218 REV. R. HARLEY ON SYMMETRIC FRODUCTS, AND 
= Lard! a? (a,2; +232.) — Sxiaia,t dx, (222+ x32?) 
— Dar7, X32. 
Consequently 
T=42 2° —5iatex, —10Dx322 — 20D 232,27, — 130222232, 
— 1602 a2727,7,4 4802, 227,7,27;+ 00D 72277 (a2; 
+ 2424) + 2503/2, (23a? + x22). 
But by commencing the calculations with Y (in place 
of T) we should have been conducted to the fifth power 
of the four-dimensioned function discussed in the above 
Essay. It is the form of that function which has enabled 
me to introduce 7 with such effect into the calculations ; 
and it is the introduction of 7 which has enabled Mr. 
Cockle (Phil. Mag., May 1859) to give a simpler form of 
root than the expression 
ia, +7™X + P?"™X, 4+ 8"X5-+ ea | 
to which he would otherwise have had recourse. 
The advantage of Lagrange’s and Vandermonde’s pro- 
cess is that the solution of their sextics would lead directly 
to that of the quintic. The disadvantages are the high 
dimensions of the symmetric functions, and the consequent 
difficulties of calculation and verification. The Method cf 
Symmetric Products, considered independently, leads only 
indirectly to the solution of a quintic. But it has the ad- 
vantage of lower dimensioned functions and of greater 
facilities for calculation and verification, >/aj(#.7;+232,) 
being a more manageable function than ¥’a#}(a2a,+ 43a) 
or 2x, (a3a2+x3v}). The unsymmetric expression for the 
roots of a quintic given by Lagrange and Vandermonde 
is undoubtedly an element of the theory of quintics as 
treated in the foregoing Essay, but the progress made will 
be tested by the simplicity of the results to which the 
Method of Symmetric Products has conducted us. If the 
difficulties of dealing with symmetric functions increase in 
more than arithmetical proportion to their dimensions, it 
