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

364 HISTORY OF THE THEORY OF DETERMINANTS
Following the ordinary division-process -for expressing the ratio
of the second function to the first as a continued fraction of the
form
Po
1+T~P2 3
1+.
and denoting the remainders in order by
l{2Ca~-1 + CX -2 + C22X?-3 +..
CoX- + C1X 23 I +
C02
{C043XC 3 + C14XCC 4 -+ C24C n-
he finds that
C01 CO,+ _2 C=1,2f-2
Po -- y Pr+, =- ' 2n-l -
C00 COCy+2 CO,2n- 2
COO C0,rC0,?'+l %0,21 -2
The second suffix of any one of the new c's is seen to indicate
the remainder-function to which the c belongs, and the first
suffix the position of the c in that remainder. To obtain
expressions for these in terms of the original two sets of c's
it is taken for granted, and with reason, that as a result of
the process we have generally
C0, s - 1 0,s - 2
Cr+1,s-1 Cr+],s-2
By using this twice upon itself, so as to lower the second suffixes
of the first column, there is found
c0,s-2 CO,s-3
CrS C,s-2 Cl,s-3 C0,s-2 (2)
Cr+2,s-2 Cr+2,s-3 Cr+l,s-2
where the second suffixes are now s-2, s-3. A page is then
occupied in ridding (2) in the same way of the elements which
have s-3 for a suffix,,the result being
rl~v r v -~ -- ~' -----— 7 ---- — ~~ ---- '~~- zn