PRO PERT IES of the CIRCLE. 69 



the fines of the angles formed at the centre O, by OP or OQj and 

 the diameters palling through the points of contact to the radius 



.im 



OP or OQj that is, = Va + ¥e -f P b +, &c. or Q^f + 



Qj + Q^ +> & c « — tne f um °f the 2 m powers of perpen- 

 diculars drawn from any point in the circumference of a circle 

 defcribed from P as a centre, with PO as radius to n number 

 of right lines, interfering each other in P, and making all the 

 angles equal, = the fum of the m powers of the rectangles A a X 

 aG, Be X er, C b X b h, &c. ; or of the rectangles G c >< c A, 

 rf X /B, h d X dC, &c. when the regular figure circumfcribing 

 the circle has an odd number of fides ; but equal to twice the 

 fum of the 2 m powers of faid fines, or to twice the fum of the 

 m powers of faid rectangles, when the regular figure circumfcri- 

 bing the circle has an even number of fides, fince the number 

 of the diameters drawn through the oppofite points of contact, 

 and making equal angles with each other, at their interfection in 

 the centre O, is only half the number of the points of contact or 

 fides of the figure. But thefe rectangles are refpectively equal 



to r — O a X r+Oa, r—O e X r -f- O <?, r— O iXr + O^&c 



a 2 2 



or r* — O a , r 2 — O e , r* — O b , &c. ; and the fum of their m lh 

 powers is n r™ — ~. r vn ' % X Oa +Oe -\-Ob -f, &c to n terms, 



+ -• ~r^. r"*-4 X Oa+Oe*-\-Ob*+ } &c to n terms +, &c. 

 1 *> 



im i>n tin 



&c. + Oa -f- Oe -\-Ob -f-, &c. to («) terms, or to 



xm iff/ 1'« . . 



Oa — O* ^ O £ — , &c. to [n) terms, according as m 



is even or odd. 



III. 



