QUADRATURE of the CONIC SECTIONS, &c. 311 
here . \y . 
Ree Duet) 
3 
Hie Ri EGTA) hues Svea ® 
T(m) — T(m+1) 
As thefe limits to R differ but little when the terms T(n),. 
T(m41) are confiderably advanced in the feries, the latter may 
be exprefled more conveniently for calculation thus 
I 
$ 
Tm ra Ty m)) T m 
R <3 Tmt) + (lens Fem oe ) oS). 
3. 3 (T (m) — Tin+1)) 
43- Let us next inveftigate a feries for the quadrature of the 
hyperbola, which may be analogous to our fecond feries for the 
rectification of the circle. For this purpofe, proceeding as at 
Art. 13. we refume the formula ——~ : 
I I 
= ——__.. s iL 
tanS Bran lS Sey 0 Ss 
and taking the fquare of each fide of the equation, get 
Eee ee Pa 
ers fun ss poets Sire 
Inftead 
be deduced one from another by the help of the common trigonometrical tables, 
It is this, 
— 2abs 
™~ “ords 
' I , I ur I iv 
— (fina + age eile Uke lgeai + 3 fin a 26+ Tm) + T41) + R):- 
The arches a’, a”, al”, al’, &c. are to be deduced one from another as follows. 
; 0 = —_ 
Take a fuch that fina om, then, fina! tan = a, fin. tan-2 fin gi! — 
2 
rom 
tan > a’, fin ai’ = tan z*? &c. The fymbols T(m), T(m+1) and R, denote. 
the fame things as in the other form of the feries. 
