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1912]                         THROUGH  A  STRATIFIED  MEDIUM, ETC.                                87
immediately assume the maximum admissible value, and retain it for such a time that the pendulum, then left free, will just reach the new position of equilibrium, after which the force is reimposed. The present solution is excluded, if it be required that the force never decrease in value. Under this restriction the best we can do is to make the force assume at once half its final value, and remain constant for a time equal to one-half of the free period. Under this force the pendulum will just swing out to the new position of equilibrium, where it is held on arrival by doubling the force. These cases have already been considered, but the analogue of the pendulum is instructive.
Kelvin* has shown that the equation of the second order
can be solved by a machine.    It is worth noting that an equation of the form (53) is solved at the same time.    In fact, if we make
yi=s^?,        Py^,........................(91)
yi     dx'           J2     dx'                               ^
we get on elimination either (90) for y1} or
for yz.    Equations (91) are those which express directly the action of the machine.
It now remains to consider more in detail some cases where total reflection occurs. When there is merely a simple transition from one medium (1) to another (2), the transmitted wave is
If there is total reflection, aa becomes imaginary, say — iaj ; the transmitted wave is then no longer a wave in the ordinary sense, but there remains some disturbance, not conveying energy, and rapidly diminishing as we recede from, the surface of transition according to the factor e~a"^(x~Xi]-From (2)
»22 = Y^ cos2 Q<i = ~ (V-2 - sin21
A.2                         AH    \ V 2
It appears that soon after the critical angle is passed, the disturbance in the second medium extends sensibly to a distance of only a few wave-lengths.
The circumstances of total reflection at a sudden transition are thus very simple; but total reflection itself does not require a sudden transition, and
* Roy. Soc. Proc. 1876, Vol. xxiv. p. 269. is to be no reflection in the original problem, the force must be such a character that when it becomes constant the pendulum is left at in the new position.    If fche object be to effect the transition between two states in the shortest possible time, but with forces which are restric never to exceed the final value, it is pretty evident that the force ms 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.