310 
ON SYMMETRIC FUNCTIONS AND SEMIN VARIANTS. 
[932 
disappear, and we have a relation between products of the form in question (i.e. of 
a quadric and a cubic seminvariant) and seminvariants of a degree inferior to 5, say 
this is a quintic syzygy. 
In particular, a non-composite final first presents itself for the weight 12, viz. 
here the finals are b 2 ce 2 , bed 3 , c 3 d 2 , the last of these is doubly composite, and it 
furnishes a reduction of bed 3 . For the weight 13, the finals are b 3 / 2 , b 2 de 2 , bc 2 e 2 , 
bd 4 , c 2 d 3 which are each of them singly or doubly composite : for the weight 14, 
they are b 2 cf 2 , b 2 c 3 , bede 2 , c 3 e 2 and cd 4 , and here the doubly composite form furnishes 
a reduction of bede 2 . For the weight 15, we have a final bcé which gives a quintic 
perpétuant. I have, in fact, in my paper “A Memoir on Seminvariants,” American 
Journal of Mathematics, vol. vu. (1885), pp. 1—25, [828], worked out the theory of 
quintic syzygies and perpétuants, and subsequently connecting this with the present 
theory of finals, I succeeded in showing that, when the doubly composite final contains 
a b, then there is not a reduction but a syzygy; we thus have 
G. F. for finals b 3 c 2 , b 3 d 2 , ... — x 7 2, 
„ „ bc 2 d 2 , ... = x 11 -T- 2.4, 
whence for the two forms 
G.F. is ¿c 7 ^2+tf 11 -=-2.4= {<r 7 (l —0 + # u }-h2.4, 
or say for $ 5 , the number of quintic syzygies G.F. is = ¿c 7 2.4. 
I further satisfied myself that the finals for the quintic perpétuants are bcOe 3 , 
and bcOe/ 2 , viz. the b, c, e, f being discrete letters, the interposed 0 denotes that 
the c and e are not consecutive letters. The conjugates of these forms contain the 
factors D 2 EF and GD 2 EF respectively, and it hence appears that the G. F.’s are 
= x 15 -r 3.4.5 and x lv -r- 2.3.4.5 ; adding these, we find 
for quintic perpétuants G.F. is = x 15 -r- 2.3.4.5, 
which expression was given in the memoir just referred to: the result was obtained 
by investigating in the first instance an expression for S 5 , the number of quintic 
syzygies of a given weight. The course of Stroll’s investigation to be presently 
given is different; he determines directly the number of perpétuants, and we may 
if we please use conversely this result to obtain the number of syzygies. 
62. The foregoing theory of reduction is independent of the form of the 
seminvariants, which may be blunt or sharp at pleasure : the actual formulae will 
of course be different, and they are very much more simple for the sharp semin 
variants, viz. here in many cases a seminvariant is found to be equal to a product 
of seminvariants of inferior degrees. I subjoin the following table of the reduction 
of the several sharp seminvariants up to the weight 12; the forms referred to are 
the tabulated forms, and to mark that this is so I write down in each case the 
numerical coefficients of the initial and final terms, viz. instead of c oo b 2 , d x b 3 , &c., 
I write c oo — b' 2 , d oc 2b 3 , &e. As appears by the table, these are for shortness denoted 
by G, D respectively, and so for weight 4, the forms are called E, E 2 , for weight 
5) F, F‘>, for weight 6, G, G 2 , G 3 , G 4 , and so on, the unsuffixed letters having thus 
an implied suffix, not 0 but 1. The table is