[ '56 ] 



the radius produced without the circle draw PO, PO', &c. then 



n n 



r C — O C = P O X P O' X P O" X &c. when P is without the circle 



n n 



and OC - PC = PO X PO' X PO" X &c. when P is within the 

 circle. 



2. Let the circumference be divided into 2 « equal parts OS, 

 S O', O' S; &c. then P c"+ O C = P S x P S' x &c. 



Demon ST RATfON. 



I. Let OC be unity, PG = a-; a, a, «", &c. the cofines of o, 

 GO', 00", &c. 



Then will PO^' = x= + i — 2 ax 



PO" = x"- + I— idx 



&c. &c. 



or 



P 0* X P 0'» X &c. = X» + I 2 tf X X x' + I 2 dx X &c, 



a 



a 



a 



&c. 



n— I 



a a 



^^rqTTl" — y f 2 X- X* + i^ + a d' J> 2^ X'. x^ + I 



&c 



+ 2 add' &c. X » • 



Now if c be the cofine of any arc, the cofine of n times that arc 



„_l „ n— 3 n— 2 ^ „ ^ "—5 n— 4 



will be 2 r — n. 2. ^ + -^ 2 f — &c. continued by 



I. 2 



fucceffively diminifhing the index of f by 2 until it becomes o or i, 



and 



