wire circuit-reduction coefficients covering loss of energy due either to damping by the imperfectly conducting wires or to imperfect reflexion at the ends of the circuit. If then ! T is the intensity of the nth train emerging from B towards C, 23 R is the intensity of the nth train reflected from B towards A, we have so we have to sum a finite-difference series. Rewriting (1) with these values of T and R, we have ก Abed B + rx A, Bb = dx A + re B, Suppose by and b, to be This is an equation to determine b. the roots; A1 A2, B1 B, being the corresponding values of A and B. We have then b1+b2=r(e+x), b1b2=(r2 — d2) ex. In = A1b2 + Ab1⁄2 (4) portional to E where E=(1+x)8T.=(1+x) {+} And from (6), 1−(b1+b2)+b1bq 1 rI。=dx { b1b2-re(b2+b2) + r2c2 But by the relations (4), whence or 1−(b1+b2)+b2b2=1−r(e+x)+ex (p2 —d2), b1b2—re(b1+b2) + r2e2 = —exd2, Thus, finally, —edrI=b2A ̧+b1A2—red Io, b2A1+b1A2=0. dlo E = (1+x) 1−r(e+x) + ex (n2 —d2)2 (7) and E If then we determine, d, and r for the different thicknesses of the electrolyte, we shall have by substituting their values in (8) a pair of simultaneous equations for the determination of e and X. This process only led to impossibilities for all values of B/B2 and the log. dec. of the wave-train that were tried. Either ex became greater than unity or negative, or e+x became negative. It may be noted that this correction did not work very successfully in Dr. Barton's case either. His method of applying it was, however, different: having formed an experimental estimate of the rate of decrease of the total energy of a wave-train by its passage along the imperfectly conducting wires, he assumed that there was perfect reflexion at both ends of the circuit, and thence calculated the correction to be applied to the experimental curve. The corrected curve obtained in this way lay below the calculated curve. Taking two points at random on his curves, I tried to calculate ex and e+x by my method. The results were again impossible. It must be remembered that in both these cases more than one of the constants of the theoretical curve were simple guesses, y and B/B, in my case were both only approximately known. In trying to make things square by the correction of (8), we may have been attempting the impossible and rightly got irrational rosults. I need not dwell at greater length on this point: possible causes of error are numerous. The non-agreement of experimental results with the equations. does not of course prove the wrongness of the latter, but only the wrongness of their application to the case. I am compelled to admit that, considering the positions of the maxima in my experimental curves are dependent on the phase-changes and on the correction we have been discussing -possibly also on other uneliminated disturbances-no great weight can be attached to the accuracy of the values of the dielectric constants for water, alcohol, &c., deduced on the assumption that such corrections were negligible *. VII. The Numerical Value of some of the previous Results. When the expressions in sections II. and III. had been obtained, the question at once arose, how far the values of b, c, f,,, and so on might be practically affected by possible variations in the rate of damping of the incident ray. In working with ordinary oscillators are changes likely to become important or to remain quite negligible? In working with light are differences likely to be appreciable between flashes and steady rays? As regards oscillator wave-trains, the damping should certainly be taken into account. The second question, however, must, I think, be answered in the negative, for the simple reason that one cannot get a quick enough flash†. Even an electric spark lasts for thousands or millions of the vibrations of violet light: the rays from it would be practically steady, not damped. The set of curves in fig. 6 is drawn for a steady ray, * Phil. Mag. vol. xxxvi. p. 539; Wied. Ann. vol. 1. P. 748. wave-length one metre. It illustrates the variation in b (the ratio of the transmitted to the incident amplitude at the surface of an infinitely thick slab), when the conductivity and dielectric constant of the slab vary. Curve (1) is drawn for Fig. 6. Curves showing the variation in "b" with the dielectric constant and conductivity of the plate, for an undamped wave-train. Wave-length 100 cms. -9 a non-conducting slab, curve (2) for a slab of conductivity 001 x 10 C.G.S. units, and curve (3) for a slab of conductivity 01 x 10 or ten times the last. These conductivities are all extremely low, that of a 5 p. c. copper sulphate or zinc sulphate solution being roughly 2 x 10. The curves show very well how rapidly the conductivity of the reflecting plate grows in importance relatively to the dielectric constant even for great wave-lengths. But if we take the case of a charge vibrating on an isolated perfectly conducting sphere, the amplitude falls to es or about of its original value in the time occupied by a complete vibration*: this gives us x=54° 44′ 8′′. Let us use this as an example of a damped wave-train to compare with the steady ray, retaining the same wave-length-1 metrecorresponding to a sphere about a foot in diameter. * J. J. Thomson, 'Recent Researches,' p. 370. 80 The curve (1) for b corresponding to a non-conducting plate would be the same in both cases. The curves (3) corresponding to a plate of conductivity 01 x 10 are shown together in fig. 7. b is at first greatest for the damped Fig. 7. 9 Comparison of the values of "b" for damped and undamped wave-trains. Conductivity of plate=01x10. Wave-length in either case-100 cms. 8=1. wave-train, but as the dielectric constant is increased this ratio is reversed. Physically speaking, this has very little meaning any actual method would measure the energy of the reflected ray, and it has been shown that the energy is a function of the phase-change, the phase-change being itself a function of the rate of damping of the wave-train. Taking the phase-changes, y, first, I calculated them for the steady ray, for the damped ray with which we are dealing, and the conductivity 01 × 10. The results are given together in fig. 8. For the lower values of the dielectric constant, for |