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

ORTHOGONANTS (VELTMANN, 1871)

291

the intermediary variables %1, 011 3,..., layingc down the
proposition that x1, X2, x31-., and e, 2, 3...are
orthogonally related if
111xI+ 112X2 + 113.3..3.= t11e1+21C2+1313+...
121x + 122X2 +193X 3+1. l2CI +l22e2 + 326 +.
131x1 + 132x2 + 133X3 +.. l3& ~ l93e2 +13e"
where the determinant of the I's on the left is meant to be unitaxial skew, and is manifestly the conjugate of that on the right.
To establish this, he has only got to solve for each variable of the
one set in terms of the variables of the other. Thus, from the
equations
X1~ V X2/J.~X3 = e i1 el:~, + U -
VX1 + X2~ XX3 = 4le+ e92-X1C {
LXI XX2 + X3 = IA X1 + 6)
there is obtained in the usual way
~-Y2$-~SQ V 1~U i — ~
x1 v$1+ 2 -~C 1 I ) X I + - 1
i~x2- AA 2-V-el + 2 (Xju - ) + 2 +A + At)
I + X2,U2 + V22 I X2X +U2+,)262 I + > 2 + IA'2 _t V
and similar expressions for x2 and x3.
We may also note that from the constitution of the unsolved
set of equations the changing of the signs of X, /A, v in the coefficients here obtained must give us the coefficients in the expressions
for e in terms of x1, xI, X3.
SIACCI, F. (1872).
[Questioni 1, 2, 3, 11. Giornale di Mat., x. p. 188, p. 360. Solution of 3 by P. Cassani, x. pp. 239-240; Solution of 1, 2, 3 by
M. Albeggiani, x. pp. 285-290.]
The first three 'questions,' sent to the Giornale on 17th May, are
special cases of a general theorem which is the subject of a