X. A Method of extraBing the Root of an In- 
jlnite Equation. By A. De Moivre, F. R. 5^. 
Theorem. 
k/W-^-'A &c. then will g V + 
+ " — 
, k'-'hBB-'2hAC—';cAAB—iiA'' , 
+ — a r 
+ — 
+ m-^ zbBD ~ hCC — 2hAE'- cB'^ 6c ABC — ^cAAD 
— 6dAABB AdAK — ^eA^B — fA' 6 / 
— - J &c. 
For the underftanding of this Series, and in order to con- 
tinue it as far as we pleafe ; it is to be obferved, i. That eve- 
ry Capital Letter is equal to the Coefficient of each preceding 
Term ; thus the Letter B is equal to the Coefficient 
2. That the Denominator of each Coefficient is always a. 
3. That the firft Member of eachNumeratorj is always a Co- 
efficient of the Series ^7 -}- hyy &c. viz,, the Firft Nu- 
merator begins with the firflCoefficientg^the Second Numera- 
tor with the Second Co-efficiem /6,and to on. 4.That in every 
Member after the Firft ,the Sum of the Exponents of theCapi- 
tal Letters^ is always equal to the Index of the Power to which 
this Member belongs : Thus confidering the Coefficient 
k-hBB-2UC-:icAJB^^Jj^, which belongs to the 
Power/, wefliali fee that in every Member bBB, 2hAC 
iCAABy dA'^y the Sum of the Exponents of the Capital Let- 
ters is 4, ( where I muft take notice, that by the Exponent of 
