742 MR DUNCAN M. Y. SOMMERVILLE ON 
this the fundamental variety. Then all the other varieties can be obtained from it by turning 
some of the groups through 5 Let us denote this operation by R. . In the fundamental variety 
the twelve pentagons occupy relatively the same positions as the meshes of the dodecahedral 
network, so that with respect to one of the groups the others can be divided into three sets: 
5 adjacent, 5 circumjacent, and 1 opposite. Now suppose the operation R to be performed upon 
one of the groups. This gives a variety 8, (fig. 37). Next suppose a second group to be operated 
upon. The adjacent ones cannot be moved, for the first operation has destroyed their symmetry. 
Operating upon the opposite one we get a variety @, (fig. 38), while operating upon one of the 
circumjacent groups we get a fourth variety y, (fig. 39). From , we cannot obtain any further 
variety, for each of the remaining groups is adjacent to one of those already operated on. From 7, 
we can obtain a fifth variety, y. (fig. 40), by turning either of the two groups which are circum- 
jacent to both. In f, and , pairs of quadrilaterals occur, 5 in the former, 10 in the latter. In 
y, and y, there occur respectively 1 and 3 groups of three quadrilaterals. 
§ 25. To every spherical network there corresponds a convex polyhedron whose 
vertices are the nodes of the network. The polyhedra which correspond to the semi- 
regular networks have for their faces regular plane polygons. These form only a class 
of convex polyhedra in general, but they are the only ones whose faces may be regular 
polygons, and which, at the same time, may be inscribed in a sphere. 
If we examine the numbers of the several polygons in the various natok above 
we find that, with the exception of the two infinite series, they can all be’ connected 
with the regular networks. ‘The series with two quadrilaterals and an n-gon at each 
point corresponds to a series of right prisms on a <8 polygonal base, the altitude 
diminishing indefinitely as 1 increases. 
The polyhedra corresponding to the other types. can’ be obtained from the regular 
polyhedra by cutting off the corners in particular ways: Thus the octahedron (fig. 23) 
bounded by triangles and hexagons can be obtained from the regular tetrahedron by 
cutting off the corners, either triangles or hexagons corresponding to vertices, according 
to the depth of the section. When the numbers of the polygons are the numbers of 
faces, lines or vertices of a regular polyhedron; it is evident in what way they corre- 
spond. In some, however, the same kind of polygon may correspond to both edges 
and vertices, then its number has to be divided into two parts, each a multiple of the 
number of edges or vertices of the regular polyhedron. 
This holds only for the unique types and the fundamental varieties of the other 
types, 7.e. those in which no two polygons of the same kind are adjacent. The other 
varieties may or may not be obtainable from the corresponding regular polyhedron. 
Those of Class B. are not, while the four derived varieties of Class C. II. may still be 
obtained from the regular dodecahedron, since the positions of the pentagons are 
unchanged. 
§ 26. We may therefore group the simple types in three divisions according to 
their morphology.* We shall use the notation 3,'6,' to denote a simple type consisting 
* A correspondence between the regular polyhedra and certain general classes of polyhedra was considered by 
C. Jordan, “ Recherches sur les polyédres,” Comptes Rendus, 1x. 400-408, lxi. 205-208, Ixii. 1339-1341, 1865-66. 
