Chapter 5

The results of the observations are expressed graphically in fig. 6.fig06Fig. 6.The upper is the curve for the observations at noon, and the lower that for the evening observations. The dotted curves representone eighthof the theoretical displacements. It seems fair to conclude from the figure that if there is any displacement due to the relative motion of the earth and the luminiferous æther, this cannot be much greater thanof the distance between the fringes.Considering the motion of the earth in its orbit only, this displacement should beThe distance D was about eleven meters, orof yellow light; hence the displacement to be expected was. The actual displacement was certainly less than the twentieth part of this, and probably less than the fortieth part. But since the displacement is proportional to the square of the velocity, the relative velocity of the earth and the æther is probably less than one sixth the earth's orbital velocity, and certainly less than one-fourth.In what precedes, only the orbital motion of the earth is considered. If this is combined with the motion of the solar system, concerning which but little is known with certainty, the result would have to be modified; and it is just possible that the resultant velocity at the time of the observations was small though the chances are much against it. The experiment will therefore be repeated at intervals of three months, and thus all uncertainty will be avoided.It appears, from all that precedes, reasonably certain that if there be any relative motion between the earth and the luminiferous æther, it must be small; quite small enough entirely to refute Fresnel's explanation of aberration. Stokes has given a theory of aberration which assumes the æther at the earth's surface to be at rest with regard to the latter, and only requires in addition that the relative velocity have a potential; but Lorentz shows that these conditions are incompatible. Lorentz then proposes a modification which combines some ideas of Stokes and Fresnel, and assumes the existence of a potential, together with Fresnel's coefficient. If now it were legitimate to conclude from the present work that the æther is at rest with regard to the earth's surface, according to Lorentz there could not be a velocity potential, and his own theory also fails.Supplement.It is obvious from what has gone before that it would be hopeless to attempt to solve the question of the motion of the solar system by observations of optical phenomenaat the surface of the earth. But it is not impossible that at even moderate distances above the level of the sea, at the top of an isolated mountain peak, for instance, the relative motion might be perceptible in an apparatus like that used in these experiments. Perhaps if the experiment should ever be tried in these circumstances, the cover should be of glass, or should be removed.It may be worth while to notice another method for multiplying the square of the aberration sufficiently to bring it within the range of observation, which has presented itself during the preparation of this paper. This is founded on the fact that reflection from surfaces in motion varies from the ordinary laws of reflection.fig07Figures 1, 2, 3 and 4 fromSupplement.Let(fig. 1 (Figure 07)) be a plane wave falling on the mirrorat an incidence of. If the mirror is at rest, the wave front after reflection will be.Now suppose the mirror to move in a direction which makes an anglewith its normal, with a velocity. Letbe the velocity of light in the æther supposed stationary, and letbe the increase in the distance the light has to travel to reach. In this time the mirror will have moved a distance.We havewhich put, andIn order to find the new wave front, draw the arcwithas a center andas radius; the tangent to this arc fromwill be the new wave-front, and the normal to the tangent fromwill be the new direction. This will differ from the directionby the anglewhich it is required to find. From the equality of the trianglesandit follows that,,or neglecting terms of the order,Now let the light fall on a parallel mirror facing the first, we should then haveand the total deviation would bewhereis the angle of aberration, if only the orbital motion of the earth is considered. The maximum displacement obtained by revolving the whole apparatus throughwould beWith fifty such couples the displacement would be''. But astronomical observations in circumstances far less favorable than those in which these may be taken have been made to hundredths of a second; so that this new method bids fair to be at least as sensitive as the former.The arrangement of apparatus might be as in fig. 2 (Figure 07);, in the focus of the lens, is a slit.are two glass mirrors optically plane and so silvered as to allow say one twentieth of the light to pass through, and reflecting say ninety per cent. The intensity of the light falling