684 BELL SYSTEM TECHNICAL JOURNAL 



are due chiefly to the fact that the system is mathematically "de- 

 generate." Owing to this circumstance there are more than one pos- 

 sible mode of vibration, more than one stationary wave-pattern, for 

 each (except the first) of the permitted energy-values. To describe 

 these it is necessary to consider both of the equations (174) and (175). 



Since the equation (175) is identical with the corresponding equation 

 derived for balls of actual physical fluids, the modes of vibration for 

 Schroedinger's atom-model are identical with the modes of vibration of 

 actual fluid spheres insofar as the dependence on angle is concerned. 

 The imaginary fluid is divided into compartments by nodal planes, 

 nodal double-cones and nodal spheres; and the division by planes and 

 double-cones is identically such as we should find in the corresponding 

 mode of vibration of an actual fluid ball ; it is only the division by nodal 

 spheres which differs. 



To the first Eigenwert, Ex (n = 1) there corresponds a single Eigen- 

 funktion of equation (174); to the second, £2, a pair; to the third, 

 three; and so forth. This multiplicity is linked with the limitation 

 upon the Eigenwerte which was foreshadowed above. In the expression 

 for the function ^ as a product of functions of the individual variables 



^(r, 6, ^) = F{r)Yi{d, <p), (178) 



if we assign an Eigenwert En to the parameter E in the first factor 



according to (174), we have still a choice of values to assign to the 



parameter / in the second factor according to (176). This choice 



however is limited. We must not take any value of / as great as 



or greater than the value adopted for n; otherwise the value of En 



would not be an Eigenwert in the sense adopted. Thus for w = 1 we 



are restricted to the choice / = 0; for w = 2 we have the alternative 



of / = or / = 1 ; for n = 3 the option of w = 0, 1, or 2, and so forth. 



Each Eigenwert En thus admits (w — 1) distinct spherical harmonics 



Yi{d, (f), ¥^{6, <p) • • • Yn-\{d, (p) as solutions of equation (175); and 



to each of these there corresponds, with each of these there goes, a 



distinct Eigenfunktion Fn, i{r) of the equation (174), which is expressed 



... . , ... 2x-V— 2mEn 4ir'^me'^ 1 



as lollows m terms 01 a variable p = ; ;' = r-^- f = r 



h nil" nao 



instead of r to make the function seem less intricate: ^^ 



X..M = const. p.'.-/-|' (^'(» + ;_ J _^), (180) 



1' The factor in parentheses in equation (180) stands for the "number of combina- 

 tions of (n + I) quantities taken {n — I — 1 — k) at a time," which is the 

 (m — / — 1 — ^)th coefficient in the binomial expansion of (a + i)'+". 



