176 ~—- REV. R. HARLEY ON SYMMETRIC PRODUCTS, AND 
The first renders X symmetric. The second does not 
enable us to satisfy the third equation of condition. But 
the roots of the third (which are, in fact, the unreal fifth 
roots of unity) satisfy them all. Let then i, ?, ® and # 
denote these roots, and 
Si) =44+1 MH4+P at? att 4;; 
then JPM=At? +H agti x+F x,, 
J@P)=Az +B dati Lett H+? as, 
SJ@/) HA +t at? H+? wy+t 23, 
and m(2) =fli) fl) fE) FO. 
It will be observed that, if from the above expressions we 
expunge the 2’s, the four horizontal rows read downwards 
are identical in value and order with the four vertical 
columus read from left to right, and that 7 and # lie in 
inverse symmetry upon, and 2 and @ around, diagonals. It 
does not appear to be possible to render 7,(z) more nearly 
symmetrical. A similar arrangement when x is prime, or 
a modification of it when x is odd, will probably be found 
available. "When all the divisors of m are even, we must 
adopt an arrangement analagous to that employed for bi- 
quadratics. (See Art. 4.) 
By actual development, we find 
fli) -f@)=Se +741) +79), 
+. f@) f@) =30? +762) +760), 
where T Ly Mgt Ly %gt+X3%y +X 2, +%5 2, 
7 2 Ut Hy Ly tL; Lat Mo %yt+%yX, 
and p(t) =i+e ; 
+. ma) ={3a*+7$() +74@)} {34+ 74@) +79O} - 
Developing and bearing in mind the known properties of 
i, and the relation 
T+T =A ao, 
we find 
w(x) = (2a)? — Sa? Bay @— (Fa, #2)? +577 
=(Sx)*-5( Sa)? Fa, v2.4 5( Za, x)? +577’. 
The function t7’ is unsymmetric relatively to x; conse- 
