( 94 ) 
Limit greater than any of the Roots of the Equation 
AT* — A a;"““ * - 4 “ = o* 
III. From this Lemma fome important Theorems 
in the Method of Series^ and of Fluxions, and the Re- 
folution of Equations are dernonftrated with great Faci- 
lity ; it is obvious that the Coefficient of the fecond 
Term of the Equation ofj,'in that Lemma is the FluxF 
ono^ the firft Term divided by the Fluxion of e\ the 
Coefficient of the third Term is the fecond Fluxion 
of that firft Term divided by fuppofing e to flow 
uniformly. The third Term is the third Fluxion of 
the firft Term divided by x x 3 ^ ; and fo on. There- 
fore fuppofing ‘ &c. = r, the 
« • • 
• • • c c 
Equation for determining j/ will be —y^ 
6 I X2. 6 ^ 
— j/ j 6cc. = O5 and hence, when ^ is near 
H 
I XXX3 e 
the true Value of at. Theorems may be deduced for ap- 
proximating to j/, and confequently to x^ which is fup- 
pofed equal toy e. . 
IV. Let A P (= x') be the Abfcifs and PM (= ^) 
the Ordinate of any Curve B L M ; and fuppofe any 
other Abfcifs A K = ^ and Ordinate K L = ^7, then 
* • • • « 
• • 
0 c 
fhall ^ ( = P M)= cy — y-\ r- y* + 
• . e X e * 
XX 3 
y. 
-.-y^ dec. 
-1 ^ - 
' X X 3 X 4 ^ 
For let .s be fuppofed equal to any Series confifting 
of given Quantities, and the Powers of x, as to A x^ + 
B a; + C a; * 6cc. and fubftituting e+y for x, we fhall 
find after the manner of the Lemma, 
3 
^ = 
