4.1© Deinonstraiions of Stewart^ s 



Thus, If n denote the number of the sides of the fig^- 

 tire ; d, the Ime drawn from the assumed point to the 

 centre of tlie circle; and R, the radius of the circle; 

 eight times the sum of the fourth powers of the perpen- 

 diculars, will be equal to 8/jR'+24n(/^R'+3«a?*. 



BEMQNSTR^ TIOJV.* 



Case I. (Fig. 10.) Let the point be taken any where, within 

 the figure ; and when without the circle; let it be denoted by V ; 

 and when within, by V. Let the same construction be made, as 

 in Proposition V. (excepting that the figure shall have more than 

 four sides ;) and let the interiections of the lines drawn perpen- 

 dicular to PQ, from all the points /, a, b, c, d, &c., when the 

 point is taken within the circle, be denoted by q', j)', &c. 

 , It may be shewn, (as in Case 1st. of Proposition V.) that V'H 

 =Pp' = R-ci+y'p'; and V'S=P^'=:R-r/+VY. And, (as 

 in Case 2d. of the same Proposition,) that yH=Pp=Yp — 

 (^d—R) ', and YS=Fq=Yq — {d—R) ; and so of all the rest in 

 either case. 



Hence, Pp'^=(V>'+R'^)'=T'//^+4V'p'^ •(R-(?) + 6V>'^- 

 (j^-d) ^+4y'j/ • {R-d) '-{(R-dy ; and Fq' ^ =Y'q H4V'^' ^ ' 

 (R-d)+GY'q' ^ • (R-d) -+ 4.Y'q '■ (R-dy+(R-d) * ; and so 

 of the rest. 



In like manner, Pp'*—(Vp—(/-li)^ =Vp''-4Vp=-(^—R) + 

 6Yp2-(rt-R)^-4V7?'(c/-Rj' + (t?-R)''; and Pq^=Yq*~ 

 AWq^^ {d-R)+QYq^ • {d-Ry-4Y'q • {d-Ry+id-Ry; and 

 ^o of the rest. 



Hence, Pp"*+P^'^4- ^. = Vy ^+VY^+ &c.+4(R-(?)' 



Y^^TY^'^^^A 6 {R-d) 2 • Vy 2+V^^+&c.+4 (R-<Z) 3' 

 V y + V^f/^^Ts^. + ( R _ cT) <» + (R _ df) '^ + .^c. 



And Pj?' + P^^ + &c. =Y2y^-JrYq*+ ^yc. - 4(rf-R) » 



Y;^Tyq^V^C' +6 (d-R) 2 'Y^'+Yq'^~+~hc. ~ 4 {d~R)^' 

 Yf^'qV^- + (cZ - R) 4 + (c?- R) " + &cc. 



Now, for the reasons assigned in Case 1st. of Proposition Y., 



(by Prop. IV.) V>'^+Vy ^+ &Lc, = —nd^ ; and (Prop. IH.) 

 Yy-{-Y'q'' +&rc.=-nd'; and (Coroll. Prop. IV.) V'p" + 



Y'q"' + ^c.=—nd^; and, (Lemma 3d.) V'p'+VY4-&c.=wJ ; 



2 



and (R-d)^-^(R-d}^-^hc.=n(R-~d)\ 



•^^■JVb.'e. The case where ihe point falls in the circumference of the 

 circlcj was considered in Proposition IV.; acd therefore is hwe omitte^v 



