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80                                  ON THE PROPAGATION  OF  WAVES                                [360
is of the opposite sign. If three vectors of equal numerical value compensate one another, they must be at angles of 120. The necessary conditions are satisfied (in the case of perpendicular transmission) if the total thickness (21) is X, in accordance with
The total thickness of the layer of transition is thus somewhat reduced, but only by a very artificial arrangement, such as would not usually be contemplated when a layer "of transition is spoken of. If the progress from the first to the second uniform quality be always in one direction, reflection cannot be obviated unless the layer be at least \ thick.
The general formula (48) may be adapted to express the result appropriate to continuous variation of the medium. Suppose, for example, that a- is constant, making 0 = 1, and corresponding to the continuity of both $ and d(f)/dc *. It is convenient to suppose that the variation commences at x = 0. Then (48) may be written
a at any point x being connected with the angle of propagation by the usual relation (3). In the special case of perpendicular propagation, a = 27ryu./X1/a] , /A being refractive index and X1; fa relating to the first medium.
A curious example, theoretically possible even if unrealizable in experiment, arises when the variable medium is constituted in such a manner that the velocity of propagation is everywhere constant, so that there is no refraction. Then a is constant,  = 1, and (48) gives
 ! =3 I _L_ aviax                                                        /Kf)\
A,     Jlcr6       ............................... (52)
Some of the questions .relating to the propagation of waves in a variable medium are more readily treated on the basis of the appropriate; differential equation. As in (1), we suppose that the waves are plane, and that the medium is stratified in plane strata perpendicular to x, and we usually omit the exponential factors involving t and y, which may be supposed to run through. In the case of perpendicular propagation, y would not appear at all.
Consider the differential equation
in which (unless &2 can be infinite) it is necessary to suppose that 0 and d<f>/da: are continuous;  A;2 is a function of x, which must be everywhere
* These would be the conditions appropriate to a stretched string of variable longitudinal density vibrating transversely.ample, 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.