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Full text of "Scientific Papers - Vi"

184                            ON THE  PASSAGE  OF WAVES  THROUGH
Again, if m be integral,
$ sin2"120 cos 20 log (2 sin 0)
= ,   1   ft f log (2 sin 0) d sin2m+1 20 4/M + 2 I
1       /*^ir
_ _---------      Sin2m 20 (1 + cos 20) dd>
4m+ 2 J0            ^^
= -,   1   nf 7Tsin2w20rf0 = --i------5 f "sin2
4m+ 2J0             r   r        4m+2J0
1       2m — 1. 2m — 3 ... 1 IT                                  .
_
= ~ 4m + 2      2m . 2m - 2 . . . 2      2
For example, if w = 0,
riir                                                                       ,77-
d0 cos 20 log (2 sin 0) = -~r , .................. (1
Jo                                               *
f i71"                                                                                                   IT
and(m = l)               cZ0 sin2 20 cos 20 log (2 sin 0) = - -^-. ............. (1
\ o                                                                                     4*
Using these lemmas, we find
(99) = oht (cos4 « — 6 cos2 a sin2 a + sin4 a)
+ A2 (30 cos2 a sin2 a - 10 sin4 a - 3 cos2 a +3 sin2 a) — ^?r cos2 a (cos2 a + 3 sin2 a) ;
and thence, on introduction of the values of A-2, A4, for the complete va this order of approximation,
-TT~( 5 cos4 a -I- 18 cos2 a sin2 a + 21 sin4 a)   ....... (1
64 v                                                            J
To carry out the calculations to a sufficient approximation witi general value of a would be very .tedious. I have limited myself t extreme cases cos a = 0, cos a = ± 1. For the former, we have
1 cty_3    f           i'M\ (3F62    /c464         W     }
" * fa ~ 2 + V +    g 47 JTF " "256 + 2^4
__   °
+                              3'  ...... '"(-
64        6.256    43.256.8' and for the latter
3             ,
4. I no-  _ I  J _ ___ L
8
__ .       __ _                           _        _ ___    _ _______ ___
dec         2V           4 / ( 16       16.16    4.16.16.16     24 . 164j
e          41309/c868
64  + 16,64.15 + 16.3.70.64.64    165.9.420 ~32~ + 4716716 ~2TT6^+ ~lĄ7W................^cos 6 — cos a)2 + cc*