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

306 HISTORY OF THE THEORY OF DETERMINANTS

and this being known to be equal to
(n + 1)+l1- n
the desired result is reached.
In the case where n = 2, 3, 4, Borsch is able to give to the
elements values which satisfy the conditioning equations. These
values are as seen in the determinants
1. 1 1.. 1 1 1.
iJ 2 13- i 12 3 4 4. --
-/3-, 1 -i-2 >/5 _- 1 -{l >5 -4-5 -.1 -,/ -2- s/6, 1 -I-5 5 IL 5 -
1 ds — 5 -I5 -
the squares of which are
27 256 3125
4' 27' 256
He might also have pointed out that in his procedure is implied
the theorem that if the elements of an n-line determinant be such
that the square of every row is equal to a, and the product of any two
different rows is equal to b, the value of the determinant is
a(a-b)-(a+ nb)A result known otherwise (see under Roberts 1864, Sardi 1868, etc.)
would thus have been brought into touch with orthogonants.
BIBLIOGRAPHICAL NOTE.
It is probable that the writings dealt with in the foregoing pages
are practically all belonging to the period 1860-1880 that are
strictly relevant to the theory of orthogonants. The inclusion of
papers on the wider subject of orthogonal transformation, though
justified at an earlier date when orthogonants had scarcely a separate
existence, would evidently be improper now unless the papers
involved some new property of the determinant in question. For
the convenience, however, of students of this branch of linear
substitution, the following titles are added: