123 
of Edinburgh, Session 1866-67. 
then add to it the sum of the others : and we have 
V == (n - 1) 
a n 
... a 
1 , re 
2 , re 
re , re 
- a l , 
... — a , 
,i 
2, 1 
re , 1 
— a, „ 
— (7 „ 
... — a „ 
1,2 
2,2 
re , 2 
-a, - a ... - a 
l,re — 1, 2, re — 1 n,n~i 
The negative signs may be removed and compensated for by the 
factor ( — l) n— 1 placed outside the Determinant, and the first row 
may be made the last by multiplying again by (-l) n—1 , which 
counteracts the former multiplication. Hence 
V = 0 - 1) 
Q. E. D. 
a. . 
OL , . . 
. a , 
1,1 
2,1 
re, 1 
a, „ 
. « „ 
1,2 
„ . 
2,2 
re, 2 
a . 
a a . . 
1 , re 
2 , re 
re, re 
When n = 3, this is the quaternion theorem in question. 
The second formula is the well-known property of the reciprocal, 
and a simple proof of it will be given below. 
The third and fourth are immediate translations of Quaternion 
expressions into Determinant forms. And I suppose there is no 
analytical difficulty in enlarging the number of Sir William R. 
Hamilton’s symbols, i, j, h (though it would introduce the concep- 
tion of ideal space of more dimensions than three), and thus ex- 
tending the reach of the “ Quaternion path” to Determinants of 
any order. 
The fifth formula, in its algebraical expression, would be 
y « 
+ 
2/1 *1 
25 03 
+ 
*1 X 1 
yi 
y-i *2 
j 
*1 *1 
*2 X 2 
yi 
2/2*2 
+ 
2/2 *2 
y « 
&C., 
2/ 2 *2 
y a 
+ 
y * 
2/1 *1 
&c., 
&c. 
&c. 
&c. 
= - 2 
x y z 
x x y x z x 
x 2 V 2 z 2 
2 
