126 Mr. A. Cayley on Poinsot's /owr new Regular Solids. 



I form now the following Table, comprising as well the ordi- 

 nary five figures as the new ones of Poinsot, and where we have 



H, the number of faces. 



S, the number of vertices. 



A, the number of edges. 



n, the number of sides to a face. 



n', the number of sides (angles) at a vertex. 



e, viz. the angles at a vertex make together e times four right 



angles. 

 e', viz. the angles which the sides of a face subtend at the 



centre of the face make together e' times four right 



angles. 

 E, viz.' the faces make together E times the spherical surface, 



the area of a stellated face being reckoned (as by Poinsot), 



each portion being taken once only. 

 D, viz. the faces make together D times the spherical surface, 



the area of a stellated face being reckoned as the sum of 



the triangles, having their vertices at the centre of the 



face and standing on the sides. 



The Table is— 



where the figures which are polar reciprocals of each other are 

 written in pairs : viz. as is well known, the tetrahedron is its 

 own reciprocal, the hexahedron and octahedron are reciprocals, 

 and the dodecahedron and icosahedron are reciprocals; more- 

 over the great stellated dodecahedron and the great icosahedron 

 are reciprocals, and the small stellated dodecahedron and the 

 great dodecahedron are reciprocals. The number which I have 



I 



