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

336j HISTORY OF THE THEORY OF DETERMINANTS
are equal to the successive determl~inants of the array
(b 0. b,,
divided by the product of the squares of the first coefficients of all
the preceding remainders and by b a"- l and by the sign-factor
The r~tl remainder, when divested of the threefold divisor here specified, a,. say, Trudi follows Sylvester in calling
the rtl residuo semplificato,* and denotes by p,., so that R,. = Pr/ar.
For example, when the originating functions A and B are
ax4 +bX3 + CX2 — dx + e,
px3 + qc2 + rX + S,
the three Sturmian remainders in their 'simplified' or disen

cumbered form are
a bc
p q X2
p q r
abc
a b
* pq
p q
p q r

'a b d'
+ 1. p r x +
p q s
ie: laQ b
cd
q r X+
r s 1
5. I p q
abcde
a b c d e
a babcde
p pqrs
* p q r" 5

a b el
Ii
p q
cd.
bce
p q
q r
r s *K

p q r s...1i

* A most natural and helpful notation for such a remainder would be
(a,.,,)
Thus, in the case here used for purposes of illustration, the remainders would be
written
a b c d e, (xa, x,1), (a b c d e) I(x,1),
* p q r (p q r8
p q r s