172 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
XIV.— On the Method of Symmetric Products, and 
its Application to the Finite Algebraic 
Solution of Equations. 
By tHE Rev. Ropert Hartey, F.R.A.S. 
Read April 5th, 1859. 
Tue following investigations were suggested by Mr. 
Cockle’s paper, entitled “ Researches in the Higher Al- 
gebra,” which I had the pleasure of communicating to 
the Society a few months ago. In that paper a certain 
function of the roots of a quintic is discussed. By an in- 
direct but effective process its author succeeds in obtain- 
ing the sextic of which that function is the root. In the 
present paper Mr. Cockle’s results are verified by a strictly 
independent method, and one which comes commended by 
its simplicity, directness and generality. Other results, 
of greater or less importance, are evolved. 
Mr. Cockle first explained his method of symmetric 
products in a series of five papers “On the Transformation 
of Algebraic Equations,” printed in the first and third vo- 
lumes of The Mathematician, a journal of which the first 
number was published in November 1848. Several papers 
proceeding from the same pen on the same subject have 
also appeared in the Mechanics’ Magazine, the Cambridge 
and Dublin Mathematical Journal, the Lady’s and Gentle- 
man’s Diary, the Philosophical Magazine, and in the 
Quarterly Journal of Mathematics. But as this memoir 
will probably fall into the hands of many who cannot con- 
