29 
1913-14.] Factorable Minors of a Compound Determinant. 
The row of M which has the three arbitrary row numbers is the same 
(viz. the first) as the column of M which has the column numbers 12 3. 
That is, the three arbitrary row numbers are associated with the column 
numbers 12 3, and it is obvious that they might have been associated with 
any of the ten combinations of the numbers 1 2 3 4 5 taken three at a time. 
Thus, for instance, associating them with the combination 2 3 4, we have 
the minor 
( 
— 
--5 
1 -- 
-45 
1-4 
1-5 
3 4 5 
1 3 4 ; 
1 35 
145 
\ 
2 3 4 
2 3 5 
12 3 
2 45 
1 2 4 
125 
3 4 5 
134! 
1 35 
145 
1 5 
1 — 4 5 1 
1 — 4 5 
1-345 
1-345 
1-345 
1 2 3 4 5 
12345* 
1 2 3 45* 
1 2 3 4 5 
1 2 3 4 5 
1 2 3 4 5 
If in M we put in the place of the dashes the same numbers as those in 
the column numbers with which they are associated, the six factors other 
than A 4 are all equal, and we have Sylvester’s theorem. 
If 1c = 4 and we take the minor 
12 34 
12 3 4 
123- 
12 3 5 
1 2 -- 
1245 
1 
13 45 
2 3 4 5 
M= 
which equals 
A • 
and put in the place of the dashes 8, 7 8, 6 7 8, 5 6 7 8, respectively, so that 
MW 
1 2 3 4- 
12 3 — 
1 2 j 
1 
1 2 3 4 5 
1 2 3 4 5 
1 2 3 451 ' 
1 2 3 4 51 
1234 
1234 
1238 
12 35 
1 2 7 8 j 116 7 8 
1245 1345 
5 6 7 8 
2 3 4 5 
we have 
M = A • 
1 2 3 4 8 
1 2 3 7 S 
[1 2 6 7 8 
1 2 3 4 5 
1 2 3 4 5 
* 1 2345 
1 5 6 7 8 
1 2 3 4 5 
which is an example of Muir’s theorem. 
In the general case the theorem is simple, though its statement is a 
little cumbersome. 
Take any one of the — A combinations (n\m \ k), (n\m\ Jc), . . . 
a 1 a. 2 
(n\m\ k), say the /3th or (n\m \ k) to start with, and arrange the set into 
a \ a /3 
groups as follows : — 
1st group containing 1 combination consisting of (n | m 1 Jc). 
a (3 
2nd group containing (m — k) l combinations, consisting of those which 
have in common the first (Jc — 1) only of the numbers of (n | m 1 1c). 
a £ 
