261 
828] A MEMOIR ON SEMIN VARIANTS. 
and it can be capitated, for instance, into 
443.2 2 
- 333.33 
- 432.2 s 
+ 2.43.2 4 
+ 3.4333.2 
+ 432 3 .2 = 0, 
the form of capitation being (for the reason mentioned above) quite immaterial. 
Observe that in every case where the sextic syzygy contains in the first instance any 
quintic seminvariants, it is assumed that each of these is expressed in terms of quintic 
perpétuants, as shown in No. 38 ; and this being done, the sextic syzygy exhibits itself 
as a syzygy containing, or else not containing, a quintic perpétuant or perpétuants. 
47. The conclusion is that from any sextic syzygy of the weight w — 6, which 
does not contain a quintic perpetuant, we can obtain by capitation a sextic syzygy of 
the weight w. The number of sextic syzygies of the weight w — 6 is (S 6 )', and the 
number of quintic perpétuants of the same weight is ((5))' : the former of these is 
(for not too large values of w) the greater; and at first sight it would appear that 
we can, by elimination of the quintic perpétuants, obtain from the (S 6 )' syzygies, 
(S 6 )' — ((5))' syzygies which do not contain a quintic perpetuant : if this was always 
the case, we should have in ($ B ) the term ($ 6 )' — ((5))', completing the series of terms, 
and the formula would be 
№ = ((0)) + ((2))' + ((3))' + ((2, 2))' + ((3, 2))' + (S 5 y + (S 6 )' - ((5))', 
leading to 
S B = 
otf’ + x vi — 2x w — x is 
2.3.4.5 6~ 
48. But this result is on the face of it wrong, for as remarked by Sylvester in 
the memoir referred to, from the mere fact that the sum 1 + 1 — 2 — 1 of the 
numerator coefficients is negative, it follows that the coefficients of the development 
ultimately become negative; and the actual calculation showing when this happens is 
given by him. And it is further to be noticed that not only the formula cannot be 
correct beyond the point at which the coefficients become negative, but it cannot be 
correct beyond the point for which (S 6 )' — ((5))' becomes negative: the sextic syzygies 
of the weight w — 6 may add nothing to, but they cannot take anything away from, 
the number of the sextic syzygies of the weight w. 
49. If for a moment we further consider these syzygies of the weight w — 6 ; so 
long as the number of these is greater than the number of quintic perpetuants of 
the same weight, we can by means of them presumably express each of the quintic 
perpetuants in terms of sextic products, viz. in the language of Capt. MacMahon, express 
each quintic perpetuant as a “ Sextic Syzygant.” The syzygy of the weight 9, above