314 NEW SERIES for the 
45. Tue 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=ab2S, and ab’S —ord’S=1, 
therefore, taking the fum and difference of the correfponding 
fides of thefe equations, we get 
2ab*S=ab2S8+1, 20rd S=ab25—13 
and hence, by dividing the latter of thefe equations by the 
former, and putting tanS inftead of pubs , we find 
ab2S—1I 
ab2S+1 
From this formula, by fubftituting s, +5, +5, &c. inftead of 
S, we obtain expreflions for tan’ s, tan’>s, tan?+s, &c. Thefe 
being fubftituted in the feries, and afterwards s put inftead of 
25, +5 inftead of s, +5 inftead of ts, &c. (fo as to produce a 
refult involying only the abfciffae correfponding to the fector s, 
and its fub-multiples) ; and, finally, the whole being divided by 
4, we fhall get 
fabs+1 2 
| abs—1r 3 
tans = 
=| = 
rabis—1_, rabis—1, 1 abis—1 
and this expreffion is analogous to our fecond feries for an arch 
of a circle, as given at Art. 17. 
46. Wr may now inveftigate 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 refpect to the firft 
of 
