328 
Transactions of the Boijal Society of South Africa. 
different, the equality of the elements of the first column is tantamount to 
the equality of z^, y^, x^, iv^ — a result not different from that obtained in 
dealing with the rows. Our conclusion thus is that the alternant 
I z°y^x2w3 ... I 0/ the nth order luill he inversely orthogonal if z, y, x, w, ... he 
the roots of the equation w'^ = a. 
10. Since a determinant that is inversely orthogonal (an ant-ortho- 
gonant say) continues to be so when the elements of any row or column 
are all multiplied by the same quantity, we may without loss of generality 
make a = l, z, y, x, ... then becoming the ?^th roots of unity. Further, by 
taking c to be a primitive nih. root of unity z, y, x, ... then become 
c, c^, g^, and we see that the ant-orthogonant thus reached may be 
written 
Q 
^271-2 
g' 
^3n— 3 
1 
1 
1 
.. i 
By passing the last row over all the others the result becomes axisym- 
metric, and is then identical with that obtained by using Sylvester's 
"rule." 
Since, in addition, the elements are all unimodular, the determinant 
reached is also, by a result of § 6, an instance of a maximum determinant. 
11. If jj be Sylvester's ant-orthogonant of the nth order, it is readily 
found by determinant multiplication that 
And as we already know that 
=zn^, 
it follows that 
^ = ^'.(-l)^^^-'^'"-^>. 
It may also be worth noting that the complementary minor of the first 
element is symmetric with respect to both diagonals. 
12. If fi^., fjig be maximum determinants of the ?"th and sth orders 
respectively, and /x„ the maximum determiaant of the (rs)th order formed 
according to Sylvester's second rule, then 
13. It would, of course, be unwise to conclude without further investi- 
gation that the determinant reached in § 9 is the only w-line orthogonant. 
As an illustration, let us inquire whether the axisymmetric determinant 
1111 
1 a b c 
1 b d e 
1 c e f 
