148 
Proceedings of the Royal Society of Edinburgh. [Sess. 
are binomial with — 1 in each case for one of the terms : in other words, 
that we may advantageously conceive Cayley putting his problem to 
himself in the form of a question as to what the elements of | x 1 y 2 z 3 . . . | 
must be in order that 
x x — 1 
Vi 
X.j 
y 2 ~ 1 
X 3 • • • • 
Vs 
* 8-1 •••* 
may be an orthogonant. And if we do this we are prepared to view his 
answer as a direction to take for the elements of | x 1 y 2 z 3 . . . | the doubled 
elements of the reciprocal * of any unit-axial skew determinant. 
For the present we confine ourselves to showing that this is not the 
only solution to the problem even in the specialised form here given. As 
for other specialised forms it must suffice to note merely that they do 
exist, for example, the skew orthogonant 
<rq 
Cl. 2 
«4 
(T 
cr 
cr 
cr 
a 2 
«4 
_a 3 
cr 
cr 
cr 
cr 
_a 3 
_«4 
a l 
« 2 
cr 
cr 
cr 
a 
_«4 
a c 
o 
a 0 
a Y 
O’ 
cr 
cr 
or 
w here cr“ = a\ F a s F F ci% 
(6) The determinant 
X ± — 1 
x 2 
x 3 
. . . 
Vi 
2/2— 1 
Vz 
• • • 
z i 
F 
*8-! 
* • • 
will be an orthogonant , if the (r,s) th element of | x x y 2 z 3 ... | be taken equal 
to 2a'a s /2a 2 . For example 
- -1 
ab 
ac 
ha 
2F 
cr 
he 
ca 
ch 
1 
cq ( 
Cd 1 
C T 
where cr = a 2 F 6 2 F c 1 . The verification of this is perfectly simple and 
need not be given. It is more important to note that the result is not 
included in Cayley’s : to suppose so would imply that a unit-axial skew 
* Not the cidjugate. 
