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XVI. On the Method of Symmetric Products, and on Certain Circular Functions con- 
nected with that Method. By the Bev. Eobeet Haelet, F.B.A.S., Corresjjonding 
Memler of the lAterary and Philosophical Society of Manchester. Communicated 
hy A. Cayley, Esp, F.B.S. 
Eeceiyed October IS, — Bead December 13, 1860. 
In a paper printed in the second part of the fifteenth volume of the ‘Manchester 
Memoh's,’ I have given a systematic exposition of Mr. Cockle’s Method of Symmetric 
Products, and its application to the finite algebraic solution of the lower equations. In 
that paper, to which I shall in future refer as my original memoir, I have also defined 
a new cyclical s^unbol, and I have by its aid succeeded in effecting the direct calculation 
of a certain sextic equation, on whose solution that of the general quintic may be made 
to depend. In an Addendum I have pointed out the connexion between the circular 
functions which occur in my own researches and those to which we are led by the 
theoiy of Lageaxge and Vaxdeemoxde, and, by means of the cyclical process, I have 
given a neat expression for the first coefficient of Lageaxge’s reducing equation. These 
researches I have followed up in an article “ On the Theory of Quintics,” in the third 
volume of the ‘ Quarterly Joui-nal of Pure and Applied Mathematics.’ My present pur- 
pose is not to repeat, but to endeavour to generalize and extend former results. I shall 
therefore content myself with a very brief resume of my investigations, referring the 
reader for details to the above w'orks. Mr. Cockle’s earlier researches on the subject 
were published in a series of five papers “ On the Transformation of Algebraic Equa- 
tions,” printed in the first and third volumes of ‘ The Mathematician'*.’ 
Section I. 
The Method of Symmetric Products, and a New Application of it to the Solution of the 
Lower Ecpiations. 
1. Any n symbols A’,, x^, x^, . . . x^ may be regarded as the roots of an equation of the 
form 
(«, h,c, ... \x, Vp=a{x—x,){x—x^){x—x 3 ) . . . {x—x^ = ^\. 
* The first paper of the series appeared in ‘The Mathematician’ for March 1844. Mr. Cockle’s sub- 
sequent contributions to the subject will be found in the ‘ Mechanics’ Magazine,’ the ‘ Cambridge and 
Dublin Mathematical Journal,’ the ‘Lady’s and Gentleman’s Diary,’ the ‘Philosophical Magazine,’ ‘Lioxr- 
tille’s Journal,’ the ‘Quarterly Journal of Pure and Applied Mathematics,’ the ‘Manchester Memoirs,’ 
and elsewhere. 
t Throughout the whole of this paper I adopt Mr. Catlev’s quantical notation. In the ordinary nota- 
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