40 
Proceedings of the Royal Society of Edinburgh. [Sess. 
the three Sturmian remainders in their “ simplified ” or disencumbered 
form are 
a 
b 
c 
a 
b 
d I 
a 
b e 
V 
2 
+ 
<M 
8 
V 
T p “H 
p s 
V 
<1 
r 
p 
<1 
s \ 
p 
<1 • 
a 
b 
0 
d 
e 
a 
b 
c 
d . 
a 
b 
0 
d 
a 
b 
o e 
V 
d 
r x + 
V 
q s 
p 
d 
r 
s 
P 
d 
r . | 
V 
d 
r 
s 
p 
d 
r 
s 
abode 
abode 
a b c d e 
p q r s 
p q r s 
p q r s 
p q r s 
the removed encumbrances being the factors 
a 
b 
c 
2 
P 
d 
1 
p 2 
V 
d 
r 
p 2 
a 
b 
0 
2’ 
p 2 
a 
b 
0 
d 
e 
2 
P 
d 
a 
b 
0 
d 
p 
d 
r 
p 
d 
r 
p 
d 
r 
s 
p 
d 
r 
s 
. 
respectively. The general expression for the factor, a r , connecting the 
simplified and unsimplified forms of a remainder is readily got (pp. 138-139) 
in Sylvester’s way by using the fact that the product a v _ x a r is equal to the 
square of the first coefficient of p r . For, this is the same as saying that, if 
we denote the first determinant of 
by D r , we have 
and these lead to 
and 
where 
j (a 0 , ... , a m ) r j| 
I (6q , . . . , & n ) || 
l 2 2 
— T)2 
= j . . . • Dt-i 
a, 
a 2ju,+l 
1 d m • • • ’ 
a 1 = (~ . b 
m—n+ 
0 
