QUADRATURE of the CONIC SECTIONS, &c. 309 



T I 



ries of quantities tan- s, tan ~ s, &c. is greater than half the 



40 



term before it ; and as thefe, multiplied by the fractions -, -, 



2 4 



&c. refpe&ively conflitute the terms of the feries, each term of 

 the feries, under either of its forms, is greater than one-fourth 

 of the term before it. 



40. Again, from the formula tan- Sr: -tanS (1 -J-tan 2 - S) 



2 2 2t 



~ , 2 tan tS , 2I0 jr-ii 2 tan 7 S 



we find ' = 1 + tan - S, and fimilarly, ^- = 



tan S 2 tan ± b 



1 -+• tan 1 -- S. But from the nature of the hyperbola 



4 



,i_ , zi c , r 2 tan IS 2 tan ~ S , 

 1 + tan 2 - S < 1 + tan - S ; therefore r%- < — - — 2 —-, and 



4 



tan t S tan S 



hence tan I S < tan * . Therefore, putting — s inftead of S, 

 4 tan b a » 



and multiplying by — , we have 



— tan — s 



T I„2« 4« 1 1, 



— tan -5- s < 1 — x — tan — s, 



An 8 « t 1 , 2« A.n 



* - tan — s T 



» 2 » 



from which it appears, that each term of the feries, following 

 the fecond, is lefs than a third proportional to the two terms 

 immediately before it. So that, upon the whole, it appears, that 

 the limits of the rate of convergency of our firfl feries for an 

 hyperbolic fector, are the fame as thofe of our firft for an arch 



Qjl 2 of 



