773] 
ON THE 8-SQUARE IMAGINARIES. 
371 
Write as before e 1 = e 2 = e 3 = + ; then, disregarding the lines (such as the first line) 
which contain the symbol 0, and writing down only the signs as given in the third 
and fourth columns, these are 
«6 
E 4 
€ 7 
~ *5 
“ € 7 
- e 4 
- *6 
«5 
- *4 
- E 0 
— €g 
€ 7 
«4 
€7 
€ 5 
- e g 
€7 
«4 
€g 
- «G 
~ «4 
— €7 
- «6 
E 5 
+ 
“ e 5 e 6 
— 
— e 4 e 7 
+ 
— € 5 € 7 
— 
~ e 4 e 6 
— € 4 € 7 
- 
*4 
«G 
€5 
e 7 
- *4*6 
- 
+ 
- E 6 
- E 4 
€ 7 
~ *5 
- «5*7 
+ 
+ 
- E G«7 
- 
- E 4«S 
- e 4 E 5 
- 
— C 6 e 7 
+ 
We hence see at once that the pairs of signs in the two columns respectively cannot 
be made identical: to make them so, we should have e 6 = e 4 , e 7 = — e 5 , e 7 = e 4 , that is, 
e 4 = e 6 = e 7 = — e 5 , which is inconsistent with the last equation of the system — e 6 e 7 = +. 
Hence the imaginaries 1, 2, 3, 4, 5, 6, 7, as defined by the original conditions, are 
not in any case associative. 
If we have e 1 = e 2 = e 3 = + and also — e 4 = e 5 = e 6 = e 7 = 9, that is, if the imaginaries 
belong to the 8-square formula, then it is at once seen that each pair consists of 
two opposite signs; that is, for the several triads 123, 145, 167, 246, 257, 347, 356 
used for the definition of the imaginaries, the associative property holds good, 
12.3 = 1.23, etc. ; but for each of the remaining twenty-eight triads, the two terms 
are equal hut of opposite signs, viz. 12.4 = -1.24, etc. ; so that the product 124 of 
any such three symbols has no determinate meaning. 
Baltimore, March 5th, 1882. 
47—2