Mr. A. Cayley on Poinsot's foia- new Regular Solids. 127 



called D is reciprocal to itself; this is not the case for Poinsot's 

 E ; and I have not been able to define E in such a manner as to 

 enable me to form the definition of a reciprocal number E' : this 

 may be possible, but in the mean time it seems better to dis- 

 card E altogether, and use instead of it the number D. 



Euler's well-known relation applying to ordinary polyhedra is 

 S + H = A + 2. 



Poinsot in his memoir has (by an extension of Legendre's de- 

 monstration of Euler^s theorem) obtained the more general rela- 

 tion, eS + H = A + 2E, 



which, however, does not apply to the two stellated figures where 

 e' is different from unity ; the general form is 



eS + e'H = A-f2D, 



which applies to all the nine figures. This applies to all poly- 

 hedra, regular or not, which ai'e such that e has the same value 

 for each vertex, and e' the same value for each face. To prove 

 it, we have only to further extend Legendre's demonstration. 

 If for any face, stellated or not, the sum of the angles is s, and 

 the number of sides n, then, according to the foregoing mode of 

 reckoning, the area of the face (measured in right angles) is 



s + 4e' — 2?j. 

 Now the sum of all the faces is D times the spherical surface, 

 = 8D. But the sum of the term s is equal to the sum of the 

 angles about each vertex, =4eS; the sum of the term 4e' is 

 = 4e'H, the sum of the term 2n is four times the number of 

 edges, =4A. Hence 4eS + 4e'H-4A = 8D, or eS-l-e'H = 2D. 

 I remark that the small stellated dodecahedron and the great 

 dodecahedron are descriptively the same figures, and that, if we 

 repi'esent the vertices by a, b, c, d, e, f, g, h, i, j, p, q, and the 

 faces by A, B, C, D, E, F, G, H, I, J, P, Q, then the relations 

 of the vertices and faces is shown by either of the following 

 Tables :— 



