120 
MR. A. CAYLEY’S SECOND MEMOIR UPON QUANTICS. 
which does not terminate ; the number of irreducible covariants is therefore infinite. 
Substituting the preceding values, the expression for the number of the asyzygetic 
covariants of the degree 6 is 
coeff'.^®in(l —x^)~^{l—x*) ^(l —x^) ^(l— — a;‘®)®(l — a;®)’(l — — a? 
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— 
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„, M^, M^, .. 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 0. We may in like manner form a covariant such that its leading term is lower 
than the leading term of every other covariant except Gj ; or rather we may form a 
system of such covariants, since if O2 be a covariant having the property in question, 
<I>2+A:0, will have the same property, but k maybe determined so that the covariant 
shall not contain the leading term of Gj, i. e. we may form a covariant 02 such that 
its leading term is lower than the leading term of every other covariant excepting 
0,, and that the leading term of 0i does not enter into 02 ; and there is but one such 
covariant, 02. Again, we may form a covariant 03 such that its leading term is lower 
than the leading term of every other covariant excepting Gj and 02, and that the 
leading terms of 0i and 02 do not either of them enter into 03 ; and there is but one 
such covariant, Gj. And so on, until we arrive at a covariant the leading term of 
which is higher than the leading terms of the other covariants, and which does 
not contain the leading terms of the other covariants. We have thus a series of 
covariants 0 ,, 02, 03, &c. containing the proper number of terms, and which 
covariants may be taken to represent the asyzygetic covariants of the degree and 
order in question. 
