272 



NEW SERIES for the 



Let the fums of the correfponding fides of thefe equations 

 be taken, and obferving that the feries 



+ 



+ 



2 tan | a 4 tan £a 3 tan £ a 



• • • ™r* 



>«— 1 



tan 



a 



2«— i 



is found in each fum, let it be rejected from both; and the re- 

 fult will be 



2" tan — 



2" 



taL-^ -gtan^+Itan^ + |tan-^ + ^tan-I«... 

 + - tan t \ 



2" 2«/ , 



the number of terms of the feries in the parenthefis being », 

 and hence we have 



I ,!„ I r I \ I ,1, I , 



zz -f — tan - a 4- - tan - a 4- - tan - a 4- 



2 24 48 8 



2" tan 



a 



2 n 



tan# 



—, tan — a . . . -f- _ tan — . 



10 10 2" 2" 



6. Now, 2tani-^ is the perimeter of a figure formed by 

 drawing tangents at the ends of the arch a, and producing 

 them till they meet; and 4tan^# is the perimeter of a figure 

 formed by bifecting the arch a, and drawing tangents at its ex- 

 tremities and at the point of bifection, producing each two ad- 



joining tangents till they meet ; and in general 2 n tan — is the 



perimeter of a figure formed in the fame way, by dividing the 



arch 



