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

PERSYMMETRIC DETERMINANTS (SYLVESTER, 1851) 331
where x1, 2,... are the roots, and therefore by
18, So, so s1 S81 82
81 82 1- 82 83
2 83 84....
where s,. = x + +... + x. All this, however, is practically
implied in Cayley's paper of 1846 (August).*
SYLVESTER, J. J. (1851, May).
[EssAY ON CANONICAL FORMS: Supplement to a "Sketch of a
Memoir pn Elimination, Transformation, and Canonical
Forms," 36 pp., London. Or Collected Math. Papers, i.
pp. 203-216.]
In giving a preliminary notice of his general method for
reducing odd-degreed functions to their canonical form, Sylvester
says he based his method on the proposition that every one of
the n-line minor determinants of the array
T1 T2 T3...
T2 T3 T4... Tn+2
T3 T4 T5... Tn+3
T,, T+ Tn+2... 2n
vanishes if
T r-irsC l, y-i7]sfL+i r'-i ~s+i
T = a -ibl + 2 cb +. +.. cn-1 1-1.
This, which he hastily calls "a beautiful and striking theorem,"
and which he generalises in Note B of an Appendix, arises from
the simple fact that each determinant is the product of two
zeros, Tj being
~-i ) r-i ~ - 1 ~'~ -j s1. '
^? -^2,-. a.nt. E -, < b 0+.,b2 b )n- ~)
It is of more importance, therefore, to recall that it was in
* The proposition Borchardt is concerned with is of course that The equation
f (x)=0 has as many pairs of imaginary roots as there are changes of sign in any
one of the three series mentioned.