An Upper Limit for the Value of a Determinant. 
329 
can only be inversely orthogonal when 
a, b, c, d, ej= -i, (-if, {-if, (-iy, (-if, (-if, 
= -i, -1, i, 1, -1, -i. 
To ensure inverse-orthogonalism the products which must be equal are 
adf+ 2bce - c^d - ¥f~e'^a, h(-bf - c - e + b +f + ce), 
- df -ce - be + cd + bf +e'^, c(be + c + d- b -e~ cd), 
bf + + ae - be - af - ce, d(af+2c-a-f — c^)j 
-be - bc-ad + b^ + ae + cd, e(-ae~c-b + a + e + bc), 
a(df+ 2e-d-f- e% f{ad + 2b-a-d-b^); 
and since the equality of the 8th and 10th products is tantaniount to the 
equality 
d _2h-b^ — a 
f 2c -c^ _ a' 
it is evident that the said two products will be equal if we make c — b and 
f=d. Doing this we next see that the equality of the 3rd and 6th products 
is tantamount to 
a(e — d) = b^(e — d) ; 
and, since the taking of e = d is excluded by the fact that this would cause 
both products to vanish, we are forced, in order that the two may be 
equal, to take a = b^. It will be found, however, that this taking of a, c, 
f=b^, b, d makes certain others of the products equal — that, in fact, there 
only remain five to be dealt with, these now taking the forms 
b\d-e)(d + e-1), - (d - e)(d + e - 2b), {d-e)b{l-b), 
d{b - l)(bd + d- 2b), - e{b - l){be + e - 2b). 
Eecalling again the fact that none of the factors here visible can be 
allowed to vanish, we see that the equality of the first three products is 
tantamount to 
d + e = 2b(b + 1)1 (b- -\-l) = b- + b, 
and that therefore the said products can only be properly equal when 
b = -1, - d. 
As, however, their common value is then —4:d, and as the 4th and 5th 
products have this value also, our final result is that 
1 
1 
1 
1 
1 
1 
-1 
-1 
1 
-1 
d 
-d 
1 
- 1 
-d 
d 
is inversely orthogonal whatever d may be, the product of any element by its 
