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

AXISYMMETRIC DETERMINANTS (HESSE, 1872)

105

respectively, and similarly [ay, /S] in. the case where a two-line
border has been affixed.
The connecting relation
I ul I. [ap, y] = [a, 7] -[, ']-[ a, 1] - [3, y]
comes to his hand at once, on account of the member on the right
being viewable as a two-line minor of the adjugate of the second
factor on the left. In like manner, or by the mere interchange of
letters, he obtains
I 1 i [aCy, P] = [, /3] [y, ] - [a, ] [y, /3],
I ln I. * [[ y/3] = [a, ] [, /3] - [a, 3] [8, y].
Then, positing the axisymmetry of I u, he notes that
[3, d]= [3,....
and that therefore the first of the three right-hand members
is exactly the sum of the two others. There thus follows the
theorem
[a/3, y1] = [ay, P6] + [a, y/3],
or, as we may prefer to write it,
[ak3, y8] - [ay, 8] + [Ca, /3y] = 0.
A second theorem which he gives is easily remembered from this,
namely, that if we annex to the three terms on the left the outuwardly
resembling multipliers
[a, /3] [y, 8], [ay]'[/, ], [a, 8]. [/, y]
respectively, the right-hand side need not be changed. This results
from using the said multipliers on the three original equations, and
then reasoning as before.
For ourselves, we may note that the existence of the two equations side by side is a consequence of the general fact that if we have
X = a-b, -Y = b-c, Z = c-a,
it follows that we have both

X-Y+Z = 0 and cX-aY+bZ = 0.