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

426 HISTORY OF THE THEORY OF DETERMINANTS

la loi de formation est evidente; effectuant, on trouve
A,X2-l, X,(X2-22), (X2-12)(X2_32), X(A2-22)(X2-42),
(X2 - 2)(X2 - 32)(X2 - 52), X(2 - 2)(X2 - 42)(x2 - 62),
et ainsi de suite."
That Sylvester was the author of the implied theorem may be
considered proved by an entry in the index of the volume (See
p. 478), and by a statement of Cayley's in the Quarterly Journal
of Matheqzatics, ii., p. 163. Probably the title of the communication was prefixed by the editors, who, knowing of Sylvester's
papers in the Philosophical Magazine, felt themselves justified
in applying the name " Sylvester's determinants."
SCHLAFLI, L. (Nov. 1855).
[Reduction d'une integrale multiple qui comprend l'are de
cercle et l'aire du triangle spherique comme cas particuliers.
Journ. (de Liouville) de Math., xx. pp. 359-394.]
Here there appears the equation
a(,/B,*. * *, ) = 1 - COs2a cos2/3
A(3,..., ',1) 1 1 - -
cos2C
1 - Cos2'
where, in view of the contents of a subsequent paper (see under
year 1858), it would seem that A(a,3...,, ) was used for
1 cos a
- cos 1 cos/
- cos/ 1
- cos t 1 cos 71
- cos I 1

No properties, however, of this determinant are given.