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1911]             SCHLOMILCH'S THEOREM IN BESSEL'S FUNCTIONS                  25
is arbitrary over the whole of the radius. And this motion will maintain itself without external aid if outside r = TT the initial condition is chosen in accordance with (6), F (%) for values of £ greater than TT being determined by (4). A similar statement applies of course to the vibrations of a stretched membrane, the transverse displacement w replacing s in (5).
Reference may be made to a simple example quoted by Whittaker. Initially let/(r) = r, so that from 0 to TT the form of the membrane is conical. Then from (12), (14), (15)
7T2                                                                     2
IO--T,    &» = 0 (n even),    bn = -~(n odd) ;
J&                                                                                                                                                      IV
and thus
...... (16)
the right-hand member being equal to r from r = 0 to r = TT.
The corresponding vibration is of course expressed by (16) if we multiply each function J"0 (nr) by the time-factor cos nt.
If this periodic vibration is to be maintained without external force, the initial condition must be such that it is represented by (16) for all values of r, and not merely for those less than TT. By (11) from 0 to TT, F{%) = ^IT%, from which again by (4) the value of F for higher values of % follows. Thus from TT to 27r, -F(£) = ^TT (27r- £); from 2?r to STT, F(%) = fatf- 27r); and so on. From these / is to be found by means of (6). For example, from
7T tO 27T,
/•sine=7r/r                      /"8infl=l
/ (r) = r            sin 6 d6 +\           (2-jr - r sin 6) dB
'         ='
= r - 2 V(^2 - 7T2) + 2-7T cos"1 (vr/r),  ........................ (17)
where cos"1 (TTJT) is to be taken in the first quadrant.
It is hardly necessary to add that a theorem similar to that proved above holds for aerial vibrations which are symmetrical in all directions about a centre. Thus within the sphere of radius TT it is possible to have a motion which shall be strictly periodic and is such that the condensation is initially arbitrary at all points along the (7). Now if 6 be the angle between the normal to this plane and the radius vector R, r = JR, sin B, and the mean is