Scientific Papers - Vi

170
ON THE PASSAGE OF WAVES THEOUGH
since the last two integrals cancel, as appears when we write TT — n being even. .
In (35)
\ sinn ((f) + a) + £ sin71 (<j> — a) = sin'1 $ cosw a
tfi fi — j
"*—'-7—«r~" si11'1""2 ^ cos2 <£ sin2 a cos91"2 a JL , ^
+ -
n.»-l.n-2.ji-3
sinn~4 (j) cos4 <£ sin4 at cosn~4« + . . . + cosn <f> sinn a, (
and thus the result may be expressed by means of the integrals A.
if n = 2,
(35) = 4 I cZ</> sin2 </!> {sin2 <p cos2 a 4- cos2 <£ sin2 a} log (2 sin 0)
J o
= 4 f (cos2 a — sin2 a) /i4 4- sin2 a A™}...............................(
I \ y •» «; \
(35) = 41 <i«A sin4 d> {sin4 rf> cos4 a + 6 sin2 cf> cos2 </> sin3 a cos2 a Jo
+ cos4 ^ sin4 a} log (2 sin 0) = 4 {(cos4 a — 6 sin2 a cos2 a -f- sin4 a) /i8
+ (6 sin2 a cos2 a — 2 sin4 a) &6 4- sin4 a h4].............(.
(35) = 4, {(cos6 a — 15 cos4 a sin2 a + 15 cos2 a sin4 a — sin8 a) hlz -f (15 cos4 a sin2 a — 30 cos2 a sin4 a + 3 sin6 a) h10 + (15 cos2 a sin4« — 3 sin6 a) A8 + sin8« A6}.....................(.
It is worthy of remark that if we neglect the small differences be1 the tis in (39), it reduces to 4cos6«/t:2J and similarly in other cases.
When n is much higher than 6, the general expressions correspond! (37), (38), (39) become complicated. If, however, cos a be either 0, 01 (36) reduces to a single term, viz., cos'3' <£ or sinn <£. Thus at the Q (cos a = 0) from either of its forms
On the other hand, at the edges (cos a = ± 1)
\_ [^ • '
Jo
In (30), the object of our quest, the integral (35) occurs with the coeff
22.42.62...n2.........................^ (0 + a) log (2 sin <£).