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corresponding to (37). Discarding the imaginary part, we get, corresponding to (36),
M' f F=- -      (kcosnt + k'sinnt)dn..................(38)
7T   J0
Since k, k' are known functions of TO, or (a and v being given) of n, (38) may be calculated by quadratures for any prescribed value of t.
It appears from the tables that k, k' are positive throughout. When m = 0, k and k' are infinite and continually diminish as TO increases, until when m = oo , k = I, k'  0. For small values of TO the limiting forms for k, k' are
k = 1 + - .-?-__         k' =______........    (39)
TO2 (log m)2'                 m2 log TO' ............
from which it appears that if we make n vanish in (35), while Vn is given, F comes to zero.
We now seek the limiting form when t is very great. The integrand in (38) is then rapidly oscillatory, and ultimately the integral comes to depend sensibly upon that part of the range where n is very small. And for this part we may use the approximate forms (39).
Consider, for example, the first integral in (38), from which we may omit the constant part of k. We have
TT f00   cos lit dn  _ 4f7rv f  cos (4svar* t.x}dx
Writing 4u>t/a? = t', we have to consider
cos t'so, dx                                          ,    .
'o   tf(log^)2...............................(    ;
In this integral the integrand is positive from OG = 0 to as = 7r/2i', negative from 7r/2' to 37r/2', and so on. For the first part of the range, if we omit the cosine,
_J^L_ = f d l?S * = ___1 _ _ .                    /4x
^(loga;)2    J(log)2     log(2'/7r)J ...............^    }
and since the cosine is less than unity, this is an over estimate. When t' is very great, log (Zt'/ir) may be identified with logtf', and to this order of approximation it appears that (41) may be represented by (42). Thus if quadratures be applied to (41), dividing the first quadrant into three parts, we have
cos 7T/12           STT f
..______'____I    p/-\q  ____
1          5vr [
rJ + CS 1*2 [l
_ log6/7r          12   log S^/TT    log 6^/Tr            1*2    log M/TT
of which the second and third terms may ultimately be neglected in comparison with the first.    For example, the coefficient of cos (3-7r/12) is equal to
i     01     W  i     W Iog2-Mog~.log  .verse displacement (w'} of a plane elastic plate. And a surface on which the fiuid remains at rest (i/r = 0, d^/dn = 0) corresponds to a curve along which the elastic plate is clamped.