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

136 HISTORY OF THE THEORY OF DETERMINANTS

Following Cayley's paper still further, Brioschi similarly
makes clear that one of the nine-line determinants there
obtained, namely.... '.... 1 1 1.. 1.. 1. s.. '. 1.1. 1... 1... 1.. yJ3. 1. 1...:
1... 1....a
~.. z3 3 3...
1 1 1.......
may be viewed as originating in any one of the nine equations
x+ y+ z=, x + y + az = O, x + +/3z + = 0,
x + ay + z = 0, x+ay+ z = 0, x+ 3y+ z = 0,
x +,y + az = O, a + y + z = 0, 3x + y + Z = 0,
where a, /3 are the imaginary cube roots of unity, and could thus
be shown to be equal to the product of the nine left-hand
members of those equations.
SPOTTISWOODE (1851, 1853).
[Elementary theorems relating to determinants. Second edition,
rewritten and much enlarged by the author. CGelle's
Journal, li. pp. 209-271, 328-381.]
The information given by Spottiswoode regarding axisymmetric determinants appears under a variety of headings. What
little the first edition contained (pp. 33-34) as a part of ~ vi.
on "Inverse Systems" is placed in the second edition under
"Compound Determinants" (pp. 368-372). Sylvester's mode of
reaching Cayley's determinants connected with the mutual
distances of points is given under " Multiplication" (pp. 250-53);
and the chapter or " section " (~ iv.) on " Homogeneous Functions,"
which, of course has to deal with quadrics, goes so far as to