1912]

ELECTEIQAL  VIBRATIONS  ON  A  THIN  ANCHOE-EING

113

The first integral in (7) may be evaluated for any (integral) value of m. Writing |<£ = ty, we have

(o-^2a^2 cos 2i/r — m^2) cos

.(8)

The large part of the integral arises from small values of t^. We divide the range of integration into two parts, the first from 0 to -^ where ^, though small, is large compared with e/2a, and the second from ty to ^tt. For the first part we may replace cos 2-v/^ cos Zmty by unity, and sin^2 ty by •»p. We thus obtain

a?a" — m^2 a,

(log 4a/e

. ...(9)

Thus to a first approximation aa = + m. In the second part of the range of integration we may neglect e^2/4«^2 in comparison with sin^2^, thus obtaining
(a^2a^2 cos 2^ — m^2) cos 2ma/r d^r a sin ir
.(10)
The numerator may be expressed as a sum of terms such as cos^2'^1 i/r, and for each of these the integral may be evaluated by taking cos \jr — z, in virtue of
l-z
Accordingly
_CQS2
when small quantities are neglected.    For example,
' cos^2 ty dty        _n	,	r*^71" cos^4
The sum of the coefficients in the series of terms (analogous to cos^an"»/r)
which represents the numerator of (10) is necessarily a^2a^2 — 7?i^2, since this is
the value of the numerator itself when ty - 0. The particular value of
<v|/> chosen for the division of the range of integration thus disappears from
the sum of (9) and (10), as of course it ought to do.	**
When m = l, corresponding to the gravest mode of vibration specially considered by Pocklington, the numerator in (10) is
4a^2a^2 cos^4 ^ - (4a^2a^2 + 2) cos^2 ^ + a^2 a^2 +1,
R.   VI.	8