CONTEMPORARY ADVANCES IN PHYSICS 685 



The function X„, i{p) has {n — I — 1) roots, so that the corresponding 

 mode of vibration has (n — I — 1) nodal spheres. To each permitted 

 energy-value En there consequently correspond n different solutions 

 of the general equation (172), differing from one another in respect 

 of the number of nodal spheres: 



^n,iir, e, 4>) = Xn,i{p)Y,(d, ^); / = 0, 1, 2 ••• (w - 1). (181) 



Each of these describes a permitted class of modes of vibration, 

 owing to the subdivision of the spherical harmonic Fi into terms 

 according to (135). 



Allowing for the subdivision of the spherical harmonics, there are 

 (1 + 2 + 3 + • • • w) = w(w + l)/2 modes of vibration for the wth 

 permitted energy- value En. 



The equation (181) exhibits the various modes of vibration of 

 which our imaginary "fluid," the model for the hydrogen atom, is 

 capable. It would be possible to describe these with a wealth of 

 verbal detail. I hesitate to do so; for vast amounts of industry and 

 ink have been expended during the last twelve years in tracing and 

 describing electron-orbits, which are now quite out of fashion; and 

 who dares afifirm that in another five years the vibrating imaginary 

 fluid will not be demoded Yet it is altogether probable that for some 

 years to come, if not for all time, the image of the vibrating fiuid will 

 furnish the customary symbolism for expressing the data of experi- 

 ment. Therefore let me point out some features of the vibrations 

 corresponding to the first (or "lowest," or "deepest") three states 

 of the hydrogen atom : 



Normal State, n = \. One Eigenfunktion, Xi^o(p); an exponential 

 function of r, decreasing steadily from the origin to infinity, with no 

 nodal spheres. Corresponding spherical harmonic Yo{d, (p), — a con- 

 stant. The vibration consequently is described by 



^(r) = const, e-'^'^ (ao = Ji'l^Tr-me') (182) 



and is endowed with perfect spherical symmetry. 



First Excited State, n = 2 (the state into which the atom relapses 

 after emitting any line of the Balmer series). Two Eigenfunktionen 

 X2, and X2, 1 ; the first represents a vibration with a single nodal 

 sphere, the second a vibration diminishing steadily in amplitude from 

 the origin outward. The first is to be multiplied by Yq{6, <p) to obtain 

 the complete description of the vibration; Fo being a constant, this 

 mode is endowed with perfect spherical symmetry. The second is 

 to be multiplied by Fi, which is a combination of terms written out 



