270
A SECOND MEMOIR UPON QUANTICS.
[141
52. For a quintic, the number of irreducible covariants of the degree 6 is
coeff. x e in
1 + x + x 2 + 4 a? + 6æ 4 + 8x 5 + 9x 6 + \0x 7 + 12æ 8 4-1 Oæ 9 + 9æ 10 +Sic 11 +Qx 12 +4,* 13 +îc 14 +x 15 +x 1 '
(1 — af) 2 (1 — ¿c 4 ) (1 — x 6 ) (1 — ¿c 8 )
the numerator of which is
(1 + xf (1 — x + 2x 2 + x 3 + 2Æ 4 + 3& 5 + x e + 5x 7 + of + 3æ 9 + 2æ 10 + x n + 2x 12 — x 13 + x u ) ;
the first factor is (1 — x)~ 2 (1 — x 2 ) 2 , the second factor is
(1 -x)(l -x 2 )~ 2 (l -x 3 )~ 3 (1 -x*)~ 2 ( 1 -a?)- 2 (I
which does not terminate; hence [incorrect] the number of irreducible covariants is
infinite. Substituting the preceding values, the expression for the number of the
asy z ygetic covariants of the degree 6 is
coeff. x e in (1 — x)~ l (1 — A’ 2 ) _2 (l — ¿it 3 ) -3 (1 — 3 (1 -x b )~ 2 (l - # 6 ) 4 (1 - x 7 ) 6 (1 - a? 8 ) 6 (1 — A’ 9 ) 1 (1 — ^ 10 ) -9 (1 -.v 11 )- 19 ...,
which agrees with a previous result: the numbers of irreducible covariants for the
degrees 1, 2, 3, 4 are 1, 2, 3 and 3 respectively, and for the degree 5, the number
of irreducible covariants is three, but there is one syzygy between the composite
covariants of the degree in question; the difference 3 — 1=2 is the index taken with
its sign reversed of the factor (1 — x 5 )~ 2 .
53. I consider a system of the asyzygetic covariants of any particular degree and
order of a given quantic, the system may of course be replaced by a system the terms
of which are any linear functions of those of the original system, and it is necessary
to inquire what covariants ought to be selected as most proper to represent the
system of asyzygetic covariants; the following considerations seem to me to furnish
a convenient rule of selection. Let the literal parts of the terms which enter into
the coefficients of the highest power of x or leading coefficients be represented by
M a , Mp, these quantities being arranged in the natural or alphabetical order;
the first in order of these quantities M, which enters into the leading coefficient of a
particular covariant, may for shortness be called the leading term of such covariant,
and a covariant the leading term of which is posterior in order to the leading term
of another covariant, may be said to have a lower leading term.
It is clear, that by properly determining the multipliers of the linear functions we
may form a covariant the leading term of which is lower than the leading term of
any other covariant (the definition implies that there is but one such covariant); call
this ©. We may in like manner form a covariant such that its leading term is lower
than the leading term of every other covariant except ©j; or rather we may form a
system of such covariants, since if d> 2 be a covariant having the property in question,
d> 2 + &© x will have the same property, but lc may be determined so that the covariant
shall not contain the leading term of ® x , i.e. we may form a covariant © 2 such that
its leading term is lower than the leading term of every other covariant excepting
@j, and that the leading term of S 1 does not enter into © 2 ; an d there is but one such
co variant, © 2 . Again, we may form a co variant © 3 such that its leading term is lower
than the leading term of every other covariant excepting © x and © 2 , and that the
