large star in Fig. 28 and the four calculated apexes for the resultant motions at the four epochs are shown by the small circles, which necessarily lie on the circle representing the calculated aberration orbit. The observed apexes for the four epochs are represented by the small stars. The location of the pole of the ecliptic is also shown. The close agreement between the calculated and observed apparent apexes would seem to be conclusive evidence of the validity of the solution of the ether-drift observations for the absolute motion of the earth and also for the effect of the orbital motion of the earth, which hitherto has not been demonstrated.
Fig. 28. Observed and calculated apexes of the absolute motion of the solar system.
It may seem surprising that such close agreement between observed and calculated places can be obtained from observations of such minute effects, and effects which are reputed to be of such difficulty and uncertainty. Perhaps an explanation is the fact that the star representing the final result for the February epoch is, in effect, the average of 8080 single determinations of its location; the star for the August epoch represents 7680 single determinations, that for September, 6640 and that for April, 3208 determinations.
Attention is called to the fact that the results here obtained are not opposed to the results originally announced by Michelson and Morley in 1887; in reality they are consistent with and confirm the earlier results. With additional
observations, the interpretation has been revised and extended.
The model, Fig. 29, represents, to scale, the conclusions of this study of the absolute motion of the earth. The earth is represented by the ball near the top of the model and the plane of the ecliptic is the horizontal plane through the center of the earth. The cosmic component of the earth's motion, which is the absolute motion of
Fig. 29. Model illustrating the components of ether-drift.
the solar system, is directed to the apex near the south pole of the ecliptic and is represented by the arrow near the top of the model and by the rod in its prolongation below the earth. The orbital motions for the four epochs of these observations are represented by the four arrows in the horizontal plane. The four resultant motions are shown by the diagonals of the four parallelograms corresponding to the several epochs. The resultant motion in the course of a year traces on the celestial sphere the aberration orbit of the earth represented by the circle near
the bottom of the model; the four epochal positions are marked by the arrows at the bottom. This part of the model corresponds to the orbital circle on the chart, Fig. 28, and to the model of the orbit with the four globes, Fig. 24.
Probable error
A study of the numerical results as plotted in Fig. 26 shows that the probable error of the observed velocity, which has a magnitude of from ten to eleven kilometers per second, is
±0.33 kilometer per second, while the probable error in the determination of the azimuth is ±2.5°. The probable error in the right ascensions and declinations of the polar chart, Fig. 28, is ±0.5°.
Full-Period Effect
Throughout these experiments, while the attention has been given to the second-order, half-period effect, there has been present a full-period, first-order effect of comparable magnitude. The theory of the ether-drift experiment as usually given is exact but it is also abstract, being based upon simplified conditions of the apparatus which never exist in the actual experiment. What actually happens to the interference fringes depends not only upon the ether-drift effect but also upon the geometrical arrangement of the mirrors. The simple theory assumes that the mirrors at the ends of the two arms of the interferometer are perpendicular to the rays of light; this would produce fringes of infinite width, the whole field of view being uniformly illuminated, a critical condition never desired nor used in practice. In order to produce a series of straight fringes, suitable for the measurement of displacements, as shown in Fig. 7, it is necessary that one of the end mirrors be rotated about a vertical axis through a very small angle so that the two virtual interfering planes intersect. The width of the fringes and the number of fringes in the field of view are directly dependent upon this inclination of the end mirror. The angle of incidence of the light on the mirror, as used in these experiments, differs from 0° by about ±4″. The late Professor W. M. Hicks of University College, Sheffield, has given an elaborate
discussion of the theory,15 using methods which are not only rigorous but also general, applying to any adjustment whatever of the optical parts of the apparatus. In the theory of Hicks it is shown that when the periodic variation in the relative phases of the two beams of light in the interferometer takes place with the mirrors adjusted as in actual practice, there is introduced an additional effect which is periodic in a full turn of the instrument. The amplitude of this full-period effect, which varies inversely as the width of the fringes being used at the time of observation, is about equal to the amplitude of the ether-drift effect when there are eight fringes in the field of view; with the adjustment usually secured for six fringes in the field of view, the full-period effect is smaller than the half-period effect, as shown in Fig. 21.
The full-period effect, which has usually been overlooked, is present in all of the observations, including the original observations of Michelson and Morley. Hicks called attention to the latter fact and calculated its magnitude. Unfortunately, in none of the observations heretofore made have there been quantitative measurements of the width of the fringes which determines the angle of inclination of the mirror and it is not possible to use the full-period effect for a solution of the problem of ether-drift. However, the approximate number of fringes visible in the field of view has frequently been recorded. A comparison of the width of fringes thus indicated with the magnitude of the full-period effect shows a direct
Fig. 30. The full-period effect; the relation of amplitude to width of fringes.
16 W. M. Hicks, Phil. Mag. [6] 3, 9, 256, 555 (1902); Nature 65, 343 (1902); E. W. Morley and D. C. Miller, Phil. Mag. [6] 9, 669 (1905).
