348 
EEV. E. HAELET ON THE METHOD OF STMMETEIC PEODHCTS. 
is sriven by the expression 
M2 M3 
^-dt.m+dv. 
where is the symmetric product for the imperfect form treated of in the foot-note nnder 
paragraph 18, D is the differential symbol 
A I A , A I A.A, 
dxi ~ </a'2 ‘ dxc, 
and the relations 
DP=D(-i2^,^,)=-fXr=-4M, 
and others corresponding to them, hold. Mr. Cockle had further noticed that Dt is 
to be regarded as free from M, a condition substantially equivalent to the expunging of 
the portion 
of the operator y which will be presently considered. Although he had not accurately 
completed this process of derivation, yet he had made a near approach to its completion, 
when Mr. Cayley, to whom as well as to myself Mr. Cockle had communicated it, 
showed that the same results might be more immediately and conveniently obtained by 
means of the quantical calculus, and in so doing he incidentally corrected an oversight 
which I had already pointed out to Mr. Cockle. In a letter under date September 28, 
1859, Mr. Cayley called my attention to the circumstance that the several coefficients 
of the resolvent equation of the quin tic are leading coefficients of a covariant. Mr. Cockle 
had previously suggested that the symmetric product 11, or the last coefficient, was such 
a term. The test that a function of the roots may be such a term is that it is reduced 
to zero by the operation 
It is clear that this is the case with respect to each factor of the product 
Therefore it is also the case with the product itself; and since the like is true with 
respect to the other five values of 0 , it is also true with respect to any symmetrical func- 
tion of the six values. Consequently each coefficient of the sextic in & is the leading 
coefficient of a covariant. At present, however, we have only to deal with the last 
coefficient, that is, the symmetric product. 
21. n is a seminvariant reduced to zero by the operation 
* The term “ SeminTariant ” is due to Mr. Oatlet, who in a letter to me dated March 22, 1860, saj's, 
“ The meaning is a function which is reduced to zero by one only of the operators which reduce to zero 
an invariant. It is in fact the leading coefficient of a covariant. It may also be defined as a function of 
the coefificients which is not altered by the substitution of x + h for x.” Defined as functions of the coeffi- 
cients which are not altered by the snbstitution of for x, seminvariants are what Mr. Cockle (vho 
discussed such functions some years ago in the third and concluding volume of the ‘ Mathematician,’ and 
more recently in some of the other journals referred to in the foot-note under the first paragraph of this 
paper) calls “critical functions.” I may add that some years since Mr. Cockle pointed out that the 
factors of the resolvent product, and, consequently, the product itself, are critical functions. 
