Lemma 3'. 



" The values of e and n continuing as before, let m be any •whole 

 number diiferent from k, but like it confiding of many places of 

 figures ; if out of the binomial or refidual i + e be extrafted roots whofe 



indices are — and — , and if the roots be calculated true to numbers 



of places of decimals lefs by four or more than twice the numbers of 

 places of figures in either m or m; and laftly if from each root 

 be fubtrafted unity, then will the remainders be to each other as the 



indices „ and , or reciprocally as the denominators n and m." 



m 



For if I be fubtrafted from the root whofe index is — , by the two pre- 



n 



cedinff lemmas, the remainder will be + — e— — e' H f e* H — <?' 



^- -.. 



&c. which is equal to — x +e e'' + —e^ - ~e* + — -e' &c. and 



^ n — 2-3 4-5 



like manner, if i be fubtrafl:ed from the root whofe index is — the remain- 



' '»> 



der will be — x -\-e e'' + —e^ - —e* + —e' &c. and it is evident 



m, ~ 2—3 4_5 



that thefe feries are to each other as — and — , or reciprocally as n and m. 



Lemma 4. 



" The values of e and n continuing as before, if out of the quan- 

 tity I II t a root is to be extracted whofe index is — ; and unity 



being fubtrafted from the root, if the remainder is to be multiplied 

 by the denominator « ; it is required to find the moft fimple feries that 

 fliall give the produft true to a number of places of decimals, lefs by 

 three or more than the number of places of figures in n." 



From what is faid in the firft and fecond lemmas it is plain that the 



feries required is + e -ie"- ±^e ^- ,e* ±\e' &c. the laws of whofe 



continuation 



