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, 2, ... m, each element being repeatable an indefinite 
number of times by the notation 
and I write for shortness 
P'(0, 1, 2, ...m) 9 q = P(0, 1, 2 ... vi) 9 q — P (0, 1, 2 ... vi) 9 (q - 1). 
then tor a quantic of the order to, the number of asyzygetic covariants of the degree 
6 and of the order /x is 
P'(0, 1, 2 ... m)°%(vid — /*). 
In particular, the number of asyzygetic invariants of the degree 6 is 
P' (0, 1, 2 ... m) 9 ^m6. 
To find the total number of the asyzygetic covariants of the degree 0, suppose 
first that mQ is even; then, giving to /jl the successive values 0, 2, 4, ... vi0, the 
required number is 
P {\vi0) — P (|to0 — 1) 
+ P Mid — 1) — P (hn6 — 2) 
+ P(2) -P(l) 
+ -P(1) 
= P (|to0), 
i. e. when vi9 is even, the number of the asyzygetic covariants of the degree 9 is 
P (0, 1, 2 ... vi) d \m9; 
and similarly, when vi0 is odd, the number of the asyzygetic covariants of the degree 
9 is 
P (0, 1, 2, ... m) e m0 — 1). 
But the two formulae may be united into a single formula; for when md is odd \vi9 
is a fraction, and therefore P (i\m6) vanishes, and so when m.6 is even \(vi6 — 1) is a 
fraction, and P^(m9 — 1) vanishes; we have thus the theorem, that for a quantic of 
the order m: 
The number of the asyzygetic covariants of the degree 0 is 
P(0, 1, 2 ... m) e \vi6 + P (0, 1, 2, ... m) e %(vi0 - 1). 
41. The functions P (Jm0), &c. may, by the method explained in my “ Researches 
on the Partition of Numbers,” [140], be determined as the coefficients of x e in certain 
functions of x; I have calculated the following particular cases:— 
Putting, for shortness, 
P' (0, 1, 2,... vi) 9 \m0 = coefficient x 9 in fan, 
C. II. 34