SECTIONAL TRANSACTIONS.—A*. B41 
Analogously in four dimensions, a hyper-sphere is divided into a number of equal 
hyper-spherical tetrahedra by a properly chosen set of 3-spaces through its centre. 
In a certain special case, it is possible to select points within alternate tetrahedra so 
as to form the vertices of a new polytope having regular bounding solids. This is 
the case when each tetrahedron is a right (triangular) pyramid, with dihedral angles 
4m at the basal edges and 47 at the lateral edges. The new polytope is hounded by 
24+ 96 regular tetrahedra and 24 icosahedra; and its 96 vertices, along with the 
24 vertices of a regular ‘ 24-cell’, make up the 120 vertices of a regular ‘ 600-cell’. 
A further extension of this process leads to a four-dimensional space-filling, in 
which each vertex is surrounded by five ‘ 5-cells’ (regular simplexes), one ‘ 16-cell’ 
(eross polytope), and four specimens of the new polytope. 
Mrs. Boole Stott has shown that the 96 vertices of the new polytope lie respectively 
in the 96 edges of a 24-cell (reciprocal to the above-mentioned 24-cell). Each edge 
of the 24-cell is divided in the ratio t : 1, where t?=t+1. The 24 bounding icosahedra 
of the new polytope are inscribed in the 24 bounding octahedra of the 24-cell. 
The vertices of the new space-filling divide, in the same ratio, the edges of the 
regular space-filling of 24-cells. 
Mrs. Stott’s drawings were exhibited, of two solid sections of the new polytope. 
One of these was coloured for comparison with her model of the corresponding section 
of the 600-cell. 
Mr. M. H. A. Newman.—Topology and Continuous Groups. 
Among the ‘ Mathematical Problems’ enumerated by Hilbert in 1900 was that 
of showing that the analytic character of a continuous group follows from its continuity 
alone. For two-dimensional groups this problem was solved in 1910 by Brouwer ; 
he showed that from the mere continuity of the transforming function it follows 
that the group is one of those enumerated by Lie (who had of course assumed analytic 
character). Since then hardly any progress has been made with the extension to 
n dimensions. 
The recent advances in topological technique, and especially the fixed-point 
formula of Lefschetz, have provided the materials for a new attack on the problem, 
and the truth of Hilbert’s conjecture can now be asserted when the group is Abelian 
and the group-space compact: such a continuous group is necessarily the closed 
translation group : 
2,=2,+4, ((=1,2, ..,n). 
Prof. H. W. TurnspuLi.—Canonical Matrices and Matrix Equations. 
The object of this communication is twofold: (1) to give examples from a wide 
range of subjects in which various authors have utilized matrices in solving their 
problems; (2) to suggest certain lines along which the future advance in matrix 
manipulation is likely to be made. 
Examples are taken from the algebraic theory itself, besides its applications to 
differential equations, calculus, analytical geometry, electricity and theory of statistics. 
The fundamental réle played by functions f(z, y) of two matrix arguments 2, y 
in the study of functions of n matrices 1s indicated. 
‘> 
s¢ 
oP Yee 
Friday, September 25. 
- Miss M. L. Carrwrieut.—Integral Functions of Integral Order. 
4 
An integral function f(z) =f (re) is said to be of order 0 if 
lim log log M (r) 
————— =f 
t=>E9 log r 
where M(r) = max | f(rei#) | . Further, if f (z) is of order p and if 
lim log M(r) _ 4 
t>co 7P 
then f(z) is said to be of order pe and of minimum, mean, or maximum type 
according as A=0,A>0, or A=co. Suppose that f(o)=1, and let z, =r, et@', - 
5 
=t 
fell 
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