On the Theory of Aberration and the Principle of Relativity

by Henry Crozier Keating Plummer

On the Theory of Aberration and the Principle of Relativity.

By H. C. Plummer, M.A.

I. In a former paper (vol. lxix. p. 496) the theory of aberration has been discussed from the standpoint of ordinary optical theory. This suffices for the conclusion that beyond the astronomical effects as commonly understood no first-order optical effects can be put in evidence, and that, in the absence of greater experimental refinement, it is impossible for the observer to detect his own absolute motion in the ether. With this position the astronomer must be content. If we have reason to be dissatisfied with the results of the efforts hitherto made to determine the constant of aberration, we have little ground for taking into account the second-order effects. But on the other hand an optical experiment has been performed which cannot be reconciled with the ordinary theory, and we have been forced to admit that the theory, if not actually erroneous, can be no more than an approximation to the truth. Meanwhile the electronic theory of matter has been developed, and has embraced in its synthesis an explanation of this difficulty. The result is that the Principle of Relativity, with its far-reaching implications, has obtained a cardinal position in modern science. It is possible that there are some astronomers who are not familiar with the literature of the subject, and to whom an elementary account of the new ideas in physics may be of interest. Accordingly the subject will be approached from the purely optical side, and an attempt will be made to present the theory, which is a product of the last decade and is due chiefly to Professor Lorentz,^[1] in the simplest possible way.

2. According to the principle of relativity, it is impossible to find experimental evidence of the absolute motion through the ether. Let us consider the bearing of this on a very simple experiment. Let AB and AC (fig. 1) be actual lengths l' and l at right angles to one another, and let mirrors be placed at B and C perpendicular to AB, AC, so that rays of light from A will be reflected back to A. Now suppose that the whole apparatus is carried through the ether with the velocity v in the direction AB, the velocity of light being U. The time from A to B and back again will be

$t_{1}=l'/(U-v)+l'/(U+v)=2l'U/\left(U^{2}-v^{2}\right).$

But the time by the path AC'A' in the ether will be

$t_{2}=2AC'/U=2l/{\sqrt {U^{2}-v^{2}}}$

Hence if $t_{1}=t_{2}$,

$l'=l{\sqrt {1-v^{2}/U^{2}}}$

The actual lengths l', l must thus be different if the light-times are the same. Nevertheless no difference can be detected between the lengths by any test that we can apply, or the principle of relativity will be contravened. Hence we are led to admit a universal contraction of all material systems in the direction of their motion through the ether, according to the law

$l'=l\ \cos \ \beta$, where $\sin \ \beta =v/U$.

The dimensions transverse to the motion are considered unaltered.

If it were objected that the light-times considered above could not be determined with sufficient accuracy to support so astonishing an inference, the objection would apply only to the simple form of argument adopted. In practice the light-times can be compared with extraordinary precision by making the beams interfere on returning to A. In fact we are dealing, as it were, with a schematic representation of the Michelson-Morley experiment, the null result of which is the fundamental fact on the optical side which has to be explained.

3. The uniform contraction in one dimension of the moving material system, the possibility of which we have thus been led to entertain, suffices of itself for the discussion of some interesting optical problems. As an example, the position of the focus of a moving parabolic mirror may be chosen. Let BA₁B' be the mirror (fig. 2) and F₁ its focus when at rest. Let BA₂B' be the figure of the mirror when in motion towards a star situated on its axis and F₂ the simultaneous position of the point of the (contracted) apparatus which corresponds to F₁. The incident wave-front is BCB'. If CF₁ is taken as the axis of x and CB as the axis of y, the equation of BA₁B' will be

$z^{2}+y^{2}=4f(x+b)$

where A₁F₁ = f, A₁C = b. If we admit a contraction λ^-1 which will make CF₁ = λ.CF₂, the equation of BA₂B' becomes

$z^{2}+y^{2}=4f\lambda (x+b/\lambda )$.

This is a real deformation of the figure of the mirror. But if BA₃B' is the surface at which the advancing wave-front actually meets the moving mirror

$A_{2}A_{3}/v=A_{3}C/U=A_{2}C/(v+U)$

where v is the velocity of the mirror, and the virtual surface on which the wave falls becomes

$z^{2}+y^{2}=4f\lambda (1+v/U)\{x+b/\lambda (1+v/U)\}$.

