64 GEOMETRICAL RESEARCHES, 



Aj A 2 A M Aj , it follows that by putting 5" to represent the 



rectare of this %'gon, we will have f 2' = 2 rectare 

 OY B W B I + a 2 'B 1 B 2 + .... + V B w -i B w ) ; and therefore 

 rectare (^ - B M B I + ^ B 4 B 2 + + a w B^ BJ = 



s - i y. 



And since the squares of diameters of circles are four times the 

 squares of the radii, it is evident this last equation may be written : 



(A 1 0)2.sin(a 1 -B M B 1 ) + (A 2 0) 2 .sin(a 2 -B 1 B 2 ) + + (A, 0)'. 



sin (a n ' B n _ x BJ = 8 2 - 4 J'. 



But if we assume any point p in the plane, and draw the 



perpendiculars p Cj , p C 2 , p C n , to the respective sides 



Aj A 2 , A 2 A g , A n Aj ; and that we find the centres 



c , c , c , c , of the circles which circumscribe the quadri- 

 laterals p C n A 1 Cj , ^ C t A 2 C 2 , / C^^j A M C n : it is evident 



we have 



angle (^ B M B x ) = angle (c x C n 0^. 



angle (^ B x B a ) = angle (c a Cj O a ). 



angle (a, . B^ BJ = angle (c n - 0^ CJ. 



Hence it is obvious we may write the last general equation in 

 in the form (\ O) 2 sin (^ C % C x ) + (A 2 O) 2 sin (c a ' C x 2 ) 

 4- .... + (A n O) 2 . sin (C M . C^ C n ) = 8J - '. 



And from this we learn that the locus of the point O is a 

 known circle. Moreover, it is evident (see Salmon's Conic 

 Sections, pages 88 and 89) that if M indicates the centre of this 

 circle, and that p 2 represents the square of its radius, then we can 

 write the preceding equation under the form 



p* (sinc 1 .C |i 1 + sin V C i C 2 + 

 + (AjM) 2 . sin Cl . C n C x + (A 2 M) 2 . sin c g . Oj O 



