368 
[773 
773. 
ON THE 8-SQUARE IMAGINARIES. 
[From the American Journal of Mathematics, voi. iv. (1881), pp. 293—296.] 
I write throughout 0 to denote positive unity, and uniting with it the seven 
imaginaries 1, ...,7, form an octavic system 0, 1, 2, 3, 4, 5, 6, 7, the laws of com 
bination being 
0- = 0, l 2 == 2 2 = 3 2 = 4 2 = 5 2 = 6 2 = 7 2 = - 0, 
123 = e x , 145 = e 2 , 167 = e 3 , 
246 = e 4 , 257 = e s , 
347 = e 6 , 356 = e 7 , 
where e = ±, viz. each e has a determinate value + or — as the case may be ; and 
where the formula, 123 = e x , denotes the six equations 
23= e x l, 31= e x 2, 12= e x 3, 
32 = — e x l, 13 = - e,2, 21 = -e x 3, 
and so for the other formulae. The multiplication table of the eight symbols thus is 
0 
1 
2 
3 
4 
5 
6 
7 
0 
0 
1 
2 
3 
4 
5 
6 
7 
1 
1 
- 0 
e x 3 
-e x 2 
e„ 5 
— e 3 4 
c 3 7 
-6 3 6 
2 
2 
-e x 3 
- 0 
«il 
«46 
€ 5 7 
-«4 4 
-< 5 5 
3 
3 
e x 2 
-e X l 
- 0 
«6 7 
€ 7 6 
— 67 5 
4 
4 
-6 2 5 
-e 4 6 
~ e e 7 
- 0 
€ 2 1 
e 4 2 
€ 6 3 
5 
5 
e 2 4 
— e 7 6 
-€ 2 1 
- 0 
e 7 3 
e 5 2 
6 
6 
-e 3 7 
<4* 
€y 5 
— 6 4 2 
— e 7 3 
- 0 
%1 
7 
7 
e 3 6 
^5 
— e G 3 
— e 5 2 
~« 8 1 
- 0