QUADRATURE of the CONIC SECTIONS, &c. 295 



27. Let us now, however, fuppofe n indefinitely great, then 



. 3 + 4 tan* - 

 the quantity \ /— , or . _i!, becomes fimply -^, be- 



^ ' 2 4 « 2" / » fc AV A 



[ 2 tan — } 



\ 2"/ 



A A A 



caufe tan — , and confequently 4 tan*— , vaniflies, and i n tan — 



becomes A, as we have already had occafion to obferve (Art. 6. 

 and 7.). Alfo the geometrical feries 



re + & + w< +• &c - 



having the number of its terms indefinitely great, and their 



common ratio —^ will be — . Therefore, by fubftitution and 

 10 15 



tranfpofition we have 



or, fubflituting for /A, and F-j A, Seethe quantities which 

 thefe fymbols exprefs, 



A 4 



3 + 4tan ? A 14 

 tan + A "■" 15 



i 



{ ^ (4 tan 2 i A +3 tan* | A) + ^(4 tan 2 i A-f- 3 tan^A) 



+ -L (4 tan 1 i A + 3 tan 4 f A) +, &c] 



and this is one form of the feries which we propofed to inves- 

 tigate. 



28. This 



