FOUR-DIMENSIONAL TUBES 
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XI. 6] 
It will be evident that we have done no more than touch upon 
the fringe of a region as yet unexplored. The experimental 
physicist finds a difficulty in working in this unfamiliar world, 
but it is clear that to the mathematician there is a wide field of 
investigation open in examining the properties of such discrete 
tubes of force in four dimensions. In particular it is desirable 
to study the character of the calamoids in the immediate neigh 
bourhood of an electron or magneton. It will be necessary to 
take into account both “ electricity ” and “ magnetism/' but we 
may recall in this connection McLaren’s conception of matter 
as a “ hole in the aether,” i.e. a region where the electromagnetic 
field equations cease to apply. 
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6. A Five-Dimensional Theory 
In the general theory of relativity there is an apparent dualism 
of gravitation and electricity. It has been suggested that the 
inter-connection between the electromagnetic and the gravita 
tional field may be considered from a single standpoint by 
employing a five-dimensional space. Kaluza * has given a theory 
in which the ten Einstein gravitational potentials and the four 
electromagnetic potentials are expressed in terms of the 
coefficients of the line element in a Riemannian space of five 
dimensions. The equations of motion of electrical particles in 
an electromagnetic field take the form of the equations of the 
geodesics. Klein | has developed this theory and shown that 
it may be put into relation with the undulatory form of the 
quantum theory given by de Broglie and Schrôdinger (Chapter 
XVII). He further suggests that the origin of the quantum is 
to be found in the periodicity of the fifth dimension thus intro 
duced, and that the atomicity of electricity may be interpreted 
as a quantum theory law. 
Although such a mode of representation may be found con 
venient by the mathematician, it does not appear that it is able 
to throw much light on the physical aspect of the interpretation 
of the quantum. 
* Kaluza, Preuss. Akad. Wiss., Berlin, Ber., vol. 54, p. 966, 1921. 
t Klein, Zeits. f. Physik, vol. 37, p. 895, 1926; Nature, vol. 118, p. 
516, 1926. 
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