3 oo NEW SERIES for the 



31. The feries we have juft now found, may be prefented 

 under various forms. Thus, by confidering that 



1 _ cof A _ 2 fin A cof A _ fin 2 A 

 tan A "~ fin A " 2 fin 2 A ~~ 1 — cof 2 A' 



and that 



fin A 

 tan A cof A 1 2 fin A cof A 1 fin 2 A 



i 



3 — tan 2 A " ' fin : A ' ' 2 4 cof' A — 1 " " 2 1 + 2 cof 2 A 



3_ coPA 



it will appear that by due fubftitution the feries may be other- 

 wife exprefied as follows : 



1 1 {ma ,2 fin 7 # ,2 fin^^z ,2 fin T ' 7 a j_^ r 



a~~ 2 1 — co{a 3 1 -f- ,2 cof -^ 3* i-f-2cof|-tf 3* i4-2cofj' 7 tf 



And other forms might be given to it, but they would all con- 

 verge with the fame quicknefs, and each term would be lefs 

 than fth of the term before it. The feries, however, under 

 whatever form it be given, and ail others which like it require 

 for their application the trifection of an arch, are, when com- 

 pared with thofe we formerly inveftigated, of little ufe as prac- 

 tical rules ; becaufe it is well known that to determine the fine, 

 or other fuch function of an arch from a function of its triple, 

 is a problem which produces a cubic equation of a form which 

 does not admit of being refolved otherwife than by trials, or 

 by infinite feries, both of which procefles are fufficiently labo- 

 rious, and only to be employed where the object in view can- 

 not be attained by eafier means. 



32. As from the different feries we have found for the recti- 

 fication of an arch of a circle, the fpirit of our method mull be 

 fufficiently obvious, I fhall not inveltigate any others at pre- 



fent. 



