CONTEMPORARY ADVANCES IN PHYSICS 673 



nents, but as ordinal numbers so that I may reserve the subscripts 

 to distinguish the various Bessel functions from one another. The 

 function 



Z = J\{mr){C cos \9 + D sin \d){A cos mict + B sin mut) (129) 



represents a stationary oscillation of an infinitely extended membrane, 

 in which X lines intersecting one another at the origin are nodal lines, 

 and an infinity of concentric circles centred at the origin are nodal 

 circles. These lines and circles are motionless while the sections of 

 the membrane which they delimit vibrate with the frequency mtc/lir. 

 The X lines are spaced uniformly in angle; the radii ri, ^2, • • • of the 

 infinity of circles are obtained by dividing m into the roots bx^, bx^, 

 b\^, • • • of the Bessel function of order X, J\(mr). 



How then does the boundary-condition upon the finite membrane 

 enter in? Obviously, if a membrane of radius R be clamped at its 

 edge, and if it is vibrating in the manner described by (129), then the 

 edge must coincide with one of the nodal circles; the radius R must 

 be equal to one of the quantities b\^/m. Or rather, since the nodal 

 circles are to be adjusted to the size of the diaphragm and not the 

 size of the diaphragm to the nodal circles, the coefiicient m must con- 

 form to one of the equations : 



m = bx'/R, or b^VR, or ^xV^, •••• (130) 



These equations define Eigenwerte of the parameter m in the differential 

 equation of the tensed membrane. There is a double infinity of these — 

 an infinite series of them for each of the Eigenwerte of the parameter 

 X. To each corresponds a natural frequency of the membrane, and 

 to each corresponds an Eigenfunktion, the one written down in (129) 

 with the proper value of m taken from (130). The constants A, B, 

 C, and D in the Eigenfunktionen specify the amplitude of the oscilla- 

 tion, the phase of the vibrations at any given instant, and the orienta- 

 tion of the nodal lines with respect to any given axis. Any number of 

 Eigenfunktionen may coexist simultaneously; the actual distortion 

 of the membrane will be the superposition of all. Any initial condi- 

 tions imposed on z and z (and not involving discontinuities or infinities) 

 could be satisfied by adjusting the constants. 



Example of the Ball of Fluid 



Among the familiar vibrating systems the ball of fluid presents the 

 closest analogy to the atom-model for the hydrogen atom in wave- 

 mechanics, the wave-patterns in the two cases being strikingly alike. 



