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

298 HISTORY OF THE THEORY OF DETERMINANTS

and thus is able to assert that P2 and Ai are element-by-element
identical. *
Proceeding to a second differentiation, he readily obtains
ctr1Ci U + Cr,2 cs2 +~ + (trn ( cn't:' — tr
where t4. stands for the excess of A.. over the (rs)t13 element of P2,
and thence deduces (~ 3) a quite analogous series of results.
The paper concludes with a statement of the form which p,., takes
when the orthogonant employed is Cayley's of 1846.
MANSION, P. (1877).
[On a pair Qf algebraical equations. Messenger of Math., (2) vii.
pp. 57-58.1
The~equations are
al-x ca2 a3 21-$ A2 A3
b, b2-x b3 =0 B1 B2-$ B =3 0:
c c2 c3-x C1 C2 C3 —
* Here again, however, there is a more general theorem, quite unconnected with
the differentiation of the elements of an orthogonant. For we know (Hist., i.
p. 450, footnote) that if, when I aU3y2 j is an orthogonant, there be given
( a, a2 a3 kXI, 'y, Z1) Xi, Y1, Z1,
P1 P2 P3
71 32 73
( a, a2 a3 @3,X Y32 Z2) = X2, Y21, z2,
P1 P2 P3
71 72 73
then (x1, Yi, ZMX21 Y2 z2)2 (XI, Y1, Z2 X2, Y.', 2.2).
Consequently, if we were given another triad of equations, namely,
( al a2 a3 MX3, y, Z3) X3, Y;', ZV,
P1 P2 P3
71 72 73
it would follow that
x1 y z1 1 2 Xi Y] Z1 2
x2 Y2 z2 X2 Y2 Z2
x3 Y3 Z3 X. Y3 Z 3
and that, if the squaring in both eases were done by row-by-row multiplication,
the resulting determinants would be identical, element-by-element.