A Sum of Products which Involves Symmetrically the Nth Boots of 1. 176 
It is readily manifest that in tlie product the term free of w is 
^3, 
bn-1, ■ 
. . , 63 
the cofactor of the first power of u 
is 
• • 
• > CL n 
bn, ■ • 
• ' ^3' 
the cofactor of the second power is 
^1) ^2' 
bg, 63, 6/ 
and so forth. The total product thus is 
^2 
. CL,i 
1 
OJ 
0,2 . . 
b, 
b, 
bs 
bn 
bi 
b. 
bn-i . 
bn . 
bi . 
. bo 
■ h 
■ h 
1 
or 
b, 
bn 
bu-1 
bo 
\ 
\ •■ 
\ ■ 
■ bn 
■ f-n-l 
■ • bii-2 
a 
a 
a 
bn 
bn-l 
bn-2 • 
b-z 
h ■ 
■ w 
a- 
where the important thing" to note is that the rows of the square arrays are 
the circular permutations of 
hn, bn-i, ... , ?>i and b-^^, bc^, 63, ... , bn 
respectively — in other words, that the square arrays are the arrays of the 
circulant 
0(^1, bn, bn-l, ... , ^2) 
and its conjugate. 
Of course, from the nature of the case, the a's and fe's may be inter- 
changed, and thus other two expressions for the result are obtainable. 
(6) Confining ourselves, for convenience in writing, to the case where n 
is 4, let us now introduce a third factor 
Cj + C2^ + CgW^ + Cj^u^ 
the multiplicand being, as we have seen, 
^1 ^2 ^3 ^4 ' ^1 ^2 ^3 '^4 _|_ ^1 ^3 ^4 _j_ ^1 ^3 ^4 3 
61 ^4 ^3 ^2 ^2 ^1 ^4 ^3 " ^3 bo 61 64 63 63 61 
In the new product the cof actors of u^, w^, u^, are seen to be respectively 
€L-^ City a^ 
(X^ Ctc) €t^^ Ct 
Ctc) Ctf^ 6^/^ 
h 
h 
h 
> 
^1 
> 
Co 
^4 
h 
h 
h 
^4 
h 
^3 
^4 
^\ 
^2 
h 
^2 
^3 
^4 
^1 
and therefore the whole product to be 
ao 
(^4. 
h 
bs 
bo 
^1 
C3 
^4 
6., 
b. 
b. 
C4 
^1 
% 
^3 
h 
bi 
b. 
^3 
C4 
Co 
h 
h 
b-2 
b. 
^2 
C3 
^4 
1 
ft; 
ft;2 
