AlGEBRAia FORMULA. 



27 ^ 



the feries 'of- flaehts N'jN''', N'*, &c. and the ferles of arches 

 p, <p', (p'\ &c. both conftantly tend. 



• 7. Now in order to inveftigate thefe limits, I obferve, in the 

 fir ft place, that the quantities -which conftitute the feries e,e',e!',e"\ 



&C'.- -cofiVerge very laft towards o ; for^ iin^e e' rr ^ — v^^— j^ 

 = ■'■ ■ , it Is evident that e muft always be a fradion 



lefs than i, and that f' muft be lefs than the fquareofthat 



^ ff adion ; indeed, if e be fmall, e will be nearly = -. In the 



fame manner it appears that e" is between the limits of e'^ and 



— , fo that e" is lefs than e\ but greater than |- • "a«id proceeding 



in this mannerj ^e find r" lefs thaii- e^; i?" lefs than ^' *, and fo 

 ori, the .exponents 'of e forming the geonnetrical jjrogreiTidn, 

 2, 4, 8, 16, 32, &d:^' •_■ - \ ^^ "'-■ -''^-ivl/'. 



•Thus it appearsf even without taking iiito accovmt the deno- 

 minatory of the ppyyers of e, that ;he rapidity with wliich the 



quantities e e', {'.. Sec. converge tow.ards is- indeed very creat. 



-*.■'• ./ ii v.C.T. ;Vir7,r!.~:'r :ii;ajr;- oi :, .7i'' ,:■■.,- ■ •.- ..-■'.'^ ' •- 

 8. I OBSERVE, in tne next place, that the terms which doniftT- 



ttite 'the ferres 'of^ dfdie^ '^J '^,"'^'',-(^"'i Sec. ' alf6' converge- very faft 



to a ceitain -limit. For. fiho^' "tiirfi ihave aflumed fuch.a t^latioQ 



lietw^eialjli^q.ajfhesthajtjjr,^ vhn-.I)fv:. :•:,; ;' •,:,-/ '. . ., ■..:., 



,, o. Jiix:-i:M^:^^^^^.^^,., ..,,i...p',;i, 



fin W" = -y=^SEi£L===, , .h^ib-jiot 



■io hl-^Yi Vi.ribur'jp ..:. y^^ t/if.pr'F^-if'\v.:.ij -a .ox ^ 



it*4s'-e'firfent?,frbffi w'tiat^'Kiis'llj^^^a^rfin' the laft.Artlcfe'/ tttat th:^ 



deh<>nlrnatbh'df tefr"a<aiorfs,'ii«di are eq4aV^ 



firi •8(?)'y&c- eaiiVerge-q'iiic'tly-ixy^ai^d& M^^^ 



effih 2<?)-tofirtte<pV of fi'iS4ip'Wfiti'"'4?)^; '6f fiii 8/ to' fin S?)"', &c. 



**VoL. v.— P. II. Nn. approach. 



