^UADRAfURE of the CONIC SECTIONS, &c. 283 



geometrical feries, of which the firft term is -, and ratio - j 



4 4 



therefore its fum will be - ; hence, by fubftitution, and put- 



3 



ting ■« for — ^— 4- 1, we have 



5 fin* a tan 2 ^ 



I = _i (Ifec^^ + 1 fed a + 1 -fee 2 |a + 1 . fee* i a 



« fin 1 * V4 24 4 43 8 44 16 



+, &c), 

 which is the feries to be inveftigated. 



17. There is, however, another form, under which the fe- 

 ries brought out in Article 15. may be given, and which I con- 

 fider as the befl adapted of any to the actual calculation of the 

 length of an arch. This transformation will be effected, 



if, in the well - known formula, tarn A zr — , ~ — r» 



1 -f cof 2 A 



inftead of A, we fubftitute fucceffively a, -a, -.a, &c. ; we 



1 — cof - a 

 mail then obtain a feries of fractions of the form n 



1 + cof I a 



which being fubftituted inftead of their equivalents in t 



mula 



. .. 



r ' I ,2 



the for- 



+ - 



1=^ 



tan a 3 

 Vol. VI — P. II. Nn j t 