on the observing telescopewould be about one millionth of the original intensity, so that if sunlight or the electric arc were used it could still be readily seen. The mirrorsand, would differ from parallelism sufficiently to separate the successive images. Finally, the apparatus need not be mounted so as to revolve, as the earth's rotation would be sufficient.If it were possible to measure with sufficient accuracy the velocity of light without returning the ray to its starting point, the problem of measuring the first power of the relative velocity of the earth with respect to the æther would be solved. This may not be as hopeless as might appear at first sight, since the difficulties are entirely mechanical and may possibly be surmounted in the course of time.For example, suppose (fig. 3 (Figure 07))and, two mirrors revolving with equal velocity in opposite directions. It is evident that light fromwill form a stationary image atand similarly light fromwill form a stationary image at. If now the velocity of the mirrors be increased sufficiently, their phases still being exactly the same, both images will be deflected fromand, in inverse proportion to the velocities of light in the two directions; or, if the two deflections are made equal, and the difference of phase of the mirrors be simultaneously measured, this will evidently be proportional to the difference of velocity in the two directions. The only real difficulty lies in this measurement. The following is perhaps a possible solution.(fig. 4 (Figure 07)) are two gratings on which sunlight is concentrated. These are placed so that after falling on the revolving mirrorsand, the light forms images of the gratings atand, two very sensitive selenium cells in circuit with a battery and a telephone. If everything be symmetrical, the sound in the telephone will be a maximum. If now one of the slitsbe displaced through half the distance between the image of the grating bars, there will be silence. Suppose now that the two deflections having been made exactly equal, the slit is adjusted for silence. Then if the experiment be repeated when the earth's rotation has turned the whole apparatus through, and the deflections are again made equal, there will no longer be silence, and the angular distance through whichmust be moved to restore silence will measure the required difference in phase.There remain three other methods, all astronomical, for attacking the problem of the motion of the solar system through space.1. The telescopic observation of the proper motions of the stars. This has given us a highly probably determination of the direction of this motion, but only a guess as to its amount.2. The spectroscopic observation of the motion of stars in the line of sight. This could furnish data for the relative motions only, though it seems likely that by the immense improvements in the photography of stellar spectra, the information thus obtained will be far more accurate than any other.3. Finally there remains the determination of the velocity of light by observations of the eclipses of Jupiter's satellites. If the improved photometric methods practiced at the Harvard observatory make it possible to observe these with sufficient accuracy, the difference in the results found for the velocity of light when Jupiter is nearest to and farthest from the line of motion will give, not merely the motion of the solar system with reference to the stars, but with reference to the luminiferous æther itself.[1]Communicated by the Authors.This research was carried out with the aid of the Bache Found.[2]It may be noticed that most writers admit the sufficiency of the explanation according to the emission theory of light; while in fact the difficulty is even greater than according to the undulatory theory. For on the emission theory the velocity of light must be greater in the water telescope, and therefore the angle of aberration should be less; hence, in order to reduce it to its true value, we must make the absurd hypothesis that the motion of the water in the telescope carries the ray of light in the opposite direction![3]Comptes Rendus, XXXIII, 349, 1851; Pogg.Ann.Ergänzungsband, III. 457, 1853;Ann. Chim. Phys., III, lvii, 385, 1859.[4]"Influence of Motion of the Medium on the Velocity of Light." Am. J. Sci, III, xxxi, 377, (1886).[5]It may be objected that it may escape by the space between the mercury and the walls; but this could be prevented by amalgamating the latter.[6]Archives Néerlandaises, XXI, 2melivr. Phil. Mag. V, xiii, p. 236.[7]"The Relative Motion of the Earth and the Luminiferous Æther," by Albert A. Michelson. Am. J. Sci., III, xxii, p. 120.[8]It may be mentioned here that the error was pointed out to the author of the former paper by M. A. Potier, of Paris, in the winter of 1881.[9]"De l'Influence du Mouvement de la Terre sur les Phen. Lum."Archives Néerlandaises, XXI, 2melivr., (1886).

