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```82                                     ON THE PROPAGATION OF WAVES                                 [<
If B=±A, reduction ensues to the familiar positive or negative ]_: gressive wave. But if £ be not equal to ± A, (55), taking the form
<b = | (A + B) cos (pt - lex) + \ (A — B) cos (pt + lex),
clearly does not represent a progressive wave. The mere possibility reduction to the form (57) proves little, without an examination of • character of H and 0.
It may be of interest to consider for a moment the character of 0 in (( If B/A, or, say, m, is positive, 0 may be identified with lex at the quadra but  elsewhere they differ, unless m = l.   Introducing the imaginary pressions for tangents, we find
6 = Tex + M sin 2kx + \$M* sin 4kx + %M3 sin 6kx + . . . , ...... (61
where                                        M= — ^ ................................. (62
•m + 1
When k is constant, one of the solutions of (53) makes </> proportional e~ikx. Acting on this suggestion, and following out optical ideas, let assume in general