the in-and-circumscribed Polygon. 101 



J. y Kj K 



J , K j fC 



i, &, k" 2 



the quotient being composed of the constant term C, and terms 

 multiplied by k, k\ k n ; writing, therefore, k = k' = k" = 0, we have 

 C = for the condition that there may be inscribed in the conic 

 U = an infinity of triangles circumscribed about the conic V =0 ; 

 C is of course the coefficient of f 2 in VDi;, i. e. in the square 

 root of the discriminant of fU + V; and since precisely the same 

 reasoning applies to a polygon of any number of sides, — 



Theorem. The condition that there may be inscribed in the 

 conic TJ = an infinity of w-gons circumscribed about the conic 

 V = 0, is that the coefficient of f M-1 in the development in ascend- 

 ing powers of f of the square root of the discriminant of f U + V 

 vanishes. 



It is perhaps worth noticing that ft =2, i. e. the case where 

 the polygon degenerates into two coincident chords, is a case of 

 exception. This is easily explained. 



In particular, the condition that there may be in the conic* 

 ax 2 + by 2 + cz 9 = 

 an infinity of w-gons circumscribed about the conic 

 x 2 + y 2 + z 2 = 0, 



is that the coefficient of f ra_1 in the development in ascending 



powers of f of 



V(\ + aZ){l + b1;){\+cP) 



vanishes; or, developing each factor, the coefficient of f"" 1 in 

 (l+ i«f-i«T+ ^« 3 ? 3 - 4«T+&c.)(l+ iif-fcc) 



(l + |cf-&c.) 



vanishes. 



Thus, for a triangle this condition is 



a 2 + b 2 + c 2 -2bc-2ca-2ab=0; 

 for a quadrangle it is 



a 3 + b 3 + (?-bc 2 -b 2 c-ca 2 -c 2 a-~ab 2 --a 2 b + 2abc=0; 

 which may also be written 



(b + c— a)(c + a — b)(a+b— c) = 0; 

 and similarly for a pentagon, &c. 



* I have, in order to present this result in the simplest form, purposely 

 used a notation different from that of the note above referred to, the quan- 

 tities aa?+by*+ cz 2 and aP+jJ*+& being, in fact, interchanged. 