The results of the observations are expressed graphically in fig. 6.

fig06Fig. 6.

Fig. 6.

The upper is the curve for the observations at noon, and the lower that for the evening observations. The dotted curves representone eighthof the theoretical displacements. It seems fair to conclude from the figure that if there is any displacement due to the relative motion of the earth and the luminiferous æther, this cannot be much greater thanof the distance between the fringes.

Considering the motion of the earth in its orbit only, this displacement should beThe distance D was about eleven meters, orof yellow light; hence the displacement to be expected was. The actual displacement was certainly less than the twentieth part of this, and probably less than the fortieth part. But since the displacement is proportional to the square of the velocity, the relative velocity of the earth and the æther is probably less than one sixth the earth's orbital velocity, and certainly less than one-fourth.

In what precedes, only the orbital motion of the earth is considered. If this is combined with the motion of the solar system, concerning which but little is known with certainty, the result would have to be modified; and it is just possible that the resultant velocity at the time of the observations was small though the chances are much against it. The experiment will therefore be repeated at intervals of three months, and thus all uncertainty will be avoided.

It appears, from all that precedes, reasonably certain that if there be any relative motion between the earth and the luminiferous æther, it must be small; quite small enough entirely to refute Fresnel's explanation of aberration. Stokes has given a theory of aberration which assumes the æther at the earth's surface to be at rest with regard to the latter, and only requires in addition that the relative velocity have a potential; but Lorentz shows that these conditions are incompatible. Lorentz then proposes a modification which combines some ideas of Stokes and Fresnel, and assumes the existence of a potential, together with Fresnel's coefficient. If now it were legitimate to conclude from the present work that the æther is at rest with regard to the earth's surface, according to Lorentz there could not be a velocity potential, and his own theory also fails.

Supplement.

It is obvious from what has gone before that it would be hopeless to attempt to solve the question of the motion of the solar system by observations of optical phenomenaat the surface of the earth. But it is not impossible that at even moderate distances above the level of the sea, at the top of an isolated mountain peak, for instance, the relative motion might be perceptible in an apparatus like that used in these experiments. Perhaps if the experiment should ever be tried in these circumstances, the cover should be of glass, or should be removed.

It may be worth while to notice another method for multiplying the square of the aberration sufficiently to bring it within the range of observation, which has presented itself during the preparation of this paper. This is founded on the fact that reflection from surfaces in motion varies from the ordinary laws of reflection.

fig07Figures 1, 2, 3 and 4 fromSupplement.

Figures 1, 2, 3 and 4 fromSupplement.

Let(fig. 1 (Figure 07)) be a plane wave falling on the mirrorat an incidence of. If the mirror is at rest, the wave front after reflection will be.

Now suppose the mirror to move in a direction which makes an anglewith its normal, with a velocity. Letbe the velocity of light in the æther supposed stationary, and letbe the increase in the distance the light has to travel to reach. In this time the mirror will have moved a distance.

We havewhich put, and

In order to find the new wave front, draw the arcwithas a center andas radius; the tangent to this arc fromwill be the new wave-front, and the normal to the tangent fromwill be the new direction. This will differ from the directionby the anglewhich it is required to find. From the equality of the trianglesandit follows that,,or neglecting terms of the order,

Now let the light fall on a parallel mirror facing the first, we should then haveand the total deviation would bewhereis the angle of aberration, if only the orbital motion of the earth is considered. The maximum displacement obtained by revolving the whole apparatus throughwould beWith fifty such couples the displacement would be''. But astronomical observations in circumstances far less favorable than those in which these may be taken have been made to hundredths of a second; so that this new method bids fair to be at least as sensitive as the former.

The arrangement of apparatus might be as in fig. 2 (Figure 07);, in the focus of the lens, is a slit.are two glass mirrors optically plane and so silvered as to allow say one twentieth of the light to pass through, and reflecting say ninety per cent. The intensity of the light falling on the observing telescopewould be about one millionth of the original intensity, so that if sunlight or the electric arc were used it could still be readily seen. The mirrorsand, would differ from parallelism sufficiently to separate the successive images. Finally, the apparatus need not be mounted so as to revolve, as the earth's rotation would be sufficient.

