102 Note on the Porism of the in-and-circumscribed Polygon, 



Suppose the conies reduce themselves to circles, or write 

 U=# 2 + y 2 -R 2 =0 

 V=(a?-a) 2 + j, 2 -r 2 = 0; 

 R is of course the radius of the circumscribed circle, r the radius of 

 the inscribed circle, and a the distance between the centres. Then 



fU + V=(f+l, f+1, -ZRS-S' + a*, 0, -a, 0)(*,y, I)* 

 and the discriminant is therefore 



-« + l) 2 (P 2 + r 2 -fl 2 )-(f-fl)a 2 

 = -(1 + f)(r 2 + f(r 2 + R 2 --a 2 ) + £m 2 ). 

 Hence, theorem — 



"The condition that there may be inscribed in the circle 

 a? 2 -f y 2 — R 2 =0 an infinity of ra-gons circumscribed about the 

 circle {x — «) 2 -f y 2 — r 2 =0, is that the coefficient of f" 1 in the 

 development in ascending powers of f of 



\/(l + S)(r* + Z (r 2 + R 2 -a 2 ) + fR 2 ) 

 may vanish." 



Now (A + Bf + CP)*= 



or the quantity to be considered is the coefficient of f 71 " 1 in 



where, of course, 



A=r 2 , B=r 2 + R 2 -a 2 , C = R 2 . 

 In particular, in the case of a triangle we have, equating to 

 zero the coefficient of f 2 , 



(A-B) 2 -4AC=0; 

 or substituting 



(a 2 -R 2 ) 2 -4r 2 R 2 =0, 

 that is, 



(a 2 - R 2 + 2Rr) (a 2 - R 2 - 2Rr) = 0, 

 the factor which corresponds to the proper geometrical solution 

 of the question being 



a 2 -R2 + 2Rr = 0, 

 Eider's well known relation between the radii of the circles in- 

 scribed and circumscribed in and about a triangle, and the 

 distance between the centres. I shall not now discuss the mean- 

 ing of the other factor, or attempt to verify the formulae which 

 have been given by Fuss, Steiner and Richelot, for the case of a 

 polygon of 4, 5, 6, 7, 8, 9, 12, and 16 sides. See Steiner, Crelle, 

 vol. ii. p. 289 ; Jacobi, vol. iii. p. 376 : Richelot, vol. v. p. 250 ; 

 and vol. xxxviii. p. 353. 

 2 Stone Buildings, July 9, 1853. 



