MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
117 
43. For a cubic, the number of asyzygetic invariants of the degree 6 is 
l 
coefficient in 
1— a:'*’ 
i. e. there is a single irreducible invariant of the degree 4. 
44. For a quartic, the number of asyzygetic invariants of the degree 6 is 
coefficient in 
i. e. there are two irreducible invariants of the degrees 2 and 3 respectively. 
45. For a quintic, the number of asyzygetic invariants of the degree d is 
coefficient :r» in (i, ■ 
The numerator is the irreducible factor of 1— i. e. it is equal to (1 — x®) 
-j-(l — and substituting this value, the number becomes 
l _^6 
coefficient in 
(1 — a?‘*)(l — a?®)(l— — a?'®)’ 
i. e. there are in all four irreducible invariants, which are of the degrees 4, 8, 12 and 
18 respectively; but these are connected by an equation of the degree 36, i.e. the 
square of the invariant of the degree 18 is a rational and integral function of the 
other three invariants ; a result, the discovery of which is due to M. Hermite. 
46. For a sextic, the number of asyzygetic invariants of the degree 6 is 
coefficient ** in t 
the second factor of the numerator is the irreducible factor of 1— i. e. it is equal 
to (1 — .r®®)(l— x®)(l— x®)(l— j:^)-r(l— .r’®)(l — .r*®)(l— ir®)(l— .r); and substituting this 
value, the number becomes 
coefficient a?®® in (i -x'^){\.-x%l -a;®)(l — — a?*®)’ 
i. e. there are in all five irreducible invariants, which are of the degrees 2, 4, 6, 10 and 
15 respectively; but these are connected by an equation of the degree 30, i.e. the 
square of the invariant of the degree 15 is a rational and integral function of the 
other four invariants. 
47 . For a septic, the number of asyzygetic invariants of the degree 0 is 
„ . 1 — x^ + 2x^ — x^^ + 5x^^+2x^‘^-i-6x^^ + 2x^^ + 5x^'^ — x^^ + 2x'^ — x'^ + x^^ 
coefficients in ’ 
the numerator is equal to 
(1 _^6)(1 _a^8)-2(i -a:‘®)(l 
where the series of factors does not terminate ; hence the number of irreducible inva- 
MDCCCLVI. R 
