86 
[242 
242. 
SECOND NOTE ON POINSOT’S FOUR NEW REGULAE 
POLYHEDRA. 
[From the Philosophical Magazine, vol. xvn. (1859), pp. 209—210.] 
The Note on Poinsot’s four new regular Polyhedra (February Number, p. 123), 
[241], 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 projecting 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 + H = A + 2E, 
nor of course my further generalization, eS+e'H = A + 2D] but the latter part of the 
memoir relates to a generalization, in a different direction, of Euler’s original formula, 
S + H = A + 2 : viz. Cauchy’s theorem is—“ If a polyhedron is partitioned into any 
number of polyhedra by taking at pleasure, in the 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 $ + i/ = A+ P + l; that is, the sum of the number of vertices and the 
number of faces exceeds by unity the sum of the number of edges and of the number 
of polyhedra.” 
For P = l, we have Euler’s equation S±H=A + 2; and for P = 0, we have a 
theorem relating to the partition of a polygon; viz. if the polygon is divided into H 
polygons, and if S be the number of vertices, and A the number of sides, then 
S + H = A + 1; from which it is easy to pass to Euler’s equation, S + H = A + 2, for