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The first integral in (7) may be evaluated for any (integral) value of m. Writing |< = ty, we have
(o-2a2 cos 2i/r  m2) cos
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 sin2 ty by p. We thus obtain
a?a"  m2 a,
(log 4a/e
. ...(9)
Thus to a first approximation aa = + m. In the second part of the range of integration we may neglect e2/42 in comparison with sin2^, thus obtaining
(a2a2 cos 2^  m2) cos 2ma/r d^r a sin ir
The numerator may be expressed as a sum of terms such as cos2'1 i/r, and for each of these the integral may be evaluated by taking cos \jr  z, in virtue of
when small quantities are neglected.    For example, ' cos2 ty dty        n           ,             r*71" cos4
The sum of the coefficients in the series of terms (analogous to cosan"/r) which represents the numerator of (10) is necessarily a2a2  7?i2, 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
4a2a2 cos4 ^ - (4a2a2 + 2) cos2 ^ + a2 a2 +1,
R.   VI.                                                                                                                                     8ain the large term, arising in the integral when is small, and the finite term, but we  may reject  small quantities.    Tin Pocklington finds