SECTIONAL 'TRANSACTIONS —A*. 279 
Alternative possibility of obtaining unified theory of gravitation and 
electromagnetism by treating them as statistical properties of systems 
obeying a quantum theory. 
Dr. J. H. C. WuiTEHEaD.—Projective relativity (12.10). 
Generalised projective geometry, according to this presentation, depends 
upon the idea of a geometric object determined by sets of components and 
a transformation law. Whereas in affine geometry a geometric object 
(e.g. a scalar function or a contravariant vector) has one set of components 
in each coordinate system, a projective invariant has an infinity of sets of 
components. The transformation from one set of components to another 
in the same coordinate system can be explained in terms of a geometrical 
process analogous to projection in classical projective geometry. This 
explanation involves the use of an additional variable and, as when using 
homogeneous coordinates in the classical projective geometry of 7 dimen- 
sions, the formalism is that of (x + 1)-dimensional affine geometry. 
The power of this treatment is largely due to the closeness with which the 
formalism copies the (n + 1)-dimensional affine and Riemannian theories. 
In particular this applies to projective relativity, and if the ideas referred to 
above can be elucidated it should not be necessary to introduce a great deal 
of formal detail in the discussion of relativity. It will probably seem best 
to concentrate the formal work into a derivation of the equations of motion 
of a charged particle. ‘Taking for granted the formule which are obtained 
by the standard methods of Riemannian geometry, this should involve only 
a short calculation in the course of which many of the special features of the 
theory will be underlined. 
Monday, September 10. 
Prof. J. A. CaRROLL.— Some applications of Fourier transforms (10.0). 
(1) The equation 
+1 
oe) — a| I(z + Bt)g(t)dt : : oer er) 
i 6 
regarded as an integral equation for I(z), can be solved ‘ operationally’ by 
regarding z as an operator, and on writing down the equation of which (1) is 
the operational equivalent (image equation) and rearranging the terms, 
the operational form of the rearranged equation is the solution of (1), 
namely— 
ioe) 
Iu) = zi) "ote | aa O(2)\dz dx. « (2) 
2m 
where c is a suitably chosen path, and 
+1 +1 
t= | g(t)dt, G(x) =a| e-Ptx o(t)dt. 
a 
—1I —1I 
This enables the validity of solution of (1) by the elementary method of 
Taylor expansion of J(z +- Pt) and reversion of the series obtained to be 
tested. 
Solution of the form (2) is troublesome to use when O(z) is known 
numerically for real values of the argument only. If g(t) is assumed known, 
