942] 
ON SEMIN VARIANTS. 
363 
covariant not obtainable by mere derivation from the forms of the next inferior degree; 
and it is important to verify that there are sharp eovariants not thus obtainable by 
mere derivation. I remark that the notion of a sharp covariant does not present 
itself in Clebsch, and that it will be presently explained. 
Passing from a covariant to its leading coefficient which is a seminvariant, the 
statement may be applied to seminvariants; viz. it is to be verified that there are 
sharp seminvariants not obtainable by mere derivation from the seminvariants of the 
next inferior degree. But we have to introduce the notion of “ extent ” so as to 
connect the seminvariant with a quantic of some particular order, thus, if the highest 
letter of the seminvariant is f, we say that the extent is = 5, and thus connect it 
with the quintic 
(1, b, c, d, e, f\x, yf. 
The notion “ sharp ” applies to the seminvariants of a given weight. Suppose, 
for instance, the weight is = 8; we have a series of initial or non-unitary terms 
i, eg, df, e 2 , &c., and a series of final or power-ending terms e 2 , cd 2 , b 2 d 2 , c 4 , &c., and 
we denote by eg — c 4 (where observe that here and in all similar cases the — is not 
a minus sign, but is simply a stroke), the whole series of terms (including eg and c 4 ) 
which are in counter-order (CO) subsequent to eg, and in alphabetical order (AO) 
precedent to c 4 ; and so in other cases. This being so, arranging the seminvariants 
with their final terms in AO, we have the seminvariants 
i — e-, 
eg — cd-, 
df — b 2 d 2 , 
e- — c 4 , 
&c., 
viz. we have a seminvariant i — e 2 containing all or any (in fact, all) of the terms of 
this set as above defined; a seminvariant eg — cd 2 containing all or any of the terms 
of this set, a seminvariant df — b 2 d 2 containing all or any of the terms of this set; 
and so on. These are sharp forms; a seminvariant ending in e 2 , must of necessity 
have the leading term i, and thus belong at least to the octic 
(1, b, c, d, e, f g, h, iffx, y)\ 
a seminvariant ending in cd 2 must of necessity have a leading term as high as eg, 
and thus belong at least to the sextic 
(1, b, c, d, e, f g\x, y) 6 . 
Any linear combination of these would be a seminvariant i — cd 2 , belonging to the 
octic, but it is not a sharp form; the final term cd 2 does not of necessity imply 
an initial so high as i (in fact, as we have seen, it only implies the lower initial eg): 
and so in other cases. 
For the quintic (1, b, c, d, e, f\x, yj, we have (for the weight S and degree 4) 
the seminvariants df — b 2 d 2 , and e 2 — c 4 , this last belongs, of course, also to the quartic 
(1, b, c, d, e\x, y) 4 , it is, in fact, the squared quadrinvariant (e — 4bd + 3c 2 ) 2 . I wish to 
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