324 Proceedings of the Royal Society of Edinburgh. [Sess. 
XYIII. — Vanishing Aggregates. By Professor William H. Metzler. 
(MS. received June 1, 1917. Read July 9, 1917.) 
In 1888 Dr Thomas Muir gave the following theorem : — * 
Theorem A. — If any two determinants A and B of the ?ith order be 
taken, and from these two sets of determinants be formed, namely, first, a 
set of nCr determinants, each of which is in r rows identical with A and in 
the remaining rows with B ; and, secondly, a set of the same number of 
determinants each of which is in r columns identical with A and in the 
remaining columns with B, then the sum of the first set of determinants is 
equal to the sum of the second set. 
Let the ath and /3th be two complementary selections of a and b 
respectively of the n numbers 1, 2, . . . n (where a + b'=n) f and let 
H"j) be the determinant formed by taking for its ath selection of 
a rows the ath selection of a rows from A, and for its /3th (i.e. the 
complementary) selection of b rows the /3th selection of b rows of B ; i.e . 
the rows from A occupy the same positions in A(^ 6 ^ that they do in 
A, and similarly for those from B. 
Let A(“ represent the determinant formed similarly from the 
columns, then the theorem may be stated symbolically thus : 
Theorem A. 
'a b\ ^ a fa A 
a b 
2A 
a (3 
= 2A 
where there are 
\n 
a | b 
determinants on each side of the equation. 
The object of this paper is to extend this theorem so as to involve k 
instead of two determinants. - 
Let a + 5 + c-f- . . . + k = n, and let the ath, /3th, yth, . . . /cth selections 
of a, b, c, . . . k respectively of the n numbers 1, 2, 3, ... n be a set of 
'a b c k 
va /3 y 
complementary selections. Let A 
K 
be the determinant 
formed by taking for its ath selection of a rows the ath selection of a rows 
from the determinant A, for its /3th selection of b rows the /3th selection 
of b rows from the determinant B, and so on, so that the rows taken 
from the determinants A, B, C, . . . K occupy the same positions in 
* Proc. B.S.E . , vol. xv, p. 103. 
