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1914]                   HIGH  ORDER TO THE WHISPERING  GALLERY                      217
Since ka is supposed to be decidedly less than n, Dn and Dn' are given by (18), (20); and, if we neglect the imaginary part,
= ~sinh/3............................(33)
Dn (ka)
Thus (32) becomes              -iM^-r = — ^~r sinh /3.   ........................(34)
Jn(ha)        a-k                                          ^    '
the right-hand member being real and negative. Of this a solution can always be found in which ha — n very nearly. For* Jn (z) increases with z
from zero until 0 = w + '8065?r, when Jn'' (z) — Q, and then, decreases until it
vanishes when £ = n + l'8558^. Between these limits for z, Jn'/Jn assumes all possible negative values. Substituting n for ha on the right in (34), we get
- -—sinh/3,    or    — — tanh/3,    ..................(35)
crn                                     cr
while cosh /3 = p. The approximate real value of ka is thus n simply, while that of ka is n/fj,.
These results, though stated for aerial vibrations, have as in all such (two-dimensional) cases a wider application, for example to electrical vibrations, whether the electric force be in or perpendicular to the plane of r, 0. For ordinary gases, of which the compressibility is the same,
p/cr = /is//c2 = /A2.
Hitherto we have neglected the small imaginary part of Dn'/Dn. By (18), (20), when z is real,
T)   ' (r\                     9a-t J_ n at
•Vn V>)___

approximately, with cosh J3 = n/'z. We have now to determine what small imaginary additions must be made to ha, ka in order to satisfy the complete equation.
Let us assume ha = x + iy, where so and y are real, and y is small.   Then approximately
and           .....    Jn" (>
Since the approximate value of x is n, Jn" is small compared with Jn or Jn', and we may take
"••JSU^iv)    J»'(x) L
Jn (as) See paper quoted on p. 211 and correctionnormal motions requires that