418 
<tx 
MR. SYLVESTER ON THE RESIDUES OF ^ EXPANDED 
/3, y, I, &c., the first remainder forming- the second divisor will be easily seen to 
have for its coefficients — 
&c. 
a b c 
a b d 
a b e 
0 a (3 
1 
’ ^ ’ 
0 a (3 
i 
0 a l3 
a ^ 7 
a j3 ^ 
C6 (3 s 
Hence the coefficients in the next remainder (making- 
each of the form of the compound determinant, — 
a b c 
0 a |8 
a /3 7 
— 5 * 
cc 
(3 
7 
a 
b 
c 
a 
b 
d 
0 
0 
a 
(3 
0 
a 
7 
a 
/3 
7 
a 
13 
^ > 
b 
c 
a 
h 
d 
a 
b 
e 
a 
(3 
0 
a 
7 
0 
05 
|3 
7 
a. 
h 
a 
(3 
~7n) will be 
The compound determinant above written will be the first coefficient in the 
remainder under consideration ; the subsequent coefficients will be represented by 
writing/, <P‘, g, 7, &c., respectively in lieu of e, g. Omitting the common multiplier 
the determinant above written is equal to 
r a b e a b e a b d a b d ^ 
a 
(3 
X 
0 
05 
d - 
0 
oi 7 
X 
0 
Oi 
7 
(3 
7 
05 
/3 
s 
05 
13 § 
05 
(3 
a 
b 
C 
r a 
b 
d 
a 
b 
c 
1 
+ 
0 
05 
13 
X 
(3.0 
a 
7 ■ 
o 
1 
Oi 
(3 
1, 
1 
05 
(3 
7 
1 05 
(3 
Oi 
(3 
7 
J 
The last written pair of terms are together equal to 
a b c 
0 a ^ X 
a (3 7 
which is of the form a^A-a^j 3 ^ 0 - 7 ^)a, and the sum of the first written pair is of the 
form a"B+(ai3^^^j3^-a7j3.flyi3)a. Hence the entire determinant is of the forma"(A+B), 
showing that will enter as a factor into this and every subsequent coefficient in the 
second remainder, as previously demonstrated above. 
It may, moreover, be noticed, that this remainder, when a' has been expelled, will 
for general values of the coefficients be numerically as well as literally in its lowest 
i 
— di3cc^-j-C70i^-j-acc(^^—7'^) 
r 
