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-2a2 cos 2i/r — m2) 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 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 e2/4«2 in comparison with sin2^, thus obtaining (a2a2 cos 2^ — m2) cos 2ma/r d^r a sin ir .(10) 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 l-z Accordingly CQS2 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