( 9J ) 
in Lemma xi. w ill vanifh , and therefore n x n — i 
— n — \xn — XX -f- n — x x^— .3 
X 5cc = o j and one of the equal Values of x 
will be a Root of this laft Equation, and two of them 
will be R oots of the Equation n x ^ n— i x 
\x 2 xBa?” ’ 6cc. = o. In general, it 
appears that if the Equation a’”— 
^6cc. = o have as many Roots equal amongfl: themfelves 
as there are Units in S, then (hall as many of thofe be 
Roots of the Equation nx”'^^ n i x Aa; 
~{-n — zxBx” ^(5cc. = o as there are Units inS — i ; 
as many o f them fha ll be R o ots of the Equation 
n X n 1 X at ” ix;^ — xxAa;”“'*-|- 
n^xxn—^ S X 6cc. = o, as there are Units 
in S — X ; and fo on. 
^ II. The general Rule which Sir Ifaac Nekton has 
given \vii\\Qylrtkle de limit thus Equationum for find- 
ing a Limit greater than any^ of the Values of x im- 
mediately follows from the Lemma for it is mani- 
fefl that if e be fuch a Quantity as fubftituted in all the 
Coefficients of the Equation ofy, vi^, in Ae”'~' 
-|- B ^ * 6cc. ne — n — i xA^”~^ -f- . - 2 , 
- - “■*” I _ J2 •— ‘i.' 
xB^”""* (5cc. n X j X x 
• n — 3 
Ae * -j- ^ — 2 X ~ - X <5cc. gives the 
Quantities that refult all pofitive ^ then there being no 
Changes of the Signs of the Equation of j/ in this cafe 
all its Values mult be negative • and fince j; is always 
equal to x'--e it follows that e mult be a greater Quan- 
tity than any of the Values of that is, it muft be a 
^ Limit 