If it were possible to measure with sufficient accuracy the velocity of light without returning the ray to its starting point, the problem of measuring the first power of the relative velocity of the earth with respect to the æther would be solved. This may not be as hopeless as might appear at first sight, since the difficulties are entirely mechanical and may possibly be surmounted in the course of time.

For example, suppose (fig. 3 (Figure 07))and, two mirrors revolving with equal velocity in opposite directions. It is evident that light fromwill form a stationary image atand similarly light fromwill form a stationary image at. If now the velocity of the mirrors be increased sufficiently, their phases still being exactly the same, both images will be deflected fromand, in inverse proportion to the velocities of light in the two directions; or, if the two deflections are made equal, and the difference of phase of the mirrors be simultaneously measured, this will evidently be proportional to the difference of velocity in the two directions. The only real difficulty lies in this measurement. The following is perhaps a possible solution.

(fig. 4 (Figure 07)) are two gratings on which sunlight is concentrated. These are placed so that after falling on the revolving mirrorsand, the light forms images of the gratings atand, two very sensitive selenium cells in circuit with a battery and a telephone. If everything be symmetrical, the sound in the telephone will be a maximum. If now one of the slitsbe displaced through half the distance between the image of the grating bars, there will be silence. Suppose now that the two deflections having been made exactly equal, the slit is adjusted for silence. Then if the experiment be repeated when the earth's rotation has turned the whole apparatus through, and the deflections are again made equal, there will no longer be silence, and the angular distance through whichmust be moved to restore silence will measure the required difference in phase.

There remain three other methods, all astronomical, for attacking the problem of the motion of the solar system through space.

1. The telescopic observation of the proper motions of the stars. This has given us a highly probably determination of the direction of this motion, but only a guess as to its amount.

2. The spectroscopic observation of the motion of stars in the line of sight. This could furnish data for the relative motions only, though it seems likely that by the immense improvements in the photography of stellar spectra, the information thus obtained will be far more accurate than any other.

3. Finally there remains the determination of the velocity of light by observations of the eclipses of Jupiter's satellites. If the improved photometric methods practiced at the Harvard observatory make it possible to observe these with sufficient accuracy, the difference in the results found for the velocity of light when Jupiter is nearest to and farthest from the line of motion will give, not merely the motion of the solar system with reference to the stars, but with reference to the luminiferous æther itself.

[1]Communicated by the Authors.This research was carried out with the aid of the Bache Found.

[1]Communicated by the Authors.

This research was carried out with the aid of the Bache Found.

[2]It may be noticed that most writers admit the sufficiency of the explanation according to the emission theory of light; while in fact the difficulty is even greater than according to the undulatory theory. For on the emission theory the velocity of light must be greater in the water telescope, and therefore the angle of aberration should be less; hence, in order to reduce it to its true value, we must make the absurd hypothesis that the motion of the water in the telescope carries the ray of light in the opposite direction!

[3]Comptes Rendus, XXXIII, 349, 1851; Pogg.Ann.Ergänzungsband, III. 457, 1853;Ann. Chim. Phys., III, lvii, 385, 1859.

[4]"Influence of Motion of the Medium on the Velocity of Light." Am. J. Sci, III, xxxi, 377, (1886).

[5]It may be objected that it may escape by the space between the mercury and the walls; but this could be prevented by amalgamating the latter.

[6]Archives Néerlandaises, XXI, 2melivr. Phil. Mag. V, xiii, p. 236.

[7]"The Relative Motion of the Earth and the Luminiferous Æther," by Albert A. Michelson. Am. J. Sci., III, xxii, p. 120.

[8]It may be mentioned here that the error was pointed out to the author of the former paper by M. A. Potier, of Paris, in the winter of 1881.

[9]"De l'Influence du Mouvement de la Terre sur les Phen. Lum."Archives Néerlandaises, XXI, 2melivr., (1886).

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