1918-19.] Determinant of Minors of a Set of Arrays. 
35 
IV. — Note on the Determinant of the Primary Minors .of a Special 
Set of (n— l)-loy-n Arrays. By Sir Thomas Muir, LL.D. 
(MS. received January 21, 1919. Read February 3, 1919.) 
(1) The main specialty in the construction of the arrays in question is 
that the elements of each row are the coefficients of the powers of x in 
the expansion of a product of the form 
(x — a)(x - b)(x - c) . . . ; 
that is to say, the rows all belong to the type 
1 , 2a , 2 ab , 2 abc , . . . 
The variables from which the elements of any row are formed are n — 1 
consecutive members of a series of n(n — 1) members, the member con- 
sidered to be consecutive to the last being the first member ; for example, 
when n is 3 and the 3(3 — 1) independent variables are 
a , b , c , d , e ,/, 
the rows of the arrays are 
1 a + b ab 
1 b + e be 
1 /+ a fa. 
Further, these rows in order are taken to form all the first rows of the 
arrays, then all the second rows, and so on. Thus in the case just referred 
to the arrays are 
1 a + b 
ah 
111 b + e be I 
1 c -f- d 
ed 1 
1 d + e 
de 
1 e+f ef\ 
1 f + a 
fa 
so that the determinant proposed for consideration is then 
1 a + b i 
1 d + e 
I 1 6 + c | 
j 1 e +/ I 
1 c + d 
* 1 /+ a 
1 ab 
1 de 
1 be 
1 ef 
1 ed 
1 /« 
a + b 
ab 
d + e 
de 
b + c 
*. be 
e+f 
ef 
e + d 
ed 
f+a 
fa 
\ 
