1874.] Investigation of Determinants. 225 
Eq. (40) shows that [«, ] is always an integer (as it should 
be). It will be useful to record a few values of [e,] for 
reference. Thus— 
[6] =I, le] =1, [e,] =6, [e6] =120, [eg] eae «e(42) 
[ero] = 446,985 J ~~ 
XII. Number of Elements in a Skew Determinant. 
Denote the number of elements in a skew determinant of 
nm rows bys. Then, since a skew symmetric determinant 
is a skew determinant whose leading constituents are zero, 
‘the relation between them is the same as between [A] and 
A, so that the relations between [«, ] and «, will be the same 
as those between [E,, ] and E, demonstrated under Problems 
(III.) and (IV.) as common to all determinants. Thus, the 
values of «, may be calculated from Eq. (6), viz.:— 
& =(I+ [e]”) 
n(n —1) | 2 
E+ 5. lel tog - lel +++ +7ppnce- eI +--- 
+* [én—z] + len J..ee (43). 
A few values of «, may be recorded for reference— 
Si, G1, &,=2, €,=4, &=13, &=41, &= 220, 
&,= 1072, &= 9374, &= 60,968, &,.= 723,966 j sorte 
It is also seen that if [e”] denote the number of elements 
in a skew determinant out of whose leading constituents 
only m are finite, then changing E in Problem (IV.) into ¢, 
all the relations demonstrated under Problem (IV.) between 
E,, [E,], and [E;"] are true between «,, [e,], and [e,], so 
that [e] may be calculated from the known ¢,, [e,] by the 
formule of Problem (IV.). 
XIII. Number of Minors in a Determinant. 
By definition, a pth minor is formed by erasing £ rows and 
p columns from the original, so that evidently :— 
A pth minor is a determinant of (n—f) rows and ae 
A (n—p)th minor is a determinant of £ rows and columns 
Let ”M, be the number of pth minors that can be formed 
from a determinant of » rows. Then”M,-_» is the number 
of (~—p)th minors. 
It is obvious that, in this notation, the minor of zero 
order (6=0) being the original determinant itself, and the 
minors of mth order being zero— 
*“M, =I, and”M, =1 
in all determinants (which are finite).... (45). 
