QUADRATURE of the CONIC SECTIONS, &c. 283 
geometrical feries, of which the firft term is ? and ratio ; 
therefore its fum will be = hence, by fubftitution, and put- 
ting 
I¢ I 
-for I, we have 
fin? a tan’ a + , 
BE det Ligihg pea tat 2 
Sa apt my oe Sn uprgiae rate fees gate - fec 7% 
+, &e.), 
_ which is the feries to be inveftigated. 
17. THERE is, however, another form, under which the {e- 
ries brought out in Article 15. may be given, and which I con- 
fider as the beft adapted of any to the actual calculation of the 
length of an arch. This transformation will be effected, 
1—cof2A 
if, in’ ne well-known apg re tam: A= T+ cof2 A’ \ 
inftead of A, we fubftitute fucceflively a, Pa &c. 3 we 
jorq 1 ilviniw esi1st baoost ot gLsicof ag 
fhall then obtain a feries of fractions of the form auf 
I+ Souk aa 
which being { fubftituted. inftead of heir eapiralnts in, the for- 
mula, er ' 
{ DUE. DUR 6 @5t 
3 : biwvs 3 JA is Aa ; 
a=, —Gurt Sob Stan’ pita , tan’ sat pan ait 4 
4 16 
1 +, &e,), 
Vou. VI.—P. II. Nn it 
