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

CONTINUANTS (SCOTT, 1878)

421

which by means of its difference-equation is once more found to be
equal to
fc2 + /- b3il _ (C2 -_ c2 4ab}Ln+l
2n+1 Vc2 - 4ab
Five special cases, more or less known, are mentioned, the last
being that in which c2 = 4ab and the value of the determinant
(ab) (n +1).
In the same way he finds (p. 182) that the determinant formed
from the above by prefixing 0, 1, 1,..., 1 as a row and as a column
is equal to
tn+l- + vn+l (n +1) C(Un'+l +- vn+l)
(a+b+c)2 (a+b+c)(u-v)2
2ab(n+1) u +v _ (_ a)n+1 + (- b)n+l
a+b+c (u —v)2 (a+b+c)2 '
also, that the determinant formed from this by deleting the first
column and the last row is equal to
a4 _u- n+l Vn+l+ b(tn - 1vn)
+ (- 1)n
a+b+c (a+b+c)(u-v)
where u, v are the roots of the equation
x2 -cx+ab==0.
SYLVESTER, J. J. (1879).
[Notes on continuants. Messenger of Math., viii. pp. 187-189.]
Here the main point of interest is the use of the old 'rule'
referred to in his paper of May 1853 to provide a proof that the
number of terms in the simple continuant (a,, a,..., a,) is
+ (, + (n - 2)(n - 3)+ ( - 3)(n-4)(n-5)+
i+(n —+1 +*2.3
1.12 i - -.2.3
the various parts of this expression being shown to correspond with
the various kinds of terms obtained in following the 'rule,' viz.,
1 term containing all the elements, n-1 terms containing n -2
elements, (n —2) (nz-3) terms containing n —4 elements, and