1913-14.] Factorable Minors of a Compound Determinant. 
27 
IV. — Some Factorable Minors of a Compound Determinant. 
By Professor W. H. Metzler. 
(MS. received April 17, 1913. Bead November 3, 1913.) 
If we start with a determinant A of order n, and, using exclusive umbral 
notation, take the minor 
Mee 
(n\m\k) 
a 1 
(n | m | k) 
(n | m | k) 
a 2 
(n | m | k) 
(n I m | k) \ 
a x \ 
(n\m\/c) L 
a \ / 
m k 
of the (n — 7c)th compound of A, Sylvester * has shown that 
M = A 
(n | m) 
( n | m) 
. (A) 
Besides this, Muir j- has considered another type of minor which breaks 
up into factors. It may be obtained from M by putting k = m— 1, and in 
place of the combinations — 1), . . . 1) indicating the 
a 1 am 
selections of rows for the elements we take the combinations 12 . . . m— 1, 
23 . . . m, 34 . . . m+1, . . . mm + 1 . . . 2n — 2, where for definiteness 
of statement we suppose a = 1, and (n \ m) = 12 . . . m. Thus the theorem 
a 
given in Muir is 
12 ... m- 1 
23 ... m 
mm + 1 . 
. . 2m -2 
(n\m\m — T) 
\ l i 
(n | m | m - 1) 
1 2 
(n m 
l 
i m — i) 
m 
23 ... m + 1 
34 ... m + 2 
A . 
( n | m) 
(n | m) 
l 
l 
m . . . 2m — 3 
( n | m) 
l 
(B) 
In both these theorems the combinations indicating the selection of row 
numbers are definite. In Sylvester’s theorem they are the same as the 
selection of the column numbers. In Muir’s the first one is the same, and 
the rest may be obtained by a definite sliding process. 
The object of the present paper is to show that there are a large number 
of other minors which break up into factors, and to give a general theorem 
(C) which includes these two as special cases. 
Theorems (A) and (B) are readily proved by the method used by the 
* Philosophical Magazine , 1851. 
t A Treatise on Determinants , Art. 93. 
