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366                     THE THEORY OF  THE  HELMHOLTZ  RESONATOR                     [401
The first step is to find the velocity-potential (^) due to a normal motion at the surface of the sphere localised at a single point, the normal motion being zero at every other point.    This problem must be solved both for the exterior and for the interior of the sphere, but in the end the potential is required only for points lying infinitely near the spherical surface.    Then if we assume a normal motion given at every point on the aperture, that is on the portion of the spherical surface not occupied by the walls, we are in a position to calculate if- upon the two sides of the aperture.    If these values are equal at every point of the aperture, it will be a proof that the normal velocity has been rightly assumed, and a solution  is  arrived  at.    If the agreement is not sufficiently good  there is no question of more than an approximation  some other distribution of normal velocities must be tried. In what follows, the preliminary work is the same as in the paper last referred to, and the same notation is employed.
The general differential equation satisfied by ty, and corresponding to a simple vibration, is
where k = 27T/X, and X denotes the length of plane waves of the same pitch. For brevity we may omit k ; it can always be restored on paying atten tion to " dimensions." The solution in polar co-ordinates applicable to a wave of the 77th order in Laplace's series may be written (with omission of the time- factor)
^n = SnrnXn(r) ............................... (2)
The differential equation satisfied by %n is d2vn    2w + 2 d-vn
_ 0. _L.    __ _ __ Ai   i     .     A                                          /Q\
dr* +     r       dr+Xn~~() ...................... W
The solution of (3) applicable to a wave diverging outwards is
, ,    (     d \n tr* ^(r)=(-fTr)   ......................... W
Putting n= 0 and n = I, we have
It is easy to verify that (4) satisfies (3). For if %w satisfies (3), r^Xn' satisfies the corresponding equation for %n+1. And r~> e~* satisfies (3) when u = 0.
From (3) and (4) the following sequence formulas may be verified :
Xn(r) = -rXw(r\   ........................... (6)p. 87 (1904) ; Selent^e Papers, Vol. v. p. 141).On account of the magnitude of x we have only the one curvature to deal with. For this curvature