224 Investigation of Determinants. (April, 
Table of Values of [c"].....-(39)- 
m 
d fe) “h li ili iv v vi. vii vili ix x 
oO I 
I ° I 
2 I I 2 
3 I 2 3 5 
4 6 7 9 12 17 
5 22 28 35 44 56 73 
6 140 162 Igo 225 269 325 308 
7h 927 1067 1229 1419 1644 1913 2238 2636 
8 7469| 8396 9463| xo692| 12111 13755 15668 17996} 20542 
9 || 66748] 74217] 82613] 92076)102768| 114879] 128634] 144302] 162208 182750 
10 || 667953 | 734701 | 808918 | 891531 | 983607 | 1086375 | 1201254 | 1329888 | 1474190 1636398 | 1819184 
XI. Number of Elements.in a Skew Symmetric Determinant. 
Denote the number of elements in a skew symmetric deter- 
minant of 7 rows by [e, ], the brackets being used to preserve 
the analogy with previous notation for determinants whose © 
leading constituents are zero. 
It is shown (Salmon, Art. 37) that every skew symmetric 
determinant of odd order vanishes; the number of its ele- 
ments is therefore zero. 
It is shown (Salmon, Art. 38, 39) that every skew sym- 
metric determinant of even order is a perfect square, and 
may be expressed by {3(+ap5q.drs...+Gyz)}?, 4.€., by the 
square of the sum of terms of type Gea . Ars.» bys), CACHE 
of which is the product of ~+2 constituents, involving all 
the suffixes without repetition. It has been shown (Problem 
VII.), that the number (S,) of such produéts in a deter- 
minant whose leading constituents vanish (as is the case in 
a skew symmetric determinant, see Salmon, Art. 37) is 
n 
Sn = [n+(2?- zs 
Also, since the determinant is the square of S, different 
terms, it follows that the number of its elements [e, ] is the 
same as the number of terms in the expanded square of the 
sum of S, quantities. 
ies (number of combinations of 
let eis +} S, things two together). 
=S, +3458, .(S, —1) 
a Sn . (Sn + I), 
- nN f 
|~.(|"+2?.) >) \ when # is even... . (40). 
gti. ( | -)* 
And [e, ] =o, when » is odd, (v. supra).... (41). 
