676 BELL SYSTEM TECHNICAL JOURNAL 



This function vanishes, entailing the vanishing of the wave-motion, 

 at an infinity of discrete values of the variable mr: — the roots of the 

 function, which I denote in order of increasing magnitude by B^, B^, 

 B^, ' • '. In an infinite medium we could assign any value whatever 

 to r, and then there would be an infinity of nodal spheres, their radii 

 given by B^/m, B'^/m, B^/m, •••. If the medium is bounded by a 

 rigid spherical wall of radius R, the coefficient m must possess one of 

 the values B^R, so that one of the nodal spheres may coincide with 

 the wall. These are the Eigenwerte of the constant m, and the natural 

 frequencies of the corresponding vibrations are given by B^u/IttR. 

 The Eigenfunktionen are given by the equation (136) with the various 

 values B^IR substituted for the parameter m. 



The Eigenfunktionen of the fundamental differential equation for 

 the fluid sphere are, therefore, each a product of a radius-function 

 given by (136), with a "permitted" value for the constant m deter- 

 mined by the boundary-condition; an angle-function given by (135), 

 with "permitted" values for the constants n and s, determined by the 

 fact that the angles are cyclic variables; and a time-function given by 

 (132), with a "permitted" vibration-frequency determined by the 

 boundary-condition. Each Eigenfunktion with the indices in, n, s 

 describes a mode of vibration, in which the fluid sphere is divided into 

 compartments by 5 meridian planes, (n — s) double-cones, and a 

 certain number of spheres, upon each of which the fluid is perpetually 

 at rest; within the compartments, it vibrates with a prescribed fre- 

 quency. 



Atom-Models in Wave-Mechanics 



Case of a "String" for which the Wave-Speed is Variable, or even 



Imaginary 

 Thus far I have used the images of the stretched string, the tensed 

 membrane, and the elastic fluid to illustrate the behavior of the 



differential equation 



w2V2^ = d^^ldt\ (151) 



when the coefficient w- is a positive constant. In these examples w^ 

 is interpreted as the ratio of the intrinsically positive quantities 

 "tension" (or "pressure") and "density," and turns out to be 

 equal to the square of the speed of propagation of sine-waves in the 

 string, membrane, or fluid. In certain problems of undulatory 

 mechanics we encounter just such an equation. In some of the 

 most important applications of Schroedinger's theory, however, one 

 meets with differential equations of the type of (151), in which however 

 the coefficient ti^ depends on the coordinates and even assumes negative 



