498 ON THE KEFLECTION OF LIGHT [422
and fe*g-feL = 1 ......................(40)
mr 1—r 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.]