This is still a paraboloid, and the wave BCB' will reach its focus F₃ after a time

$t=f\lambda (U+v)/U^{2}+b/\lambda (U+v)$.

Hence no change of focus will be detected provided F₂F₃ = vt. Now

${\begin{array}{l}CF_{3}=f\lambda (1+v/U)-b/\lambda (1+v/U)\\\\CF_{2}=f/\lambda -b/\lambda \end{array}}$

so that

$F_{2}F_{3}=f\lambda (1+v/U)-f/\lambda +bv/\lambda (U+v)$

which is equal to vt if

$f\lambda .v(U+v)/U^{2}=f\lambda .(U+v)/U+f/\lambda$

or

$\lambda ^{-2}=(U+v)\left(1/U-v/U^{2}\right)=1-v^{2}/U^{2}.$

This is the law of contraction previously obtained, and the result is a complete compensation of the optical effect due to the motion of the mirror.

When a telescope is moving directly towards a star, Veltmann’s theorem^[2] shows that the motion will not affect the relative position of the focus to the first order. But a second order effect will remain outstanding, although it will be too small to be ascertained by focal settings. According to our present ideas, however, no effect of any order is to be expected, and our example shows how the compensation operates in a particularly simple case.

4. In what precedes we have contemplated only a change in one dimension of the moving system, or, as it may be expressed, a transformation of one coordinate in space. We have now to consider a related transformation to apparent or, as it is called, "local" time. Let axes be taken attached to the moving system, the measured coordinates being ξ, η, ζ. Let their position at the time t = 0 be the axes fixed in space, the corresponding coordinates being x, y, z. The motion of the system is supposed parallel to the axis of x or ξ.

We now imagine a new system of time τ, which depends not only on the absolute time t but also on the position in space. It is sufficient for our purpose to suppose that

$\tau =at-bx,\ \xi =cx-dt,\ \eta =y,\ \zeta =z.$

All optical phenomena which would be described by an observer at rest in space in terms of x, y, z and t will be described by an observer in motion in terms of ξ, η, ζ and τ. Now one result of the principle of relativity and of the constant velocity of light is that the spherical wave-front

$x^{2}+y^{2}+z^{2}=U^{2}t^{2}$

must appear to the moving observer as

$\xi ^{2}+\eta ^{2}+\zeta ^{2}=U^{2}\tau ^{2},$

for a spherical wave which actually converges to a point in reality must appear to converge to a point and to move with the velocity U. This requires

$(cx-dt)^{2}-U^{2}(at-bx)^{2}\equiv x^{2}-U^{2}t^{2},$

or

$c^{2}-b^{2}U^{2}=1,\ a^{2}-d^{2}-d^{2}U^{-2}=1,\ cd=abU^{2},$

which can be satisfied by

$a=c=\sec \beta ,\ bU=d/U=\tan \beta .$

Here β is arbitrary, but we must also have ξ= 0 identical with x = vt, or

$v=d/c=U\ \sin \beta ,$

and thus the complete transformation is determined:

${\begin{array}{l}\tau =t\ \sec \beta -x\ \tan \beta /U\\\\\xi =x\ \sec \beta -Ut\ \tan \ \beta \\\\\eta =y,\ \zeta =z,\ \sin \beta =v/U.\end{array}}$

5. The physical interpretation of this transformation is simple. The coordinates y and z are unaltered, but we have at any given time t

$\xi _{2}-\xi _{1}=\left(x_{2}-x_{1}\right)\sec \beta .$

The measured distance between two points in the direction of motion is therefore greater than the actual distance in the ratio of sec β : 1. This accords with the idea already entertained that the corresponding dimension of the material system, including any scale which may be used for making measures, is actually diminished in consequence of the motion in the ratio cos β : 1.

