1874.] - Investigation of Determinants. 227 
the reasons above. Hence the whole number of fth 
minors is— 
"Mp = "Cp +41("Cp)° —("Cp )} 
eC, ("C, +1), which is clearly an integer. || (48), 
mee ele: Laae 
a2 Cb: p=?) 
Since “C; = "C,_;, therefore in symmetric determinants— 
Siren Mig pert ta a a aes (49). 
On account of the great use of symmetric determinants 
in modern geometry, it will be useful to record some values 
of "M, for reference. 
N.B. In general "M,=1= "M,, 
"M,=3n(n+1)= "My-: - (50). 
"M,=4n.(1—1) (1 —n+2)="M 
; 
Nu—2 
Walses, Gf Mipsis ans or suls (51). 
Value of p. 
n. 
oO. I 2. 3. 4. 5. 6. bie 8. g. |10. 
I I I 
2 I 3 I 
3 I 6 6 I 
4 re, |. ro 2E Io I 
5 rT} 15 55 55 15 I 
6 eo I20 | 210 120 21 I 
b E28 231 630 630 231 28 I 
8 I | 36 | 406 | 1596 2485 1596 406 36 I 
9 I | 45 666 | 3570 8001 8001 3570 666 45 I 
Io Hee 55 |) L035) | 7260 | 22755 31878 22155 | 7260 | 1035 | 55 
XV. Number of Minors in a Skew Symmetric Determinant. 
Note that all leading minors are themselves skew sym- 
metric determinants, so that all leading minors containing 
an odd number of rows vanish (Salmon, Art. 38). Nowa 
pth minor contains (n—p) rows, and the number of leading 
pth minors in a symmetric determinant is in general 
2C, = iB aE (see Problem XIV.). Hence, in a skew sym- 
metric ee the number of leading pth minors is Zero, 
or "Cy according as (1—) is odd or even. And the number 
of non-leading pth minors will be the same as in a symmetric 
determinant in general, viz., }.{("C;)?— "C,}. Hence, 
in a skew symmetric determinant— 
