268 
A SECOND MEMOIR UPON QUANTICS. 
[HI 
the numerator is equal to 
(1 - x 6 ) (1 - a 8 )- 2 (1 - ¿r 10 ) (1 - a; 12 ) -5 (1 - a; 14 ) -4 ..., 
where the series of factors does not terminate; hence [incorrect, see p. 253] the number 
of irreducible invariants is infinite; substituting the preceding value, the number of 
asyzygetic invariants of the degree 6 is 
coefficient x d in (1 — a 4 ) -1 (1 — x 8 )~ 3 (1 — a; 12 ) -6 (1 — a; 14 ) -4 ... 
The first four indices give the number of irreducible invariants of the corresponding 
degrees, i.e. there are 1, 3, 6 and 4 irreducible invariants of the degrees 4, 8, 12 and 
14 respectively, but there is no reason to believe that the same thing holds with 
respect to the indices of the subsequent terms. To verify this it is to be remarked, 
that there are 1, 4, 10 and 4 asyzygetic invariants of the degrees in question respect 
ively ; there is therefore one irreducible invariant of the degree 4; calling this X 4 , 
there is only one composite invariant of the degree 8, viz. X 4 2 ; there are therefore 
three irreducible invariants of this degree, say X 8 , X 8 , X”. The composite invariants 
of the degree 12 are four in number, viz. X 4 -\ X 4 X 8 , X 4 X 8 ', X 4 X 8 ", and these cannot he 
connected hy any syzygy, for if they were so, X 4 2 , X 8 , X 8 , X 8 " would be connected by a 
syzygy, or there would be less than 3 irreducible invariants of the degree 8. Hence 
there are precisely 6 irreducible invariants of the degree 12. And since the irreducible 
invariants of the degrees 4, 8 and 12 do not give rise to any composite invariant of 
the degree 14, there are precisely 4 irreducible invariants of the degree 14. 
48. For an octavic, the number of the asyzygetic invariants of the degree 6 is 
no • , g ■ (1 — x) (1 + x — x 3 — x i + x 6 + x 7 + ac? + x a + x 10 — x 12 — x 13 + x 15 + ¿c 16 ) 
coefficient x e in — r- -r-r — — ’ ■ 
(1 - xj (1 - x 3 ) 3 (1 - x 4 ) (1 - x 5 ) (1 - x 7 ) 
and the second factor of the numerator is 
(1 - x)- 1 (1 - x 2 ) (1 - x 3 )- 1 (1 - x 6 )- 1 (1 - och)~ l (1 - x?)- 1 (1 - a; 10 )- 1 (1 - x 16 ) (1 - x 17 ) (1 - ¿c 18 )..., 
where the series of factors does not terminate, hence [incorrect] the number of irreducible 
invariants is infinite. Substituting the preceding value, the number of the asyzygetic 
invariants of the degree 6 is 
ooeff.# 0 in (1 - a? 2 ) -1 (l-^)- 1 (l-^)- 1 (l-^)-i(l-#6)-i(l-:r 7 )- 1 (l-# 8 )- 1 (l--# 9 )- 1 (l-:r 10 )- 1 (l-tf 16 )(l-tf 17 )(l-^ 8 )... 
There is certainly one, and only one irreducible invariant for each of the degrees 
2, 3, 4, 5 and 6 respectively; but the formula does not show the number of the irre 
ducible invariants of the degrees 7, &c.; in fact, representing the irreducible inva 
riants of the degrees 2, 3, 4, 5 and 6 by X 2 , X 3 , X 4 , X 5 , X 6 , these give rise to 3 com 
posite invariants of the degree 7, viz. X 2 X 2 X 3 , X 2 X 5 , X 3 X 4 , which may or may not be 
connected by a syzygy; if they are not connected by a syzygy, there will be a single 
irreducible invariant of the degree 7; but if they are connected by a syzygy, there 
will be two irreducible invariants of the degree 7 ; it is useless at present to pursue 
the discussion further.