QUADRATURE of the CONIC SECTIONS, &c. 313 



Now, as we have found (Art. 36.) that 2 n tan — exprefles 



2» •*■ 



twice the area of the polygon AFF'F'F" (Plate IX.), the 

 numerical value of which we have there denoted by Qj it 



follows, that 2** tan 2 — — — Q\ Moreover, the geometrical fe- 



ries - ^ 1 .... 4 is equivalent to - (1 — — V 



2 2.4 2.4* ' 2.4 n ~ £ ^ 3 ^ 4"/' 



therefore, by fubftitution and tranfpofition, we get 



_i 2 / £ 



tan 2 j 3 ^ 4") 



■h- (* tan i s + —tan 1 - 1 s + 1 tan* -S +1 tan 2 lv 



\± 2 4 4 4 J 8 ^ 2'V 



I 



4Q: 



i 



^4 2 4 4 4 J « 4» 



44. Let us now conceive n to be indefinitely great, then, as 

 upon this hypothefis, Q_ becomes s, and - (1 — —\ becomes 



fimply -, and the feries whofe terms were n in number, now 



goes on ad infinitum , we have at laft, after multiplying the 

 whole expreflion by 4, 



4 8 



1 



o: 



tan s 3 



— (tan 2 1 s + I tan 2 I j + -I tan 2 ^ + - 3 tan 4 JL J+ , &C .). 

 2 4 4 4 8 4 3 16 y 



And this is one form of the feries to be inveftigated. 



45- The 



