362 
[942 
942. 
ON SEMINVABIANTS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xxvi. (1893), 
pp. 66—69.] 
I WISH to prove the following negative: a given sharp seminvariant is not in 
every case obtainable by mere derivation from a form of the same extent and of the 
next inferior degree. The meaning of the statement will be explained. 
According to the general theory developed in Clebsch’s Tlieorde der binaren 
alcjebraischen Formen, Leipzig, 1872, the covariants of a given binary quantic / are all 
of them obtainable, the covariants of a given degree from those of the next inferior 
degree, by derivation (Ueberschiebung) of these with /; viz. if the covariants of the 
next inferior degree are P, Q, &c., then the covariants of the degree in question are 
all of them included among the forms 
(f,PH=fP), (.f,P)\ (fPY,..., 
(f Q)° , (fQ)\ (.f,Q)\.... 
&c., 
the index of derivation for (/, P) being at most equal to the degree of f or to that 
of P, whichever of these is the smaller, and so for Q, &c. The forms thus obtained 
are far too numerous; but rejecting repetitions, we have a complete system of the 
covariants of the given degree, viz. every covariant whatever of that degree is a linear 
function (with numerical multipliers) of the several distinct forms thus obtained by 
derivation. 
We can therefore, by linear combination as above, obtain all the sharp covariants 
of the given degree, but we may very well have a sharp covariant not included among 
the several distinct forms thus obtained by derivation, but only expressible as a linear 
combination of two or more such forms: or say we may very well have a sharp