1901- .] Dr Muir on the Theory of Orthogonants. 267 
that is, 
I a 2 C 2,^4: I ’ I ^1 6 *3^4 I = “ I a 1^2 C 3^4 I ' i C 3^4 I 
and this, when the original determinant is axisymmetric, becomes 
I ^l C 3^4 I 2 “ “| a f-2 C ^i i * I C 3^4 ! ’ 
or, as Cauchy writes it, 
-Y 2 = SQ. 
The first four cases of S = 0 are then considered, viz., the series- 
of equations S^O, S 2 = 0, S 3 = 0, S 4 = 0, or, as] at a 
later date they would have been written, 
A uu - i 
>“ = o, 
A zz - 
S 
A 
zu 
= °, 
Ku 
A; ^ ~ S 
Ayy ~ 8 
A^ 
Ayu 
A y Z 
Kz- 
s 
A zu 
= 0, 
Ayu 
A zu 
A uu - s 
A»-s 
^xy 
Kz 
Ku 
A X y 
Ayy - S 
A yz 
Ayu 
= 0, 
A „ 
K 
Kz- 
s 
A zu 
A xu 
A yu 
Ku 
A mm — s 
where each determinant is the complementary minor of the element 
in the place (1, 1) of the next determinant. The root in the first 
case is evidently A uu . In the second case the solution is 
s = i{ + A uu + V ( A zz - A uu ) 2 + (2A ZU ) 2 f, 
where the reality of the roots , s 2 is manifest ; and as their sum 
is A zz + A uu , it follows that s 2 - A uu may he substituted for A zz .-s 1 
in 
with the result that we have 
(A mm - s 2 )(A mm = — A z f 
and are able to conclude that the roots s 1 , s 2 of the equation S 2 = 0 
lie on opposite sides of the root A uu of the equation S x = 0 . 
* We know from a later theorem (Jacobi, 1833) that when A x is~not 0 the 
identity is 
AiB 2 | = | I • I c 3 d 4 | . 
