492 
MR. SYLVESTER ON A NEW METHOD OF FINDING 
same sign as q,.q^...qn, *• e. will never change its sign from the beginning to the end 
of one interval, nor from the beginning to the end of the other; and conseqiientU 
L and A will be a superior and inferior limit respectively to the real roots of /r. It 
will of course be observed that it is indifferent for the purposes of the foregoing 
theorem, whether ? be expanded under the form of a proper or an improper fraction, 
.... 
1 e. whether we employ the ordinary or the Sturmian process of successive division, 
for changing the signs of the residues will only have the effect of changing qi into 
and the pair of equations (+)?i= + Q remains the same whether the + or the 
- sign be prefixed to q^. The result is, that if we form the 2n quantities 
±\-b^ ±2-b^ ±3-&3 ±2-bn-X . ±l-bn 
the greatest of them will be a superior, and the least of them an inferior limit to the 
roots of fx*. 
It may be remarked, that if the successive dividends in the course of the process 
be multiplied respectively by h, ... ^ will take the form 
k^ k^ 
9i + S's + ’ 
and if we write a^x-\-b^=-^c^ ... a^x-2rh^~±:c^ 
and make Ci=l ... c„=l+A’„, 
the same reasoning as above will show the greatest and least of the 2n quantities 
±\~b^ ±[l + 'k^-h ±(1 + A„)-Z>n-I _ 
flj ’ «2 5 ’ (In 
will be a superior and inferior limit to the roots oifx. 
For greater simplicity, again, consider h...k^ to be all equal to unity ; we may 
make this addition to the theorem as above stated, viz. calling Lj Ai ; Lj Aa ; ... L„ A, 
the greatest and least values of the terms contained respectively in the series marked 
below 1, 2, 3...W, viz. — 
+ !-», ±2-ft, ±2-*3 . j:--'’— . 
’ «3 ’ ”* 
+ 1_52 ± 2-^3 _ ±2-bn-i ±l-hn 
(^') ... ’ an 
+ 1 — +2 — bn-\ ^ + 1 
(3-) ^ On-l ’ 
+ 1 A„_i ^ 
(n-l) «n 
+ 1 — b„ 
{n.) - an ’ 
* For a generalization and improved form of statement of this theorem see Supplement to the present Section. 
