186 REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
epimetrics, being more general in its conception, and 
possessing peculiar working properties. It is a symbol 
of cyclical operation. 
19. Since each root recurs at every fifth step in any 
given cycle, we have 
x(5)=x(0), and 
Bx) =x(1) +x(2) +x (3) +x(4) +x(5) =27-x0) 5 
whence, by generalization and induction, 
3 X(9) = 3x (0). 
That is, the circular function 3’,(0) is not affected in 
value by the simultaneous advancing or receding of the 
roots which it contains any number of steps (g) in the 
cycle 7, Consequently 
3 Xi (0) 2, Xe Ms 
= 5’, [xi(0) £x2(0) + x2(1) +.X2(2) + x2(3) + x2(4) §] 
= 3%", [x2(0) 2x1 (0) +x1(1) + xn (2) + 20(8 Re )5]. 
This commutative property will be found of great prac- 
tical utility in dealing with circular functions. 
20. For example. Taking the first cycle as a type of 
the rest, and omitting, for the sake of simplicity, the unit 
suffix, we have 
TT =D Ly Ly 2 Ly Bs 
=D" fay Hy(@y Vet Hy Uy +2 Xz +H, Xj +25 Xo) § 
= (af Ly y+ Ay WZ Mp4 Ly Ly Ly Ly + Uj Uy Vy +2, Ly 25). 
Now 32, #3 %;= D2, Xa #3 2,4, and applying the theorem 
XD =2"x0); 
we find 
>a, Br=>'22 7,23, and Xa, 23 43=2' Li Uys; 
"TT Hay Ly Ly Vy HD Tig y+ Le Xt Lg Xs + Hy Hs) ; 
but 
Daj ay P= I’ ai (ay Ly + Hy Uy t Ly Ls + Hy Uy +a%yXs+Xy 2s), 
*, TT! = Say Uy Wy Ly + YH} Uy Ly — BX} (Lp Ws +23 Xs) 5 
and consequently (Art. 5) 
m4(x) = (Sa)*—5 (Fax)? Fay e+ 5( Bay Aq) + 53a, Ly ya 
+5 Say ay 25-53’ a}(%y Ws; + 23 Xs) 
=a' — 5a? b+ 5ac+58? — 15d — 532i (x2 %5+ 23%). 
