418 



Prof. J. Casey on the Equations of Circles. [May 23, 



again, these results may be still further extended to conies having 

 double contact with a given conic. A very large amount of geometry 

 is embraced in the paper, and many subjects of much interest are dis- 

 cussed, showing the great fertility of the methods of investigation 

 employed. 



The following is an outline of the extensions of known theorems 

 which the paper contains. 



If a polygon of n sides, whose equations are a, /3, 7, &c, and whose 

 lengths are a, b, c, d, &c, be inscribed in a circle, the equation of the 



circle is a factor in the equation — + — +— + =0, or say in 



a ft 7 c 



: 0, •; 



and if it be on the sphere in the equation, 



The equation 2 ^^ = 0, when the polygon consists of n sides, will 



denote besides the circle a residual curve of the degree (n— 3). A 

 considerable portion of the paper is occupied with the discussion of 

 the properties of this curve. More especially in the particular case 

 n=6, where the residual curve is a cubic. The following theorem may 

 be given as an instance. " When the polygon is a hexagon the line 



5+£ + 2 + -+t+-=0, which is called its axis, is the satellite with 



abode/ 



respect to the cubic of each side of the hexagon, and also of its Pas- 

 cal's line." 



The properties of the inscribed polygon derived from the equations 

 may be reciprocated, and we get tangential equations of the form 



s( c -2ii^)=o, 



giving properties of circumscribed polygons. In this last equatio 

 A, B, C, &c, denote the angles of the polygon, and \, jul, v, &c, perpen- 

 diculars from its angular points on any tangent to the circle. The 

 same tangential equation holds both for the plane and sphere. 



If a polygon whose sides are a, ft, 7, S, &c. . . . to, be . circumscribed 

 to a circle, the equation of the circle is a factor in the polyzomal curve 



