1903 - 4 .] Dr Muir on the Theory of Continuants. 
141 
Sylvester, J. J. (1854, August). 
[Theoreme sur les determinants de M. Sylvester. Nouv. 
Annates de Math ., xiii. p. 305.] 
This communication in its entirety is as follows : — 
“Soient les determinants 
X, XI X 1 0 X 1 0 0 
IX, 2X2 3X20 
0 1 X, 0 2X3 
0 0 1 X, 
X 1 0 0 0 
4X200 
0 3X30 
0 0 2 X4 
0 0 0 1 X, 
la loi de formation est evidente ; effectuant, on trouve 
X, X 2 -l, X(X 2 - 2 2 ) , (X 2 - 1 2 )(X 2 - 3 2 ) , X(X 2 - 2 2 )(X 2 - 4 2 ) , 
(X 2 - 1 2 )(X 2 - 3 2 )(X 2 - 5 2 ) , X(X 2 - 2 2 )(X 2 - 4 2 )(X 2 - 6 2 ) , 
et ainsi de suite.” 
That Sylvester was the author of the implied theorem may he 
considered proved by an entry in the index of the volume (v. p. 
478), and by a statement of Cayley’s in the Quarterly Journal of 
Mathematics , 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.” 
[Reduction d’une integrate multiple qui comprend l’arc de 
cercle et l’aire du triangle spherique comme cas particuliers. 
Journ. de Liouville , xx. pp. 359-394.] 
Here there appears the equation 
Schlafli, L. (Nov. 1855). 
A (a, A • • ■ 1 L v) 
A (ft ... , i, V ) 
COS 2 a 
COS 2 /? 
1 — 
cos 2 £ 
1 - cos 2 ?7 
1 
