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

418 HISTORY OF THE THEORY OF DETERMINANTS

in Stern's Theorie der Kettenbruiche, the fourth section of which
is given up to the consideration of such rules (Crelle's Journ.,
x. pp. 4-7).
The other observation is to the effect that
"every progression of terms constructed in conformity with the
equation,a == a,_X - b,,_ -2 + cZ_ -...
may be represented as an ascending series of principal coaxal determinants to a common matrix. Thus if each term in such progression is to
be made a linear function of the three preceding terms, it will be
representable by means of the matrix
A B' C" 0 0
1 A' B" C"' 0
0 1 A" B"' C""
0 0 1 A"' B""
o o 0 1 A""
indefinitely continued, which gives the terms
1, A, AA'- B', AA'A"- B'A"- AB"+ C",...."
This exhausts the paper so far as determinants are concerned:
the results announced in it, one can readily own, were such as
fairly to entitle the enthusiastic author to express his belief
that "the introduction of the method of determinants into the
algorithm of continued fractions cannot fail to have an important
bearing upon the future treatment and development of the theory
of numbers."
SPOTTISWOODE, W. (1853, August).*
[Elementary theorems relating to determinants. Second edition,
rewritten and much enlarged by the author. Crelle's Journ.,
li. (1856), pp. 209-271, 328-381.]
Save the utilisation of the fact that the denominator of any
convergent of the continued fraction
bi b
a<2+
a- 3 l + )-..
i"This is the author's date at the end of the paper (p. 381). The first two
parts of the volume, however, are dated 1855, and the remaining two 1856.