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

404 HISTORY OF THE THEORY OF DETERMINANTS

NACHREINER, V. (1872).
[Beziehungen zwischen Determinanten und Kettenbriichen. (Diss.)
24 pp. Minchen.]
The starting-point here is Spottiswoode's short statement given
on p. 374 of the 51st volume of Crelle's Journal. This having been
quoted, pains are then taken to establish a so-called 'generalization,' viz.
b+__ b+2 - log, V,
ar+ar+2+,. bn
where an
V a, b1..
-1 ar+1 )r+2..
* -1 2ar+2 *.... an — )n...... *. — 1 b
but unfortunately the left-hand side of the identity is not given
as here, aS + r+l being incorrectly put * instead of its
+reciprocal.
reciprocal.
The identity
a. bl..... = al 1...
C1 a2 b2.... bc1 a2 1
c 2 a3.... 2 a3....
is next demonstrated, the procedure involving the division of the
2nd, 3rd, 4th... columns by b1, b1b2, bb2b3,.... respectively,
and thereafter the multiplication of the corresponding rows by the
same. This is used to deduce some variants of the above ' generalization,' and then the results thus far obtained are employed in
dealing with known properties of continued-fractions.
* This oversight runs through sect. i., and should be corrected in the results
marked (3), (4), (6), (7), (8).