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

344 HISTORY OF THE THEORY OF DETERMINANTS
(3) from (1) 1 1
(4) the successive convergents...... to this
q1 qq2.t'o this
continued fraction being N1/D1, N2/D.....
N, = q2 1... D, = q 1.
1 q 1.... 1 q 1..... 1 q,...... 1 q3
~.........
(5) from (1) after eliminating 2, f, g..., fr-l by repeated
substitution or otherwise
f/. = De,_fi - AX_lf;
(6) from (1) after eliminating f, f2,., f-2
f = Dr/lr -D,,-2fr;
With these facts before him he seeks to find expressions for fr,
D,, N,., or, say, for the coefficients in
A,.lxn -r + A,.,2n —1 -...,
B,.,lxr + B,.,2zxr- +....
C,,lxr-l + C,.,2r-2 +.
failing to note that, Cayley having in 1846 found such an
expression for Sylvester's substitute for f., the annexure of a
known multiplier to Cayley's result would have given him the
most important of the three expressions sought.
In the first place he deduces from (4) that the coefficient of the
highest power of x in D,, - is always - of the coefficient of
I,. and from
the highest power of x in N,-1, because q= +; and fro
(6), by equating coefficients of xn, that the coefficient of the
highest power of x in f,- is the reciprocal of the highest power
of x in Dr-i: in other words, that
C,1 = nBr,i = n/Ar,,.