Praperties of the Circle. 4iJ- 



Tlierefore, Pp'M-P'Z" +4"C. =^nd*^lOnd\'^~d)-^^nd^ 



{\\-d)^+UdiJ^-d)^-^niV.-d)\ 



In like manner, ^p*+Vq^+S^c. = ^^-nd*-Wnd^ {d-'R) ^^ 



& 

 a«/^'c?-R) 2_4n(^(ci-R) 3+n (<Z-R)^ Or, P/ *-^Vq' H^'C. 



=7r — (^4 4-10c^3R_i0(^4_^9£Z2R2-18fZMl+9rf^+4rfR2-, 

 8 



35 



_ _ _^ 



And P^^+P<^^+(^c.=»" _fZ-»-10^^4-10cZ^R+9<Z^-18(i3jj+ 



9^R2-4cZ'' + 1 2d'il-\2d%il'^ + 4^11^ -j- <f4_4c^ ^a _[- 6£/2K2_4Q'li3 + R^ 



^n-lR.*+M^R^-\-^d\ 

 8 



Henc€, S • P/?"'+P^' + ^'C.= 8 • VH^+V'S^'-yc. = 8nR-*4. 

 24ncZ^R=^+3nrf^ 



And 8 •F/?"^+P^"m^^. = 8- VHM-"VSM^4^. = 8nR^-K 

 24/^^2a24-3n<^^ 



Case II. (Fig- It.) Let the point V be taken without the 

 figure ; and the same construction be made as before. 



It may be shewn as before, that VH=P^, and VS=P5', and 

 so of all the rest. 



It is manifest also, that when the sides of the figure produced, 

 fall between the assumed point and the figure, the lines P/>, P«^, 

 Sic. will fall between P and V ; but, that otherwise, they will 

 fall on the other side of P. 



Hence, VH=Pp=VP cDYp^Yp (i> {d-V.) ; and A'S=P7= 

 VP\y> Yq=zYq(J> (<?— R) ; and so of all the other perpendicu- 

 lars. 



Then, Vp^=[Yp (J) d-\\Y=Yp'-^Yp' • (J-R) + 6Y/- 

 (^_R)2_4Vp- (rf-R)2+ {d-RY', and Vq*=:Yq^-^Yq^' 

 (V_R)4.6Vf/ • {d-Kf-i^Yq ' {d-RY + (fZ-R)-* J and so of 

 all the rest. 



Hence, P^?" + Vq^ + &^c. =V/ + Yq*^ &:c.-4 (^Z-R) • 



