174 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
whence, rejecting those values of a, and a, which render 
X symmetric, we have 
@+a+1=0, 
and a, which denotes a, or a indifferently, is obviously an 
unreal cube root of unity. This root we shall in future 
designate by «. Let 
S(E)=2 +E Lyte 2s, 
then AQHA +e a+ € Xs, 
and m(xv) =f(e) -f(e) = (22)? - 3832, Xp. 
4. When n=4, assume (cyclically) 
KX =X] +A, y+ Ay X34 Ay Hy, 
Xy=a2) + Az Ly+ As X3+ Ay X, 
Xy=2,4+ As Lo+ Mh Ly+ Ay LX, 
and 13(v) =X, Xq Xz. 
In this case the conditions of symmetry are 
1=q 4 4s, 
Sa= Fay A, =A; Ay + A, A3+ Ay dg= Gj Az + A, Gz+ Ay G5, 
and Saat xa, = e+ 3a, ay as. 
The first and second indicate that a,, a, and a; may be re- 
garded as the roots of an equation of the form 
a@—Aa+Aa-1=0; 
and, combining the first four, we have 
Da Ya, dg= Fj a, 4+ 3a, dy dg=2%a+3; 
or A?=2A+8; and A=8, or —-1. 
The first value gives 
(a-1)'=0, 
and thus renders X symmetric. We therefore reject it, 
and, adopting the last, find 
a@+@—a—-1=0. 
The roots of this cubic are —1, 1 and —1; and all the 
above equations of condition are satisfied. Let 
Xy=@— Xy+ Asy- 2%, 
Xy=a + %— X3—X4, 
and Xs=2,— %y— Wet 24 5 
then 13(x) Pe why + (Sx) -435e 32, a+ 83> 2, Xs Xs. 
