168 ON THE PASSAGE OF WAVES THROUGH [375
In (24) we may, of course, replace sin 6 by cos 6 throughout. If both sin 6 and cos 0 occur, as in
f^sin'16cosm#log(2sin9}d0, .....................(26)
Jo
where n and m are even, we may express cosm 0 by means of sin 0, and so reduce (26) to integrals of the form (24). The particular case where m = n is worthy of notice. Here
sin'1 6 cos" 6 log (2 sin 0) d0 = I * sin51 6 cos« 6 log (2 cos 6) dO
7i
................. (27)
A comparison of the two treatments gives a relation between the integrals //. Thus, if n = 4,
We now proceed to the calculation of the left-hand member of (17) with ' = (If— y2)"* or, as it may be written,
f& fj~, r / ^'iLr\ p«z /xi r4 frnra ~]
QjlJ f , iKr\ T n . Hi i iii I a K> i n /ow\
., jipttf [(r+'°8 T) J° ^ + "?• - ¥A* s"+2nT6« * -'' • I'(2|S)
The leading term has already been found to be
..............................(20)
In (28) r is equal to ± (y — 77). Taking, as before,
y = 6 cos 6, y-b cos a, we have
Rikb }
7 + log -7- -f log + 2 (cos 0 - cos a) i «70 {M (cos (9 - cos a)}
£262 (cos 6 - cos a)2 k*b4 (cos 0 - cos a)4 3 JcRb6(cosd- cos a)" 11 _
22 2T42 ' 2+ ¥74>*76a~~~.......' I)"~ '
............<«0)
As regards the terms which do not involve log (cos 6 - cos a), we have to deal merely with
(.*U)
where w is an even integer, which, on expansion of the binomial and integration by a known formula, becomes
— 1 . w - 3 . % — 5 . . . 1 n.n — In, — 3.71 — 5...1 2 + ~T2~ u-2.7i-4...2COS a
W-7...1 1 .M_v
.-e..'XB'" + -+corf'g- -(32).ng at different parts of the width will arrive in various phases, of which due account must !><• takem The disturbance is no longer circularly symmetrical as in (16). But if, as is usual in observations with the microscope, we restrict ourselves to