At a given position in the ether

${\frac {\partial \tau }{\partial t}}=\sec \beta ,$

which means that a stationary clock made to synchronise with passing clocks keeping "local" time must be accelerated in the ratio sec β : 1 when compared with the standard time of space t. But on the other hand,

${\frac {d\tau }{dt}}=\sec \beta -v\ \tan \beta /U=\cos \beta ,$

which means that a "local" clock moving through the ether with the velocity v has a rate retarded in the ratio cos β : 1 when compared with the "standard" clock. The result of the transformation is that we have established a consistent system of apparent time which is such that if we imagine luminous clock-faces at all points of the moving system indicating local time, those which are at equal measured distances from a given station will appear to show the same time, and this a time differing from the local time of the station by an amount equal to the apparent constant distance divided by U.

6. We now see that if the laws of optics relative to the moving system, expressed in terms of ξ, η, ζ and τ, are formally the same as the laws for a stationary system, expressed in terms of x, y, z and t, there will be no possibility of detecting the fact of motion by any optical experiment made with apparatus which is carried with the system.

Now we have from the transformation of § 4,

${\begin{array}{l}{\frac {d\xi }{d\tau }}=\left({\frac {dx}{dt}}-U\ \sin \beta \right)/\left(1-{\frac {dx}{dt}}\sin \beta /U\right)\\\\{\frac {d\eta }{d\tau }}={\frac {dy}{dt}}\cos \beta /\left(1-{\frac {dx}{dt}}\sin \beta /U\right)\\\\{\frac {d\zeta }{d\tau }}={\frac {dz}{dt}}\cos \beta /\left(1-{\frac {dx}{dt}}\sin \beta /U\right)\end{array}}$

for the relative velocities. Let W be the absolute velocity of a ray which is parallel to the axis of x or ξ in a moving medium which has the refractive index μ when at rest. Then

${\frac {d\xi }{d\tau }}=(W-v)/\left(1-Wv/U^{2}\right).$

But this is μ^-1U if the apparent ray-velocity is unaltered by the motion. Hence

$W-v=\mu ^{-1}U\left(1-Wv/U^{2}\right)$

or

$W=\mu ^{-1}U+\left(1-\mu ^{-2}\right)v/\left(1+\mu ^{-1}v/U\right).$

The second member on the right is the apparent drift of plane light-waves which are normal to the direction of motion. To the first order it agrees with Fresnel’s expression (1 - μ^-2)v and is consistent with Fizeau’s experiment, which cannot be performed accurately enough to verify it to a higher order. For the rest we have a general explanation of the null effect of optical experiments without supposing that the ether is carried along in the neighbourhood of the Earth without relative motion.

7. According to the electromagnetic theory a plane wave of light depends upon two periodic vectors, the components of which contain the factor

$\sin \ 2\pi w\{t-(ax+by+cz)/W\}$,

where a, b, c are the direction cosines of the wave-normal and W is the wave-velocity. To the observer moving with velocity v, the corresponding factor is

$\sin \ 2\pi w'\{\tau -(a'\xi -b'\eta +c'\zeta )/W'\}$.

The transformation of § 4 gives the relations

${\begin{array}{rl}w&=w'(\sec \beta +a'U\ \tan \beta /W')\\\\wa/W&=w'(a'\ \sec \beta /W'+\tan \ \beta /U)\\\\wb/W&=w'b'/W'\\\\wc/W&=w'c'/W'\end{array}}$

or, if the transformation be reversed (which only requires the sign of β to be changed),

${\begin{array}{rl}w'&=w\ \sec \beta (1-aU\sin \beta /W)\\\\a'/W'&=(a/W-\sin \beta /U)/(1-aU\ \sin \beta /W)\\\\b'/W'&=b\ \cos \beta /W(1-aU\ \sin \beta /W)\\\\c'/W'&=c\ \cos \beta /W(1-aU\ \sin \beta /W)\end{array}}$

The latter equations give

$1/W'^{2}=\left\{1-2aW\ \sin \beta /U-\left(1-a^{2}-W^{2}/U^{2}\right)\sin ^{2}\beta \right\}\div W^{2}(1-aU\ \sin \beta /W)^{2},$

or

$1/W{}^{2}=\left\{1+2a'W'\ \sin \beta /U-\left(1-a'^{2}-W'^{2}/U^{2}\right)\sin ^{2}\beta \right\}\div W'^{2}(1+a'U\ \sin \beta /W')^{2},$

which express the apparent wave-velocity in a moving medium in terms of the absolute, or vice versa. For a vacuum we put W = W' = U, and we have

