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

CONTINUANTS (SYLVESTER, 1853)

415

nants of a symmetrical matrix. Thus suppose n =5; if we write down
the matrix
A1 1 0 0 0
1 A2 1 0 0
0 1 A3 1 0
0 0 1 A4 1
0 0 0 1 A5
(the mode of formation of which is self-apparent), these successive
coaxal determinants will be
1, i A 1 A 1 0, 1 0 0, etc.,
I A2 1 A2 1 1 A2 1 0
01 A 0 1 A3 1
0 0 1 A4
i.e.
1, A1, AA2 -1, AA2A3-Al-A A,
A1A2A3A4 - A1A2 - A1A4 - A3A4 + 1,
A1A2A3A4A5 - A1A2A5 -AA A- AA4A, - A1A2A3
+ A5 + A3 + Al.
It is proper to introduce the unit because it is, in fact, the value of
a determinant of zero places, as I have observed elsewhere."
After using this as an aid to prove his proposition regarding
Sturm's theorem, he returns to his new determinant in the
following words:"I may conclude with noticing that the determinative [determinantal? ] form of exhibiting the successive convergents to an improper
continued fraction affords an instantaneous demonstration of the
equation which connects any two consecutive such convergents as
N_, and N,
Di, Di'
namely, N-Di -Ni_iDi = 1.
For if we construct the matrix which for greater simplicity I limit to
five lines and columns,
A 1 0 0 0
1 B 1 0 0
01100
O 1 C 1 0
0 0 1 D 1
0 0 0 1 E