216 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
have been conducted with the greatest care, and verified 
in a variety of ways. I believe that the equation in @ 
will be found to have an important bearing on the theory 
of quintics; and therefore I have thought it desirable to 
place its accuracy beyond all reasonable question. 
46. We have seen how steadily the Method of Sym- 
metric Products mounts up to the higher equations, and 
in the course of our investigations we have been con- 
ducted to most significant results. Guided therefore by 
the analogy afforded by the lower equations, we might 
next consider the application of the method to quintics. 
But my professional occupations will not permit me at 
present to enter into this interesting discussion. I may 
however remark that even if the method fail to achieve 
the solution of the general problem, it will probably help 
to settle a controversy in which mathematicians of the 
greatest eminence have taken opposite sides, and to throw 
light upon the question, respecting which so much has 
been written, of the possibility or impossibility of express- 
ing a root of the general equation of the fifth degree by a 
finite combination of radicals and rational functions. 
ADDENDUM. 
The resolvent product (@) for a quintic equation is 
immediately connected with the last coefficient of the 
equation of the fourth degree which occurs in Lagrange’s 
theory, the coefficient in question being by the present 
theory determined as the root of an equation of the sixth 
degree. And, since it is part and parcel of Lagrange’s 
theory, that when any one root of the equation is known 
(ex. gr. if there is a root zero, which is the case considered 
in the second section), the roots of the proposed quintic 
can be determined by radicals, the form of the results 
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