I t 



V' fin n n''n 



-f 



e e e Sec 



(B) I. —e. —e. ~-^. ~-^e. ~-^e. &c. 



fC)i.+ —e. - --^.«+ — ^^ - 1-c.^ + ^e.' Sec. 



By multiplying together tlie terms of the foft feries above marked A, 

 according to the method of the binomial theorem, a new feries will be 

 prodaced, which will give that root true to any number of places of de- 

 cimals : but in the third and following terms of the feries A, the fraftion 



— is fo fmall, that it may be omitted without caufing any error within the 



prefcribed number of places, and this will reduce the feries A, to the 

 fecond feries above marked B, and the terms of it, multiplied according 

 to the method of the binomial theorem, produce the third feries marked 

 C, which therefore is the feries required. In it the firft term i& 1, and 

 the reft arc formed by the following laws. 



ift. In the 2d, 3d, 4th, and following terms are the feveral powers 

 off, whofe indices are i, 2, 3, &c. the natural numbers. 



adly, The co-efEcient or uncia of any term is — divided by the index of e 



n 



in that fame term : for it is — divided by the denominator of the laft of the 



n •' 



fraftions In the feries B, that were multiplied together in order to pra- 

 duce that term. 



3dly, The two firft terms of it arc pofltive , becaufe the two firft 

 terms in B are pofitive : but the following terms are alternately nega- 

 tive and pofitive ; becaufe the 3d and following terms of B being all 

 negative, in the 3d and following terms of C there will be alternately an 



t)dd and even number of negative factors. 



And 



