340 SECTIONAL TRANSACTIONS.—A, A*. 
The observations of P were found to fit the tables of Jeffreys better than those 
of Zéppritz or of Mohoroviéic. 
For distances greater than 85°, S.P.S was found in all the records. It was a well- 
marked phase, and the time-curve was well determined in all cases except 22 TIT 1928 ; 
here it was present but not clearly marked. 
S, was well defined except in the case of 13 XI 1925. 
S.P.S—P was found to be greater for 18 VII 1928 than for the two earthquakes 
for which greater distances were considered ; thus the time-curves differed. 
In all the records of 24 X 1930, P was very small or absent, whereas PP and SS 
were large and well marked phases. In the records of 18 VII 1928 and 13 XI 1925, 
P was clearly marked, but PP and SS were not very conspicuous. For the epicentres 
adopted the ranges of distance considered in the three cases overlapped. Thus 
the ratio of amplitudes PP : P differed greatly for distances which appeared to be the 
same. The epicentres could not be very accurately determined, but various con- 
siderations showed that it was almost certain that the great differences which were 
observed in PP: P actually occurred for the same distance. 
Prof. da Costa Loso.—wWNew theories of Physics, resulting from the phe- 
nomena of Radio-activity. 
DEPARTMENT OF MATHEMATICS (A*). 
Thursday, September 24. 
Prof. D. M. Y. Sommervitie.—Isohedral and Isogonal Generalisations of 
the Regular Polyhedra. 
In Max Briickner’s discussion of isohedral (gleichfldchig) and isogonal (gleicheckig) 
polyhedra (Vielecke und Vielflache, pp. 140 ff.) two faces of a polyhedron are defined 
to be equal when they are either directly or inversely congruent and the dihedral 
angles at corresponding edges are equal; two vertices are equal when the spherical 
polygons, which they form on unit spheres with centres at the vertices, are either 
directly or inversely congruent and the lengths of corresponding edges are equal. 
With these definitions all possible isogonal polyhedra are obtained by truncating the 
corners and edges of the regular n-sided prism, the octahedron and hexahedron, and 
the icosahedron and dodecahedron, in a way similar to that in which the semi-regular 
(Archimedean) bodies are obtained from the regular polyhedra ; the isohedral polyhedra 
are the polar reciprocals of these. An isogonal polyhedron has always a circumscribed 
sphere, and an isohedral polyhedron has an inscribed sphere. 
Among these isogonal and isohedral polyhedra there occur generalisations of the 
regular tetrahedron, hexahedron and octahedron, the tetrahedron (with opposite 
edges equal) being both isogonal and isohedral; there occur also isogonal icosahedra 
and isohedral dodecahedra, but not isohedral icosahedra or isogonal dodecahedra 
(except the regular ones). 
It is shown in this paper that, if the conditions of equal corresponding dihedral 
angles be omitted, there are isohedral icosahedra; and, if the condition of equal 
corresponding edges be omitted, there are isogonal dodecahedra. These in general 
have neither a circumscribed nor an inscribed sphere. 
Mr. H. 8. M. Coxerer.—A new Uniform Polytope in Four Dimensions. 
The cube has nine planes of symmetry: three parallel to faces, and six through 
pairs of opposite edges. These nine planes divide a concentric sphere into 48 spherical 
triangles, each of angles }7, 47,47. We can suppose these triangles to be alternately 
‘white ® and ‘black’; each white triangle being surrounded by three black ones, 
and each black by three white. It is possible to select a point within a white triangle 
in such a way that the points similarly situated in all the 24 white triangles are the 
vertices of a polyhedron whose faces are regular. This polyhedron is the snub cube, 
one of the thirteen ‘ Archimedean solids’. The snub dodecahedron can be derived 
similarly from a different net of spherical triangles. 
