A MEMOIR ON SEMIN VARIANTS. 
262 
[828 
obtained, will serve as an example: 54 is not a quintic perpetuant, but ignoring this, 
it is by the syzygy in question expressible in the form 
54 = 43.2 
- 33.3 
- 432 
- 3.333, 
viz. as a Sextic Syzygant, inasmuch as on the right-hand side we have terms 43.2 
and 33.3, of the degree 6, which exceeds the degree 5 of the seminvariant 54 in 
question. Referring back to the definition of reduction, No. 25, observe that this is 
not a reduction of the seminvariant 54. It may be remarked that for the weight 19 
we have 15 sextic syzygies: the number of quintic perpetuants is — 3 : so that while 
it is conceivable that the 15 equations might be such that they would fail to 
determine the 3 perpetuants, it is prima facie very unlikely that this should be so. 
I have in fact ascertained that the equations are sufficient for the determination; 
that is, that (weight 19) each of the three quintic perpetuants is a sextic syzygant. 
So in the case w = 23, the number of the sextic syzygies is = 28, and that of the 
quintic perpetuants is = 5; here also the 28 equations are sufficient to determine the 
5 perpetuants, viz. (weight 23) each of the 5 quintic perpetuants is a sextic syzygant. 
50. Supposing that for any given weight w — 6, each of the quintic perpetuants 
is a sextic syzygant : this implies that the number of sextic syzygies (S 6 )' is at least 
equal to the number ((5))' of quintic perpetuants (for each expression of a quintic 
perpetuant as a sextic syzygant is in fact a sextic syzygy) : and not only so, but it 
further implies that the number of the sextic syzygies, which do not contain a quintic 
perpetuant, is precisely equal (Sf — ((5))' : for if besides the equations which serve to 
express the perpetuants as syzygants, we have any other sextic syzygy, then either this 
does not contain a quintic perpetuant, or it can (by substituting therein for every 
quintic perpetuant its value as a sextic syzygant) be reduced to a syzygy which does 
not contain any quintic perpetuant. 
51. In the general case, we have (S s )' sextic syzygies of the weight w — 6, and 
((5))' quintic perpetuants of this weight : but it may happen that certain of the 
quintic perpetuants do not enter into any of the sextic syzygies ; and those which 
enter, may do so in definite combinations : by elimination of these combinations of 
perpetuants we obtain (it may be) a sextic syzygies not containing any quintic per 
petuant ; and the remaining (S 6 V — a equations will then serve to express each of 
them a quintic perpetuant, or combination of quintic perpetuants, as a sextic syzygant. 
The number a is at most = ((5))', or taking it to be = ((5))' — ((0))', the number of 
sextic syzygies not containing any quintic perpetuant will be = ($„)' — ((5))' + ((0))', 
that is, the number of sextic syzygies not containing any quintic perpetuant will 
be equal to the whole number (S 6 )' of sextic syzygies diminished by some number 
((*>))' — ((&))'> which is less than or at most equal to the whole number ((5))' of 
quintic perpetuants of the weight in question w — 6. But as already mentioned, I have 
not been able to obtain the expression of the function (6), = 
G. F. of the number ((d))'. 
Cambridge, England, 17th March, 1884. 
m (x) 
2.3.4.5’ 
which is the