b is a function of the period and damping of the wavetrain as well as of the properties of the reflector. If the wave-train be not damped, we have so the expression for b remains unaltered in form, b being still a function of the period. But if the conductivity of the electrolyte or a be zero, the angle becomes likewise zero, p is unity, and we get the ordinary value for b, namely, 1 βε (15) the period and damping of the incident ray having vanished from the expression. The phase-change is zero for either a perfect insulator or a perfect conductor; becoming indefinitely small in the first case, and in the second. r The refracted wave is dealt with in exactly the same Thus for the transmitted ray at the dielectric-electrolyte 0 2 surface, the change of phase is -', and the ratio of its amplitude to that of the incident ray or c is C, like b, is a function of the period and damping of the incident ray; like b it remains a function of the period even for an undamped ray; and like b it becomes a function of the dielectric constants only, if the "electrolyte " be a perfect insulator, the expression in (17) becoming We have now sufficient data to determine the speed of propagation and wave-length in the electrolyte, before going on to deal with the reflexions and refractions at the second surface. Referring back to equation (4) for X, and substituting the values we have determined for B2, P, and have X2=cA1eï(0/2—4')¿(−P1+ip2)† ̧n√ßæ{cos(x+0/2)—i sin (x+0/2)} ≈ ̧ or retaining only the real terms n cos 9, we V2= Hence the speed of the wave in the electrolyte is x and being both functions of a, and of the period, and being a function of the rate of damping of the wave-train, P λε λ is (like b and c and the phase-changes) a function of period, damping, and conductivity, except when the latter is zero when we have simply III. The Phenomena at the Second Surface of the Plate. We now proceed to the inverse case where the wave is passing from the electrolyte into the dielectric. If X, represent as before the electric force in the electrolyte, and X, represent the electric force in the second dielectric, we may now write X2=B1 ep(t−q) + B1'eP(t+qz), (22) (23) the wave in the electrolyte consisting of a direct and a reflected train, and that in the dielectric of a direct train only. Applying the interface conditions as in the last section, we get for 20 B1+ B1'=A3, dX2 dX3 Carrying out the differentiation of (25) and putting z=0, and rationalizing, we obtain finally for the transmitted ray so that the phase-change for the wave emerging from the electrolyte is (equation 16), and the reduction factor 2 f= (28) √ 1+1+2r cos & Ө B in (27) is the same as that for Α1 The expression for B1 in (8) except for the reversal of sign, so the phase-change at the second surface of the electrolyte is the same (-) in magnitude and sign as at the first surface, and the reduction factor If the electrolyte be replaced by an insulator, f takes the ordinary form holds. In the general case there is no such relation. IV. The Intensity of the Transmitted Ray. (30) We have now all the data necessary for treating the general case of a wave-train passing through a plate of finite thickWe will first take the incident wave-train and follow ness. its history in detail to obtain the expressions for the rays emergent after 0, 2, 4, . . . &c. internal reflexions. Let the incident wave-train be given by A ep(t- NB1z) from the time t=0 onwards, p being a complex quantity as defined by (9). Passing through the first surface and reaching the second the wave becomes (d being the thickness of the electrolyte) A cept-qd)i(0/2-4'), If we take the instant at which the incident wave first strikes the plate as the origin of time, the wave-train (I.) begins to emerge at time The wave reflected from the second surface at the time that (I.) emerges is -A.bce −pqd p(t+qz) i(0/2—4'−4), which becomes on reflexion at the first surface A. b'c e 2-2pqd p(t−qz) i(0/2—4'—¥) ̧ e and emerges from the second surface at time 3t, as Ab2cfe-3pqd p(t- NB12) (0/2—24'-24) (II.) This is the second emergent ray, or the ray emerging after two internal reflexions. The third emergent ray will be emerging at time 5t2; and so on for the others. The (n+1)th emergent ray, or the ray emerging after 2n internal reflexions, is |