1903-4.] Dr Muir on the Theory of Continuants. 
137 
The next “corollary,” viz., 
(^1 J • • • J a P + 1 5 • • • } a p+f)( a 1 5 • • • 5 a p ) a p + 1 5 • * • 5 ^p+ifc) 
(^'l , . . . , a p , a p ^ , ... , <^p+gr)(<7i , . . . , a p , a p+ j , ... , 
( “ ) p { (^p+l 5 • • • j a p+f)( a p+ 1) • • • > a P+Jc) — ( a p+ 15 • • ‘ > a p+g)( a p+ 1) • • • J ^P+fc 
is clearly incorrect, it being impossible for the value of the left- 
hand side to be independent of the elements a v a 2 , . . . , a ? . 
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 
Annates de Math., v. (Sept. 1851) pp. 357-358, to the effect that 
if T, z+1 == aT n - be the generating equation of a recurrent 
series, then 
T n+\ - flTVT n+i + 5T W 
b n 
is a constant with respect to n. Of course the more natural form 
of this expression is 
Tn+l — T w T, +2 . 
b n 
the numerator of which being 
is successively transformable by means of the recursion-formula 
into 
b 
l T 
*-n 
T„+i 
52 
T n 
b 3 
T 
x n - 2 
T 
x n - 1 
T 
1 x n- 1 
T„ 
, ! t „_ 2 
T 
J -n—i 
} 
T 
x n- 3 
T 
x n - 2 
so that the constant in question is 
T, T 2 
T 0 T, 
This, however, Sylvester does not show.* 
* An interesting extension of this is given by Brioschi in the Nouvelles 
Annates de Math. : xiv. (Jan. 1854) p. 20. f 
