m terms 
with the elements 0, 1, each element being repeatable an indefinite number 
of times by the notation 
P(0, 1,2, 
and I write for shortness 
P'(0, 1, 2, ..m)®^=P(0, 1,2 — P(0, 1,2 ...mYq—l. 
Then for a quantic of the order m, the number of asyzygetic covariants of the degree 
i and of the order ^ is ^ 
In particular, the number of asyzygetic invariants of the degree 6 is 
P'(0, 1,2 ..my^mO. 
To find the total number of the asyzygetic covariants of the degree d, suppose first 
that md is even; then, giving to fjb the successive values 0, 2, the required 
number is 
— P(-|m^— 1) 
+ V{\m & — 1 ) — P(^m^ — 2) 
H-P(2) -P(l) 
H-P(l) 
= P(-|m^), 
i. e. when is even, the number of the asyzygetic covariants of the degree 6 is 
P(0, 1,2 ..my ; 
and similarly, when m0 is odd, the number of the asyzygetic covariants of the degree 
^ V{0,1,2, ..my\{rn&—l). 
But the two formulae may be united into a single formula; for when md is odd \ni9 
is a fraction, and therefore P(^m^) vanishes, and so when m6 is even \{m6—l) is a 
fraction, and 1) vanishes; we have thus the theorem, that for a quantic of 
the order m , — 
The number of the asyzygetic covariants of the degree 6 is 
P(0, 1,2 ...my^md-\-V{0, 1,2, ..my\{md—^\). 
41. The functions V\md, &c. may, by the method explained in my “Researches 
