280 NEW SERIES for the 



Let the Turn of thefe equations be taken, as in the inveftigation 

 of the firft formula, and obferving that the feries 



i,i ,i 





2 2 tan*^rt 4 tan 7^ 8*tan a ±tf „•«__,* 2 « 



2 T * 8 2 2 »— 2 tan 



2« — 1 



is found on both fides of the refulting equation, let it be reject- 

 ed from both ; then we obtain 



2 a 

 2 2n tan — 



2" 



4 2 4 4 4 J 8 ' 4 4 16 



.1 * ^ /I ■ 1 , 1 , 1 1 1 \ 



H tan (- + h 1+ r . . + J. 



^ 4* 2" V2 2.4 2.4 2.4^ 2.4"— x ' 



Now it appears, that one part of this expreffion, viz. 



1+ JL+-L,+ -I + -1_, 



2 2.4 2.4 2.4 J 2.4"— * 



is a geometrical feries, the firft term of which is -, the laft 



£ 



term , and common ratio - ; therefore its fum is 



2.4— «' 4 



Alfo, fince 2 n tan — is the expreffion for the perimeter of a 



2 



polygon, formed by dividing the arch a into 2 n — l equal parts, 

 by drawing tangents at the points of divifion, and producing 



the 



