116 
MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
on the Partition of Numbers,” post. p. 34, be determined as the coefficients of in 
certain functions of a; ; I have calculated the following particular cases : — 
Putting, for shortness, 
P'(0, 1, 2, coefficient x^ in <pm, 
then 
then 
then 
1 — + 
^ (1— a;®) 
(l-a;2)2(l-a;3)(l-a?4)(] -x^) 
97= 
1 — x^ + 2x^ — x^^ + 5x^^ + 2x^^ + 6x^^-f-2x^^ + 5x^^ — x^-\-2x ^‘^ — + 
(1 —x)[\ + x — x^ —x'^ x^ + x"^ x^ -{■ aP x^^ — x^^ + x^^ + x^^) ^ 
{\ — x‘^Y[\.—x^)'^{\—x'^){\—oP){\—x'^) ’ 
P(0, 1, 2, ..my^m6=- coefficient of a?® in 
1 
X)(l— 
. l+a?"* 
. 1—x + x'^ 
— x)\i — x‘^)[l — oP) 
\+x^-\- 6x‘^ + 9x^ + 1 2x^ + + 6x^'^ + + x^^ 
\p5 — - 
{1 —x^y(i—x'^)(i—x^){i—x^) ’ 
P(0, 1, 2, ..my^(m^—l)= coefficient of x n ^ppi, 
. ^ x + sP 
'r/'3 — n — -2\2 /i _ ^4’ 
■^^b = 
(1 —x’^yfi —x^ 
x + Ax^ + ^sP+lOx’^ — + +Ax^^ + !p^ 
{l — x‘^)'^[\ — x'^){\—x^){ \ —x^) 
And from what has preceded, it appears that for a quantic of the order ^/^, the number 
of asyzygetic covariants of the degree d is for m even, coefficient x^ in and for m 
odd, coefficient x^ in and that the number of asyzygetic invariants of the 
degree & is coefficient a?® in ipm. Attending first to the invariants, — 
42. For a quadric, the number of asyzygetic invariants of the degree ^ is 
1 
coefficient a?® in 
l-x^’ 
which leads to the conclusion that there is a single irreducible invariant of the 
degree 2. 
