Waves Associated with Moving Corpuscles

Abstract

BY appropriate general treatment, it may be shown that solutions involving the path time are invariant, that is, can exist in any set of space co-ordinates. Starting, then, with the solution for the electron velocity ripple in terms of the transit half-angle¹, it may be shown, by analogy with van der Pol's treatment of waves in n dimensions², that this ripple satisfies certain conditions. The most important of these is that the ripple satisfies a five-dimensional wave equation, the wave velocity being that of the particle, provided the frequency of the ripple is such that θ « √2, where ç is the transit half angle (= ωτ/2). It will be convenient to regard this ripple as that part of the particle vibration which corresponds to free, as distinct from forced, oscillation. The factor ½ must be associated with the fifth dimension, as follows: where c₅ is the velocity appropriate to the waves.