Note on TJnimodular and other Per symmetric Determinants. 
99 
which also being unity entails a unit multiplicand. The property of com- 
binatory numbers which underlies the multiplication is — 
- ^(2A; + 5),+2.(A; + 2)6 + 
(10) The second result, 
p ^Iq 3^ ho 7^ 
= (-1) 
2k 
/ -"-0 ^ £_2 i 8 \ _ -1 
1 1' 3' 5'T' ' ■ ■ • y~ ' 
can be established in similar fashion, the multiplier now being — 
-1 
1 
1 
-3 
1 
~1 
6 
-5 
1 
1 
-10 
15 
-7 
1 
of which the Ic^^ row is — 
(-ly-Hl, - (7^ + 2),, .... , (-1)^-1(27.- 2),,_2}, 
and the property of combinatory numbers underlying the multiplication 
being: — 
2h-i 
(27.-1);. 
h-k- 
Viewed in connection with the first result the second may be put thus : 
If every element (2r — 1),. of the first ijersymmetric he divided hy (2r — 1) 
the determinant is unchanged in value. 
(11) The third result, notwithstanding the peculiar law of formation of 
its determinant, can also be established in the same way, the multiplier 
and product being respectively 
1 
1 1 
1 -1 
1 -2 
1 2 
1 3 
1 
1 
3 
3 
1 
1 ] 
4 -1 
1 . 
1 1 
2 1 
3 3 
6 4 
1 
1 
4 
10 10 5 
1 
1 
5 
1 
1 1 
with laws of formation as curious as that of the multiplicand. The ^^^^ row 
of the former is best thought of as written from right to left, thus : 
— (7»: — 4)/._6, (A; — 3)/._5, (A- — 3)/--4, 
-(Z;-2)/,_3, -{h-2)k-2, (7.--l)/,-i, 
