The theory of determinants in the historical order of development, by Sir Thomas Muir.

264 HISTORY OF THE THEORY OF DETERMINANTS

"The comparison of these three systems gives either
A = 0 1
* [12]=]21]... [ln]=[?l]
[21]=[12] *.. [2n]= [n2]
[nl] =[l [n2] [2] [2n]... *
or
[11] =0 [12] + [21] = 0. l.. [In ] 0
[21]+[12]= o [22] =... [2 ]+ [n2]=0] o
[nIl] + [I] 0 [2] + [2] =0... [lnnt] - 0
and consequently either a symmetrical skew determinant of an even
order or a determinant of an odd order vanishes."
What the first half of the sentence asserts to be proved is the
proposition that If A be a zero-axial skew determinant, then
either
(1) A= 0 and [rs] =[sr],
or (2) [rr] = and [s] = - [sr].
In this there is evidently included the assertion that A zero-axial
skew determinant either vanishes itself, or all its principal
coaxial minors vanish: but what Spottiswoode finds in it is the
much wider assertion that Either all even-ordered or all oddordered zero-axial skew determinants vanish. If however his
accuracy be granted up to this point, there is little objection to
the cogency of the next step in the reasoning, which is worded
as follows:
"But since it is found on trial that for n =1, 3,..., A vanishes,
while for n = 2, 4,..., it does not, the following theorems may be
enunciated:"Theorem XIV. A symmetrical skew determinant of an odd order in
general vanishes, and the system has for its inverse an unsymmetrical skew
system.
"Theorem XV. A symmetrical skew determinant of an even order does
not in general vanish, but the system has for its inverse a symmeetrical skewz
system?."
The only difficulty to be raised is in regard to the name given
to the "inverse system " in the first case. " Unsymmetric skew "
is clearly inappropriate when, as we have seen, [rs] = [sr]; and