PROPERTIES of the CIRCLE. 69 
the fines of the angles formed at the centre O, by OP or OQ; and 
the diameters pafling through the points of conta¢t to the radius 
OP pe 0Q; con ‘s. Ns ae 4 we a Ps” “% &e. Wu ae” vf 
OF as Od +, &c. = the fum of the 2m 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 x number 
‘of right lines, interfecting each other in P, and making all the 
angles equal, = the fum of the m powers of the reCtangles A a X 
aG, Bexer, Cb x bh, &c.; or of the rectangles Ge x ¢ A, 
rf Xx fB,4d xX dC, &c. when the regular figure circumfcribing 
the circle has an odd number of fides; but equal to twice the 
fum of the 2 ™ powers of faid fines, or to twice the fum of the 
m powers of faid rectangles, when the regular figure circum{cri- 
bing the circle has an even number of fides, fince the number 
of the diameters drawn through the oppofite points of conta¢t, 
and making equal angles with each other, at their interfection in 
the centre O, is only half the number of the points of contaét or 
fides of the figure. But thefe retangles are refpectively equal 
' tor—Oa X r+Oa, r—Oe Xr+O084, r—ObXr+O04, &e. 
iba ies} ein 
or P—Oa, r—Oce, r°—O4), &c.; and the fum of their m"™ 
m1 SS ea ee 
powers is 77?" 7. r"-* x Oa +O¢ +06 +, &c. tom terms, 
pene et al 
5 r+ x Oa" + O2'-+08'+, &e. to m terms +, &c. 
m 
ale 
zi —2m —2mN —2m 
&e. + Oa + Oc +00 - +, &E. ‘to (2) ‘terms, or to 
—21m —2m 
—2N 
—Oa —Oe —Ob —, &c. to (”) terms, according as m 
is even or odd. 
III. 
