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

ORTHOGONANTS (HERMITE, 1853)

313

There is thus obtained
f(x:,L,...) = f(201-,e, 202-2,...),
=4f(0, 2,.../ - 2( + 2+.. + 2..
so that in order to have f(xl, x,... )=f(i,,.. ) it is seen to
be necessary that
Zf v/
~+ +'"* = 2/(0, 2,... ).
Now this condition is manifestly satisfied by putting. = 0,, but
"la maniere la plus generale de la verifier en exprimant les
quantites C en 0 sera de faire
= or + i ers,
s=l -0s5
les indeterminees X etant assujetees a la condition h,, = -Xs..'
This of course implies that
Xd2r = 0r.X1;
r - 2 XE /r s aEos
and there have thus been obtained in their general form the
two sets of equations with which Cayley started in his special
case.
For those who may wish to pursue the subject of "automorphic transformation" farther than these papers of Hermite's
we may note that the actual expression of the x's in terms of
the C's was given by Cayley in a paper dated 24th May 1854,*
and that he extended his result to a bipartite quadric function
in a paper dated 10th December, 1857.t
Another problem, which in the early history of orthogonants
we have seen to be of interest, namely, the simultaneous trans* Cayley, A., " Sur la transformation d'une fonction quadratique en elle-m~me
par des substitutions lineaires," Crelle's Journ., 1. pp. 288-299; or Collected Math.
Papers, ii. pp. 192-201. See also Brioschi in Annali di Sci. mat. e fis., v.
pp. 201-206.
t Cayley, A., "A Memoir on the Automorphic Linear Transformation of a
Bipartite Quadric Function," Philos. Transac. 1. Soc. (London), cxlviii. pp. 39-46
or Collected Math. Papers, ii. pp. 497-505.