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

ORTHOGONANTS (SPOTTISWOODE, 1851) 307
SPOTTISWOODE, W. (1851).
[ELEMENTARY THEOREMS RELATING TO DETERMINANTS.....
viii + 63 pp., London.]
Following Cayley, Spottiswoode places the construction of an
orthogonal substitution at the opening of his section (~ 9) on skew
determinants. The mode of treatment differs from Cayley's in
being verificatory rather than investigative. Starting with the
two sets of equations
11101 + [1202 + A1303 + 1404 = X
and
11101 + 12102 + 13103 + [4104-
he does not seek to ascertain therefrom the linear substitutions
connecting the x's with the $'s, but bringing forward the coefficients of these substitutions as found by Cayley, namely,
2L_1 2L1 2L13 2L1
A 1 A A L
he affirms that by using as multipliers, along with the first set of
equations, the elements of the various columns of this array in
succession we shall have
(2L,1 + 2L9. 2L1 2L,,
(2Lll_)X + X2 + X3 + X4 el,
and that by using along with the second set of equations the
elements of the various rows of the array in succession we shall
have
(2LII)1 + 2L 2L + 2L1e )
At the close of a preceding section (~ 6) he devotes three pages
(pp. 35-37) to an investigation of the conditions under which