Mr. A. Cayley on Poinsot's /om* new Regular Pohjhedra. 209 



particular may very well account for what has beeu called latent 

 heat. 



It has been supposed in the theory, that the atoms from which 

 the secondary waves are propagated, and those on which they are 

 incident, are completely at rest. It is, however, evident that the 

 amount of reflexion must be in some degree influenced by the 

 mobility of the atoms, being less as the mobility is greater. 

 Hence, as it may be supposed that when a substance is passing 

 from a state of fluidity to a state of solidity the stability of the 

 atoms is increasing, it is conceivable that the total atomic repul- 

 sion during this state of transition may be on the increase, 

 although the temperature of the substance, and consequently the 

 intensity of the waves propagated from each atom, may be de- 

 creasing. In this manner the expansion of water, with a dimi- 

 nishing temperature near the temperature of conversion into ice, 

 may be explained. 



Cambridge Observatory, February 18, 1859. 



XXXIII. Second Note on Poinsot's four new Regular Polyhedra. 

 By A. Cayley, Esq.^ 



THE Note ou Poinsot's four new regular Polyhedra (February 

 Number, p. ]23) was written without my being acquainted 

 with Cauchy's first memoir, " Recherches sur les Polyedres " 

 {Jour. Polyt. vol. ix. pp. 68-86, 1813), the former part of which 

 (pp. 68-76) relates to Poinsot's polyhedra. Cauchy considers 

 the polyhedra, not as projected on the sphere, but in solido ; and 

 he shows, very elegantly, that all such polyhedra must be derived 

 from the ordinary regular polyhedra by producing their sides or 

 faces. The reciprocal method would be to produce the sides or 

 join the vertices; and, adopting this reciprocal method, and pro- 

 jecting the figure on the sphere, we have the method employed 

 by Poinsot, and explained and developed in my former Note. 

 Cauchy does not at all consider Poinsot's generalized equation, 

 eS + II = A + 2E, nor of course my further generalization, 

 eS-f-C''II = A + 2D j but the latter part of the memoir relates to 

 a generalization, in a difi'erent direction, of Euler's original for- 

 mula, S + H = A + 2: viz. Cauchy's theorem is — "If a polyhe- 

 dron is partitioned into any number of polyhedra by taking at 

 pleasure, in tlic interior of it, any number of new vertices, and 

 if P be the total number of polyhedra thus formed, S the total 

 number of vertices (including those of the original polyhedron), 

 and A the total number of edges, then S + H = A + P + 1; that 

 is, the sum of the number of vertices and the number of faces 



* Communicated by the Author. 



