314 NEW SERIES for tbe 



45. The fame feries, however, may be given under another 

 form, better adapted to calculation. For fince, by the nature 

 of the hyperbola 



ab* S + ord* S — ab 2 S, and ab 2 S — ord* S zz 1, 



therefore, taking the fum and difference of the correfponding 

 fides of thefe equations, we get 



2 ab 2 S — ab 2 S -f *> 2 ord* S = ab 2 S — 1 5 



and hence, by dividing the latter of thefe equations by the 



1 o 



former, and putting tan S inflead of , , we find 



tan S — 



abS 

 ab 2 S — 1 



ab 2 S -j- 1 



From this formula, by fubftituting s, -jj, ^ x, &c. inflead of 

 S, we obtain expreffions for tan 2 j, tan 2 7 s, tan 2 J j, &c. Thefe 

 being fubflituted in the feries, and afterwards s put inflead of 

 2 s, 4- s inflead of j, ^ s inflead of -i- s, &c. (fo as to produce a 

 refult involving only the abfciflae correfponding to the. fector s t 

 and its fub-multiples) ; and, finally, the whole being divided by 

 4, we fhall get 



[ ab s + 1 2 



1 J abj — 1 3 



* Z 1 aD i J — 1 1 1 ab ^ j — 1 , 1 abi-x — 1 f « . 



and this expreffion is analogous to our fecond feries for an arch 

 of a circle, as given at Art. 17. 



46. We may now invefligate what are the limits to the rate 

 of convergency of this feries, as alfo the limits to the fum of all 

 its terms following any afligned term. With refpedl to the firfl 



of 



