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"78                                  ON THE  PROPAGATION  OF  WAVES                                       [c
should be noticed when the layer is in effect abolished, either by supposi /j, = yu,mj or, on the other hand, /u, = jj,lt
In the second case (42), corresponding to the polarising angle, becomes
In general for this case
The second fraction in (45) is equal to the thickness of the layer transition simply, when we suppose ^ = fa.
-P    ,,         -     s,                    1_______,_________
Further,    g_S=------~~;^---------------73---------->    ......(46,
Xl    ^     ^        coS'1-^_Bin^1
the difference of phase vanishing, as it ought to do, when /A = /ZI} or /j,m, again, when 6  0.
It should not escape notice that the expressions (10) and (11) ht different signs when ^ and #2 are small. This anomaly, as it must appi from an optical point of view, should be corrected when we consider 1 significance of 8"  8'. The origin of it lies in the circumstance that, in < application of the boundary conditions, we have, in effect, used differ* vectors as dependent variables to express light of the two polarisations. ] further explanation reference may be made to former writings, e.g. " On 1 Dynamical Theory of Gratings*."
If throughout the range of integration, //- is intermediate between 1 terminal values fa, p.m, the reflection is of the kind called positive by Jair The transition may well be of this character when there is no contaminati On the other hand, the reflection is negative, if /* has throughout a va! outside the range between /^ and pm. It is probable that something of t kind occurs when water has a greasy surface.
The formulae required in Optics, viz. (43), (44), (45), (46), are due, substance, to Lorenz and Van Ryn. The more general expressions (41), (< do not seem to have been given.
There is no particular difficulty in pursuing the approximation fr (32), etc. At the next stage the second term in the expansion of the
* Roy. Soc. Proc. A, 1907, Vol. LXXIX. p. 413.d as quantities of the first order.    Adding togetl the column of equations constituting the first members of (32), (33), etc., find