SECTIONAL TRANSACTIONS.—A™*. 281 
Mr. E. A. MaxweLit.—Some examples in the theory of surfaces (11.10). 
The paper gives an illustration by example of certain general properties 
of surfaces. 
Denote by F*7(°C")? a surface of order 30, having as o-fold curve the 
rational quartic °C‘ (of order four and genus zero). ‘The canonical surfaces, 
defined in similar notation by F°%¢-4(°C*)¢-!, are invariant for birational 
transformation. Now the curve °C* lies on a unique quadric 9, meeting 
the two systems of generators in three points and one point respectively ; 
each ‘ three ’-generator necessarily lies on a canonical surface, which there- 
fore degenerates into 9, together with a variable part. ‘The curve of inter- 
section of © with F°¢ is Noether-exceptional (i.e. a fixed part of every canonical 
surface), and, in fact, consists of 20 straight lines, generators of 9. In 
accordance with general theory, these may each be transformed to a simple 
point of a birationally equivalent surface : 
The cubic surfaces through the curve °C* may be represented by the 
prime sections of a threefold locus V,°[6] of order five in six dimensions. 
The points of °C? correspond to the generators of a rational ruled surface 
R?° of order ten on V; the ‘three’-generators of » correspond to the points 
of aconic c. ‘The given surface corresponds to the surface of intersection 
of V by a primal of order o ; this latter meets c in 20 points, each of which 
corresponds to a Noether-exceptional curve. 
Similar results are given for the surface F%°(?C*)*, the only other surface 
of this type. 
Sir A. S. Eppincton, F.R.S.—Theory of electric charge and mass (11.45). 
The following principles (amongst others) are employed in the theory : 
(1) Indistinguishability.—A system of two particles No. 1 and No. 2 is 
described dynamically by giving as a function of the time the probability 
distribution of two sets of coordinates, q, g’, together with an interchange 
variable 9 such that cos?0 is the probability that the particle at g is No. 1 
When the particles are regarded as distinguishable (and always distin- 
guished without uncertainty) 9 is constrained to be zero. ‘The Coulomb 
energy of electrons and protons is the momentum conjugate to 0. 
(2) Metrical Tensor.—The tensor g,, giving the metric of macroscopic 
space must arise out of the fundamental conceptions of wave mechanics, 
and not as an extraneous datum. It is the energy tensor of the @ priori or 
standard probability distribution of the particles. Since this distribution 
itself provides the metric of the space in which it is represented, the space 
automatically appears as uniform and isotropic. 
(3) Idealisation—The most elementary equations of quantum theory 
which contain the definitions of charge and mass refer to highly idealised 
conditions. Formally the ideal uniform conditions prevail throughout the 
universe, since if the momentum of a particle is prescribed, its position 
in the universe is entirely unknown. ‘The equations therefore apply strictly 
to spherical space and to hyperspherical phase-space. 
(4) Curvature—Mass must arise out of curvature of space-time in 
- quantum theory as it does in relativity theory. Curvature (by rendering 
space finite) limits the possible uncertainty of position, and therefore gives 
a minimum uncertainty and corresponding minimum expectation value of 
momentum. 
(5) Comparison Distribution.—Observationally the effect of mass is mani- 
fested in the study of the combined probability distribution of a particle 
and a physical reference body, but mathematically mass is defined as a 
