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

ORTHOGONANTS (VOSS, 1877)

301

The existence of equal roots in the equation L(p) = 0 and the
evanescence of systems of minors of L(p) are next considered, the
k-line minors being disentangled, so to say, and drawn out in series
by bordering L(p) by k rows and k columns with a common minor
of k2 zeros; for example, if n be 5 and k be 2, the determinant
formed is
a11-p a12 a13 a14 a15 bnl b12
a2 a22-p a23 a24 a25 b21 b22
a31 a32 a33-p a34 a35 b31 b32
a41 a42 a43 a44-p a45 b41 b42
a51 a52 a53 a54 a55 —p b51 b52
Cll C1 C13 C14 C5
011 1x2 13 14 015
21 c22 C23 C24 25
this being expressible as an aggregate of 102 terms each composed
of three factors,-a two-line minor of the b's, a two-line minor of
the c's, and a three-line minor of the original.* Multiplication by
I a, | is then performed, with a result analogous to that obtained
when the unbordered L(p) was multiplied by I a,,, the product
being of the same form as the multiplicand and s/p taking the place
of p. A theorem is thence readily deduced, and several minor
results follow.
IGEL, B. (1877).
[Ueber die orthogonalen und einige ihren verwandte Substitutionen.
Denksch. d. k. Acad. d. Wiss. (Wien), xxxix. pp. 29-40.]
Instead of beginning as usual with
x = XI1+X2+ A3}
y = mC+g.+g2 \. ex+yL+Z2 = 2 + 2+62+
z = v1+ '21 + v"3
* Such an aggregate can be most appropriately viewed as a bilinear form; thus,
denoting the 3rd compound of the original by I AlA22... Ao,lo I, the determinants
of the b-array by -B, B2,..., B and the determinants of the c-array by
C C 2, C.., C0O, we have the interesting alternative expression
B10 B,.... B1
All A12.... Ai,lo Co1
A21 A22.... A2, 10 C5
Alo,l A1o,2.... Aio,jo Cl.