1917-18.] Note on the Construction of an Orthogonant. 149 
determinant could be a multiple of a 2 + 6 2 + c 2 while its adjugate not only 
vanished but had almost the highest possible index of nullity. 
In three points it differs from Cayley’s, namely, (1) in form it is 
axisymmetric, (2) the number of its arbitrary parameters is n instead of 
\n(n — 1), and (3) its value is +1 when n is odd and — 1 when n is even. 
(7) The determinant 
x 1 — 1 Xc, x 3 .... 
Vi y 2 ~ l Vs 
will be an orthogonant , if the square of | x^Zg ... | be equal elementally 
to its duplicant ; that is to say, if 
s • • • 
X-^ -f~ X 
yi + %2 
Z 1 +X 3 
X 2 + Vl X 3 + : -l • • • • 
y 2 + y 2 y-3+ z 2 •••• 
z 2 + y 3 z 3 + z 3 
> 
where the matrix of the determinant on the right is the sum of the matrices 
of | x 1 y 2 z 3 ... | and its conjugate. As before, the proof is merely a straight- 
forward application of the definition. 
(8) If we use the generalised definition of an orthogonant, namely, that 
in which the square of each row is /ud instead of 1, the necessary modifica- 
tion in the preceding is the substitution of ju for 1 in the diagonal elements 
of the asserted orthogonant and the affixing of ptasa multiplier to every 
element of the duplicant. 
(9) That our orthogonant of § 6 satisfies the condition just laid down is 
easily verified ; and that Cayley’s does so is a fact that is to be found in 
Spottiswoode, although we can scarcely say that its discoverer’s way of 
recording it has helped much to make it known. 
We shall now use the condition to test the validity of a third solution. 
(10) Strange to say, this third is connected with the same source as 
Cayley’s, namely, the adjugate of a unit-axial skew determinant. To the 
setting forth of it, however, it is necessary to partition the matrix of 
the said adjugate in accordance with the several degrees of the terms of 
its elements : for example, the unit-axial skew determinant of the 4 th 
order being 
1 
a 
P 
i 
— a 
1 
7 
€ 
-P 
- 7 
1 
8 
-t 
— € 
-8 
1 
