QUADRATURE of the CONIC SECTIONS, &c. 301 



fent. Before leaving this part of our fubject, however, it may 

 be proper to obferve, that the fecond feries may be deduced 

 from the firft, arid the third from the fecond, and fo on with 

 refpedt to innumerable others of the fame kind, by the fluxional 

 or differential calculus. 



For refuming the firft feries 



-f - tan - a -J- - tan - a -f 1 tan I a -f, &c. 



a tan a 2 2 4488 



if we take the fluxion of each term, confidering a as a variable 

 quantity, we have 



a % tan a ^4 2 4' 4 4 8 



and hence, changing the figns, and rejecting da from each 



term, and putting 1 + tan" -a for fee 2 - a. we find 

 r ° n n 



[ -U tan i" + ? tan 4* + ^ tan g a +' &c -) 



In this expreflion, inftead of the numeral feries - -f- ~-f- L-j- & c 



4 4 2 4 3 ; 



) 



(which is a geometrical progreffion having its common ratio - 



4 



fubftitute its value, viz. -, and the refult is 



3 



which is identical with the formula found at Art. 15. 



P p 2 From 



