QUADRATURE of the CONIC SECTIONS, &c. 289 



Or, compute the feries of quantities t, f, t", f", &c, one from 

 another by means of the formulae 



_ I — COf a , r </\-\-t— I _ Vl+t'— I o 



i + cof^z •i + f + i • I _|_r+i 



Then will 



f 1 1 + cof a 1 



4 1 — cof a ~ f ~ 6 

 J_ J 



where T( OT ), T(»j+i), and S denote the fame as before. 



I consider this fecond feries for the rectification of the 

 circle (under either of its forms), as preferable to the other fe- 

 ries given at Article 12. for two reafons ; firft, becaufe of its 

 oreater rate of convergency, and, next, becaufe the quantities 



coftf, cof- a, cof- a, &c., alfo the quantities t, t\ t\ &c. can 

 2 4 



be deduced more eafily one from another than the feries of 

 tangents tan a, tan - a, tan - a, tan - a, &c. 



2 4 Q 



23. That we may inveftigate another- feries, let 



us 



refume the formula r = „ . , A — - tan 4- A \ the cubes 



tan A 2 tan -A 2 



of both fides of the equation being now taken, the refult is 



-taniAJ- — itan 3 ! A= 



tan J A"" 8tan 3 iA SltaniA 2 -* 8 2 



To 



