682 BELL SYSTEM TECHNICAL JOURNAL 



and we meet the problem of finding modes of vibration and stationary 

 wave-patterns. 



If -E is supposed positive, the wave-speed is everywhere real. 

 Boundary-conditions of the usual sorts {e.g., the prescription that the 

 fluid shall be confined within a rigid spherical wall of given radius) 

 might be imposed, and then Eigenwerte of the constant E could be 

 calculated, and from these the wave-patterns and natural frequencies 

 of the fluid. If no such boundary-conditions were prescribed, the 

 equation (172) could be solved with any value of E. 



If E is supposed negative, the whole state of affairs is changed. 

 The wave-speed is now real within the sphere of radius — e^/E, zero 

 over this sphere and imaginary beyond it. This recalls the case of 

 the "string" proposed as an analogy for the linear oscillator, for which 

 the wave-speed was real along its central segment and imaginary from 

 each end of its central segment onwards to infinity. There are im- 

 portant differences: in the present case, the variable r assumes positive 

 values only, and the wave-speed at r = is infinite though real. 



In the case of the "string" with variable and at some points 

 imaginary wave-speed, we found that the law of variation of wave- 

 speed could be so chosen that the "string" enjoys a natural mode of 

 vibration with a stationary wave-pattern and a natural resonance- 

 frequency. This was done by selecting any of a series of Eigenwerte 

 for a parameter of the differential equation. Here we shall do likewise. 



Essaying for the function ^ in (172) a solution in the form of a 

 product of a function of 6 and <p exclusively by a function of r exclu- 

 sively, we arrive in the familiar way at differential equations: 



cosec dl4-A sin ^^ ) + :^ ( cosec 9^)] = - \Y. (175) 

 [ad \ ad / dif \ dip ] ] 



The equation (175) is the identical one which we encountered in the 

 case of the ball of fluid. Here, as there, the fact that the variables d 

 and ip are cyclic requires Eigenwerte of the constant X: 



X = /(/ + 1), / = 0, 1, 2, 3, 4, ••• (176) 



Equation (174), however, is not the same as the corresponding equation 

 (133) of the prior case ; here we find the difference between the fluids of 

 actual experience and the "imaginary fluid" which is to serve as ma- 

 terial for the atom-model supplied by wave-mechanics for hydrogen. 

 If in that equation (174) one were to assign an arbitrarily chosen 



