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

ORTHOGONANTS (SIACCI, 1872)

293

71
7i1

a2
32 +X2
72

73

1 + a(Xl
= ao X2
a(3X3

/~lXl YiXl
1+ /32X2 y2X2
/3X3 1 + 73X3

a3 X
+ X3
- x1X2X3

a1 a2
R 1
a2
a3S

a3
/33
73

/:

1
~2 + - '/?
X2
/3 73 +
x3

and taking the fourth, we have

1
71
61,C(
3'i
-a
-a
a2
a3
14

/32-1 3 34 X
72 73 -1 74
62 63 64- 1I
1 /1 7i 61
2 -32 - Y2 -82
3 -/33 1-73 -63
4 -4 -74 1 --1 13, 7 61
2- 1 72 62
/33 73-1 83
34 74 64-1

O1 13
71 y
81 2

2 3 a4
2 /33 34
2 73 74
64 84

/33
73-1
83

34
74
84,-1

The second manifestly follows at once from the third by putting all
the h's equal to 1 and all the k's equal to -1. As for the third
itself, we first multiply by one of the orthogonants involved, the
result being
h2/1 + 'k2f1 h2/32 + 12Y2 h2033 + 7123 i31 2 33
h371 + kC31 h 3,2 + /32 h373 + k3V3 7I1 72 Y3
hi + nk1l\a k1\X3 k71>Xy
= 7c2xa h + 7k2Z/L3 k24 7Y
kTsira ksLvr/ h^-{-C Svy
zva ~ 703 3EV3 h3 + k73',V
and then we note the difference in the product which multiplication
by the other orthogonant I Xu23 I would have brought about.