(ec aS 
XV. 
ON SERIES IN SPECTRA. 
By ARTHUR W. CONWAY, M.A., F.R.U.L, 
Professor of Mathematical Physics, University College, Dublin. 
[Read, Fesxuary 19; Received for Publication, Frpruary 26; Published, 
Marcu 25, 1907.] 
In the line spectra of various elements series of lines whose 
frequencies are connected by formule have been discovered. 
These formule have been put in various forms. A typical one 
is the formula of Kayser and Runge, in which the frequency 
of any line of the series is given by A + Bn* + Cn“, where n is 
one of the natural numbers 3, 4, 5, &., and A, B, and C are 
constants depending on the substance in question. For example, 
in the case of hydrogen, C = 0 and B = —- 44, which is Balmer’s 
formula. it is probable that these formule require other terms 
to complete them, and that they represent only the first three 
terms of the expansion of a certain function. 
According to modern ideas the atom is made up of positive 
and negative electrons in equilibrium, or in steady motion, and 
the spectrum is formed by the electromagnetic waves due to the 
motion of these electrons. The electrons may be sufficiently 
numerous to form a practically continuous body, or they may be 
comparatively few in number. In this paper an attempt is made 
to offer an explanation of a series on the supposition that it is due 
to the motion of one electron. A very slight modification will be 
required to deal with the case in which the number of electrons is 
small: for example, comparable with the atomic weight. 
In the first place, let us consider a field of force isotropic with 
respect to a fixed point, and such that the electric force is along 
the line joining the point in question to the fixed point, and equal 
to F'(r) where 7 is the distance from the fixed point. Then, if 
SCIENT. PROC. R.1).S.. VOL. XI., NO. XV- 2A 
