296 
ON THE THEOREMS OF THE 2, 4, 8, AND 16 SQUARES. 
[763 
but in which it is to be understood that each duad is affected by a factor ± 1 
which is to be determined; say the factor of 12 is e 12 , that of 34, e^; and so in 
other cases. It is however assumed that e 12 , €34, e g6 , e 78 ; e 13 , e 14 , e 15 , e 16 , e 17 , e 18 are 
each =4-1. 
We have then on the right-hand side triads of terms such as, 2 into 
64263412.34 4- 64360413.24 4 64463314.23, 
which triad ought to vanish identically, as reducing itself to a multiple of 
12.34- 13.24 + 14.23; 
viz. we ought to have 
£l2 e 34 — e 13 e 24 — e l4 e 23 j 
or, using now and henceforward when occasion requires, 12, 34, &c. to denote e X2 , 634, &c. 
respectively, we have 
12.34 = +&, 
13.24 = - k, 
14.23 = + k, 
where k, = +1, has to be determined (in the actual case we have 12 = +1, 34 = +l, 
13 = 1, 14=1; and therefore the first equation gives k = 1, and the other two then give 
24 = - 1, 23 = + 1). 
We have in this way triads of values corresponding to the different tetrads 
1234 
1256 
1278 
1357 
1368 
1458 
1467 
2358 
2367 
2457 
2468 
3456 
3478 
5678, 
which can be formed with the several lines of the formula. Thus we have from the 
first line 1234, 1256, 1278; then from the second line (not 1324 which in the form 
1234 has been taken already) 1357, 1368, ...; and finally from the last line 5678.