304. NEW SERIES for the 
35. From the nature of the hyperbola, we have 
ab S = ab*2$-+ord’+S, ord S=2ab*Sord2S; 
2 2 2 2 
therefore, by divifion, 
ys peace ie) t ord> 3S 
ordS ~ 20rd+S ' 2 ab+8’ 
thadisyollbs tales td 
tans 2tan> 
Tuis laft formula exprefles a property of the hyperbola per- 
fe@tly analogous to that of the circle (Art. 5.), from which we 
have deduced our firft four feries for the rectification of an 
arch. Therefore fimilarly, putting s, 55 * s, &c. fucceflively 
4 
inftead of S, and multiplying by the feries of numbers 1, = ; 
&c. we have as in that article 
I I I I 
es —-+-= tan -s, 
tan s tan 2.§ 2 2 
I I I I 
Sees) Se et i, 
Pte" tans yp 4 
I I I I 
juts vas pallid eb pal Ze, 
4tanis 8 tan 3s 8 8 
&e. 
Tuis feries of equations being fuppofed continued until their 
number be ”, by proceeding exactly as in Art. 5. when treating 
of the circle, we obtain 
I 
—_— 
— 
2” tan mo 
