ALGEBRAIC FORMULA. 277 



the feries "of- fluents N'/N*, N'*, &c. and the feries of arches 

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



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

 firft place, that the quantities which conftitute the feries e^e\e^,e'" r 



&.c. converge very faft towards o ; for, fince e — 



_ 1 -/i. 



1 + /1 — 



_ — , it is evident that e muft always be a fraction 



lefs than 1, and that e' muft be lefs than the fquareofthat 



fraction; indeed, if e be fmall, e' will be nearly zr -. In the 



fame manner it appears that e" is between the limits of e' 1 and 



—j fo that e" is lefs than £ 4 , but greater than ^- ; and proceeding 



in this manner, we find e" lefs than <? s , e IV lefs than e l % and fo 

 on, the exponents of e forming the geometrical progreflion, 

 2, 4, 8, 16, 32, &c. 



Thus it appears^ even without taking into account the deno- 

 minators of the powers of <?, that the rapidity with which the 

 quantities e. e', .e". x &c. converge' towards o is- indeed very great. 



8. I observe, in die next place, that the terms which -confti- 

 tute the feries of arches fftyfW <p" ! , &c. alfo converge very faft 

 to a certain limit. For. fince we have affumed fucli a relation 

 between th^acches thap 



fin 4 <p' = , ?*** , •' ' ■' 



fin W" — n ? , ^ n8 ^ 



it is evident, from wha^t has'bee f n^arfin the laft Article/ diat the 

 denominators 'Of the 5 fraction's, 1 which are equal' to' fin 2'<p',firi 4$% 

 fin 8 cp'V&C. ebriVerge 'quickly towards unity,' fo that the ratios 

 of Tin 2<p to firf*<$ of fm 4^ to 'M%f, 6f fiti 8/ to fin 8<p'", &c. 

 Vol. V.— P. II. N n approach 



