ON SEMINVARIANTS. 
23 
[801 
801] 
xix. (1883), 
some results 
{a, b, c$x, y)\ 
the indefinite 
nvariants, we 
zero by the 
— c 3 , &c. 
of a certain 
the weights 
'rding as the 
0, 2, 3, 4,..., 
longing to a 
n p; viz. in 
’ a covariant 
0 — w; hence 
f p, is thus 
the semin 
is equal to 
efficient of a 
n every case, 
Take A as belonging to the quantic (a,...~§x, y) n ; corresponding to such quantic, 
we have an operator A of the same rank n, viz. 
A = 2bd a + cd b for n = 2, 
= Sbd a + 2 cd b 4- dd c „ 3, 
= 4 bd a + Scd b + 2 dd c + edd „ 4, 
Operating with A on A, we have a series of terms 
A, A A, A 2 A, ..., A n0 ~ 2w A, 
but the next term A n0 ~ 2w+1 A, and of course every succeeding term, is = 0, and this 
being so, the coefficients of the covariant (A, B, ..., K^x, y) n6 ~ 2w are 
(1, ¿A, 1 LA^..)i, 
or what is the same thing, each coefficient is obtained from the next preceding one 
by the formulas 
B = \AA, C = ±AB, D = ±AC,.... 
The coefficients A and K, B and J, ... are derived one from the other by reversal 
of the order of the coefficients of (a, b,...\x, y) n , with or without a change of sign, 
and thus it is only necessary to calculate up to the middle coefficient, or pair of 
coefficients; and we obtain, moreover, a verification. 
Calculating in this manner the covariant 
(A, B, ..., K) p6 ~ 2w , 
which belongs to the quantic (a, ...$#, y) p , if we herein change a, b, c, ... into ax + by, 
bx + cy, cx + dy,... we obtain the covariant belonging to the quantic (a, ...][x, y) p+1 ; 
and in this covariant making the like change, or what is the same thing, in the first- 
mentioned covariant changing a, b, c,... into (a, b, c§x, y) 2 , (b, c, d^jx, y) 2 , (c, d, e§x, y) 2 ,... 
we have the covariant belonging to (a, ...$y?, y) p+2 ; and in like manner we obtain the 
covariant belonging to the quantic (a, ...Qx, y) n of any given order n. 
In particular, if w = ^p6, that is, if the given seminvariant be an invariant of 
(a, ...$#, y) p , then we obtain the series of covariants directly from A by therein 
changing a, b, c, ... into ax + by, bx + cy, cx + dy,... and in the result making the like 
change; or what is the same thing, in A changing a, b, c, ... into (a, b, c\x, y) 2 , 
(b, c, d^x, yf, (c, d, e§x, y) 2 , ... : and so on until we obtain the covariant for the 
quantic (a, ...][x, y) n of the given order n. 
A seminvariant which cannot be expressed as a rational and integral function of 
lower seminvariants is said to be irreducible. The theory is distinct from that of the 
irreducible covariants of a quantic of a given order; for instance, as regards the cubic 
(a, b, c, dQx, y) 3 , we have the irreducible covariant (invariant) 
a 2 d 2 + 4ac 3 + 4 b 3 d — 6 abed — 3 b 2 c 2 , 
but this is not an irreducible seminvariant; it is 
= (ac — b 2 ) (ae — 4bd + 3c 2 ) 
— a. (ace — ad 2 — b 2 e — c 3 + 2 bed), 
or, what is the same thing, there is not for the quartic (a, b, c, d, e$x, y)\ or for 
the higher quantics, any irreducible covariant having this for the leading coefficient.