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Full text of "Scientific Papers - Vi"

498                                  ON  THE  KEFLECTION  OF  LIGHT                                [422
and                              fe*g-feL =         1         ......................(40)
mr     1r     l+(m  l)r                                 /
When m tends to infinity, <pm approaches unity, and ^rm approaches zero.
For many purposes, equations (38), (39) may conveniently be written in another form, by making b = e&, a = ea.    Thus
<pm               "fym                       1
sinh mft    sinh a    sinh (a + m/ ?             *                  1
..................(41)
                         - -                             C42)
sinh ft    sinh a    sinh(a + #)' .....................      '
where in Stokes' problem a and ft are real, and are uniquely determined in terms of r and t by (44), (46) below*.
If we form the expression for (1 + r2  P)/2r by means of (42), we find that it is equal to cosh a.    Also
sinh2 a =--------------~~--------------,     ...............(43)
from which we see that, if r and t are real positive quantities, such that ?' + t < I, sinh a is real.    Similarly, sinh ft, sinh (a + ft) are real.
Passing now to my proper problem, where r and t are complex factors, represented (when there is no absorption) by (13), we have
,        1-t-r2 i2    cos p                                   /AA.
cosh a = X------= -r~,    .....................(44)
2?-           sin 6                                         '
so that cosh a is real.    Also
If we write a = ax + iau, ft~fti + i@2, where ax, or2; &, /32 are sinh a = sinh aa cos a2 + i cosh Oj sin a2> cosh a = cosh a: cos 2 + i sinh c^ sin 2.
Since cosh a is real, either j or sin aa- must vanish. In the first case, sinh 0,  i sin aa, and (45) shows that this can occur only when sin2 6 > cos2 p. In the second case (sin 2 = 0), sinh2 a = sinh2 al} which requires that sin2 6 < cos2 p.
Similarly if we interchange r and t,
,   _     l+i2~r2       sino                        /.^
co.h0-5-jjJ......................(46)
so that cosh /3 is real, requiring either /3a = 0, or sin /32 = 0.    Also
sinh2/3 = S-^-l............................(47)
COS20
* Except as to sign, which is a matter of indifference.   It may be remarked that his equation (13) can at once be put into this form by making his a and /3 pure imaginaries. of (11) is zero when n = 0, which is not the case when a absolutely = 0. W. F. S.]