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1912]                          THROUGH   A  STRATIFIED  MEDIUM,  ETC.                                79
must be retained, while the s's are still sufficiently represented by the first terms. The result, analogous to (37), (38), is
1 - I  <r.     —~ .das + i — l  —doc
•"i          O'lCLm          JO       JO      &                     Q>mJ 0   °"                         (Aft\
a  r
crdx .doe + i — I   crdx
0                                 °"?)i J 0
in which the terminal abscissas of the variable layer are taken to be 0 and d, instead of X-L and «m_x. I do not follow out the application to particular cases such as cr = constant, or cr sin2 6 = constant. For this reference may be made to Maclaurin, who, however, uses a different method.
The second case which allows of a simple approximate expression for the reflection arises when all the partial reflections are small It is then hardly necessary to appeal to the general equations: the method usually employed in Optics suffices. The assumptions are that at each surface of transition the incident waves may be taken to be the same as in the first medium, merely retarded by the appropriate amount, and that each partial reflection reaches the first medium no otherwise modified than by such retardation. This amounts to the neglect of waves three times reflected. Thus
.          _     e-2ft, (x*-xj   as (afc-ay    .
An interesting question suggests itself as to the manner in which the transition from one uniform medium to another must be effected in order to obviate reflection, and especially as to the least thickness of the layer of transition consistent with this result. If there be two transitions only, the least thickness of the layer is obtained by supposing in (48)
+ a,    ft +, and                                         2a2 («2 — as-^) = TT ; .............................. (50)
and this conclusion, as we have seen already, is not limited to the case of small differences of quality. In its application to perpendicular incidence, (50) expresses ' that the thickness of the layer is one-quarter of the wavelength proper to the layer. The two partial reflections are equal in magnitude and sign. It is evident that nothing better than this can be done so long as the reflections are of one sign, however, numerous the surfaces of transition may be.
If we allow the partial reflections to be of different signs, some reduction of the necessary thickness is possible. For example, suppose that there are two intermediate layers of equal thickness, of which the first is similar to the final uniform medium, and the second similar to the initial uniform medium. •Of the three partial reflections the first and third are similar, but the secondo guide the lantern-plates into position, and thus to ensure their subsequent exact superposition by simple mechanical means.