QUADRATURE of the. CONIC SECTIONS, &c. 317 



therefore - — i-ii, and hence 



Vi— t 1 — t" 



, _ 1— y/l— ? 



* — ■ , 



1 -f- v/i — * 



which is the formula required. 



48. The refult of the whole inveftigation of this fecond fe- 

 ries. for the area of an hyperbolic fecltor, may now be collected 

 into one point of view, as follows. 



Putting s for the area of the fe&or, let its correfponding 

 abfcifla be denoted by the abbreviated expreflion ab s ; alfo let 

 the abfciflae correfponding to the other fectors which are its 

 fub-multiples be denoted fimilarly. 



Compute the feries of quantities abfj, ab \ s, abf s, &c. 

 from ab s, and one another, by the formula 





abiS=V abS+I 



2 2 



Then fhall 





ab s -f- 1 2 

 ab j- — 1 3 



7 = - 



f 1 ab 7 j — 1 . 1 ah \ s — 1, 1 ab -£ j — 1 

 ] 4abij+ 1 4 2 ab^j+i 4 ab 8 4+i 



I L • • • • + T M + T(« + ,) + R 





J 



where R denotes the fum of all the terms following the term 

 T( m + 0, and this fum is always contained between the limits 



-Lt (w , + , and T W0 



15 T(„)— TCm + .i)' 



R r 2 i being 



