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

CONTINUANTS (SYLVESTER, 1853)

421

The next "corollary," namely,
(a1,..., ap, a, a+, f)(l,..., ap, p+lI,..., ap+k)
- (a,,..., ap, p+,.., * p+a )(al,... p, Cp l,., ap+7.
(-)P(p,* * *..., ap+f)(ap+,, aCp+k) - (aIp+l, * * *, ap+g)(ap+i,,., ap+)}
is clearly incorrect, it being impossible for the value of the lefthand side to be independent of the elements ca, a,..., ap.
Further, as the author gives no accompanying word of comment,
the difficulty of suggesting the true theorem is increased. A
"sub-corollary " is appended dealing with the case where all the
a's are equal, and leading up, not without some misprints or
inaccuracies, to a theorem of Euler's quoted from the Nouvelles
Annales de Math., v. (Sept. 1851), pp. 357-358, to the effect that
if T,+1 aT T- bTn_ be the generating equation of a recurrent
series, then
-c. a T TT,0,T,,+ + bT,
is a constant with respect to n. Of course the more natural form
of this expression is
Tn+1 -TnTn+2
the numerator of which being
Tn+1 TjT+2
Tn Tn+i
is successively transformable by means of the recursion-formula
into
bJT) T T b)2 T T V Tn -
b T T ~n+l b2T1 Tn-1 b 2T T T2 nTn-l T n Tn-2 Tn-i Tn-3 Tn,2,...
so that the constant in question is
T, T2
FTo T1 ~
This, however, Sylvester does not show.*
*An interesting extension of this is given by Brioschi in the Xouv. Annales
de Math., xiv. (Jan. 1854), p. 20.