I 
FOUR-DIMENSIONAL TUBES 
155 
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Nativity, Iiasrjij 
cental principle’ 
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>stitiitions.” 
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xi. 4] 
“ It must be pointed out that such things as length, velocity, 
energy, momentum, are not absolute, but relative, i.e. they are 
not attributes of the physical reality, but relations between this 
reality and the observer. Consequently the laws of conservation 
are not laws of the real world, like the law of gravitation, but 
of the observed phenomena. There is, however, one law which, 
already, before the days of relativity, had come to be considered 
as the most fundamental of all, viz. the principle of least action. 
Now action is absolute. Accordingly this principle retains its 
central position in Einstein’s theory. It is even more funda 
mental than the law of gravitation, since both this law, and the 
law of motion, can be derived from it. The principle of least 
action, so far as we can see at present, appears to be the law 
of the real world.” * 
The connection between this principle and the quantum 
theory has been emphasized by Jean Becquerel,f who points 
out that the quantum is an element of action and consequently 
the theory of relativity demonstrates that Planck’s constant is 
an invariant independent of the system of reference. 
4. Tubes of Force in Four Dimensions 
At the Edinburgh meeting of the British Association in 1921, 
Prof. E. T. Whittaker J pointed out that the ordinary electrostatic 
and magnetic tubes in three dimensions were dependent on the 
relative state of rest or motion of the observer, and went on to 
discuss the properties of electromagnetic tubes of force in four 
dimensions. The author suggested the use of these tubes in 
quantum theory. Such a tube, or “ calamoid,” would involve 
both the electric and the magnetic vector and would satisfy all the 
requirements of the relativity theory. “ Further, in the four 
dimensional world it is action, not energy, which is conserved, 
so that the field appears open for a direct application of the 
quantum principle. The experimental physicist may feel some 
what appalled at the prospect of such a solution of his difficulties, 
but it may yet be necessary to invoke a four-dimensional tube 
of force as the unit brick from which a universe may be 
constructed.” § 
We proceed to consider further the details of this forecast. 
Let x, y, z, t denote the co-ordinates of a point in the four 
dimensional hyperspace. The word surface may be used to 
* W. de Sitter in Bird’s Relativity and Gravitation, p.217 (Methuen). 
t Jean Becquerel, Le Principe de relativité et la théorie de la gravitation, 
P- 93- 
l E. T. Whittaker, Proc. Roy. Soc. Edin., vol. 42, p. 1, 1921. 
§ H. S. Allen, Nature, vol. 108, p. 341, 1921.