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Allen, H.S. (1915) Numerical Relations between Electronic and Atomic Constants. Proceedings of the Physical Society (London), 27, 425-431.
https://doi.org/10.1088/1478-7814/27/1/331

has been cited by the following article:

• TITLE: Solutions of the Exponential Equation yx/y = x or lnx/x = lny/y and Fine Structure Constant

  AUTHORS: Sabaratnasingam Gnanarajan

  KEYWORDS: Exponential Equation, Lambert W Function, Fine Structure Constant

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.5 No.2, February 17, 2017

  ABSTRACT: In this paper, we study the equation of the form of which can also be written as . Apart from the trivial solution x = y, a non-trivial solution can be expressed in terms of Lambert W function as . For y > e, the solutions of x are in-between 1 and e. For integer y values between 4 and 12, the solutions of x written in base y are in-between 1.333 and 1.389. The non-trivial solutions of the equations and written in base y are exactly one and two orders higher respectively than the solutions of the equation . If y = 10, the rounded nontrivial solutions for the three equations are 1.3713, 13.713 and 137.13, i.e. 100.13713 = 1.3713. Further, ln(1.3713)/1.3713 = 0.2302 and W(-0.2302) = -2.302. The value 137.13 is very close to the fine structure constant value of 137.04 within 0.1%.