(173) 
NOTE ON A SUM OF PEODUCTS WHICH INVOLVES 
SYMMETEICALLY THE Ntu EOOTS OF 1. 
By Sir Thomas Mtjir, LL.D. 
(1) Each of the products in question consists of n factors of the form 
-f a,^0 + a.^B^ + . • • + 
where B is an nth root of unity ; and the sum includes 7i of these products^ 
any two of which differ merely in that the nth root of unity appearing in 
the one is different from that which appears in the other. This is the same 
as saying that if w be a primitive 7ith root of 1, the sum to be considered is- 
s=n r~n 
8=1 r—\ 
(2) As a preliminary it is necessary to recall the notation of a class 
of functions studied in 1885 under the not very appropriate name of 
" bipartites," the word being adopted in consequence of it having been 
found that bilinear forms, called " bipartites " by Cayley, were viewable as 
functions of the kind in question when of the third degree.* 
(3) The function of degree- order 2,7i is 
Oj-\ 1 Cl.), . . . , G/)i 
6i, 6o, . . . , bn 
its equivalent in ordinary notation being 
a^bi + a.^h -i- . . . -i- a>,h„ ; 
the function of degree- order 3,4 is 
CL-^ (^o Ct'^ Ct^ 
bi bey b.^ b^ 
^1 ^2 ^3 ^4 
d.2 d.^ d^ 
6] 6o 6^ 
/i 
which equals 
Gj-\ Cto tt'-i ft 0^1 W'O Qj-i Cf. A n 
h + - — ^ — '■ — -J^. + 
by ^3 ^3 b^ Cy C.^ C4 
and thus represents 
the sum of sixteen terms got by taking each element of the square array 
* 'Trans. R. Soc. Edinburgh/ xxxii, pp. 461-482, 
