QUADRATURE of the CONIC SECTIONS, &c. 317 
I / 
therefore —__—— — ! it and hence 
7 = r—z” 
I—/Y/1—?t 
Toe ar 
which is the formula required. 
48. Tue refult of the whole inveftigation of this fecond fe- 
ries, for the area of an hyperbolic fector, may now be collected 
into one point of view, as follows. 
Puttinc s for the area of the fector, let its correfponding 
abf{cifla be denoted by the abbreviated expreflion abs; alfo let 
the abfciff correfponding to the other fectors which are its 
fub-multiples be denoted fimilarly. 
Compute the feries of quantities abs, abis, abys, &c. 
from abs, and one another, by the formula 
abig—Vapd+2 
ig OS SD 2 
Then fhall 
6220ab Seb iwin ince 
— 
abs —1I 3 
i rabas—1, 1 abys—t Yabts—r 
tas gabis+1° Pabis+i-° 4g abzs+1 
ie SER ae a ae J 
where R denotes the fum of all the terms following the term 
T(m+x1), and this fum is always contained between the limits 
T*(m+1) 
I 
It, q — Timta) 
T5 Ee +1) an T(m)— L(m+1)’ 
Rr 2 é being 