${\begin{array}{l}w'=w\ \sec \beta (1-a\ \sin \beta )\\\\a'=(a-\sin \beta )(1-a\ \sin \beta )\\\\b'=b\ \cos \beta /(1-a\ \sin \beta )\\\\c'=c\ \cos \beta /(1-a\ \sin \beta ).\end{array}}$

8. The laws of stellar aberration and of the Doppler effect are at once deducible, as Einstein^[3] has shown. For a star situated in the direction making an angle φ with the direction of motion of the observer we have a = - cos φ and a' = - cos φ', where φ' is the apparent direction. Then we see that

${\begin{array}{rl}w'=&w\ \sec \beta (1+\sin \beta \ \cos \phi )\\\\\cos \phi '=&(\cos \phi +\sin \beta )/(1+\sin \beta \ \cos \phi )\end{array}}$

These expressions can, however, be simplified. For the latter equation gives

$\sin \phi '=\cos \beta \ \sin \phi /(1+\sin \beta \ \cos \phi ).$

Hence

$w'\sin \phi '=w\ \sin \phi .$

Again,

${\frac {1-\cos \phi '}{1+\cos \phi '}}={\frac {(1-\sin \beta )(1-\cos \phi )}{(1+\sin \beta )(1+\cos \phi )}}$

or

$\tan {\frac {1}{2}}\phi '={\sqrt {\frac {U-v}{U+v}}}\tan {\frac {1}{2}}\phi ,$

which puts the law of aberration in an extremely simple form. If we use wave-lengths instead of wave-frequencies we have

$\lambda '/\sin \phi '=\lambda /\sin \phi .$

This form, which connects directly the apparent wave-length with the apparent aberrational position of the star, becomes useless when φ = φ' = 0, but in this case the product of the last two equations gives in the limit

$\lambda '={\sqrt {\frac {U-v}{U+v}}}\lambda ,$

which is the new form of Doppler’s principle. It must he remembered that v is the motion of the observer relative to the medium, and that λ depends on the unknown velocity of the source of light relative to the medium. In some cases we may fairly assume that λ is constant, but λ as well as φ is originally unknown and, if the principle of relativity be accepted in its widest extent, remains unknowable.

9. The geometrical significance of the new law of aberration is interesting. In the first place we may consider the stereographic projection of the celestial sphere on the tangent plane perpendicular to the direction of motion. The form of the law of aberration

$\tan {\frac {1}{2}}\phi '={\sqrt {\frac {U-v}{U+v}}}\tan {\frac {1}{2}}\phi$

shows that the effect of aberration is simply to alter the scale of the projection. But the stereographic projection is a conformal representation of the sphere. Hence actual configurations on the sphere are only changed conformally by the effect of aberration, or in other words any small area is altered only in size and not in shape. We know that aberration merely changes the scale of a photograph of a small part of the sky, and the truth of this fact now becomes independent of the velocity (however large) of the observer. Also stars which appear to lie on a circle at any one time will continue to do so permanently.

Another geometrical representation is obtained by assimilating φ' to the eccentric and φ to the true anomaly in an ellipse whose eccentricity is v/U = sin β. This means that we view the apparent celestial sphere from the centre. Then, to pass from the apparent direction of a star to its actual direction, we must imagine the sphere transformed into an ellipsoid by contracting its dimensions perpendicular to the axis along which motion takes place in the ratio of cos β : 1. The true direction will then be inferred by viewing the ellipsoid of revolution from the focus which is in advance of the centre. Conversely, we may interchange φ and φ' if we employ the other focus. We thus see that if we consider the true celestial sphere to undergo the contraction just specified, the apparent positions of the stars are given by viewing the ellipsoid from that focus which follows the centre.

10. Secular Aberration.— In what precedes, the question of all second-order effects connected with aberration has taken a new form. But we may return to the old order of ideas to consider briefly the subject of secular aberration which arises from the motion of the solar system through the ether of space, and which was first discussed by Villarceau.^[4] It is a very simple matter when regarded from the old point of view, but it has led to some apparent confusion. It has been seen (vol. lxix. p. 505) that if the light which leaves a star at the time T is observed at the time t, the observed direction of the star coincides with the actual direction of a fictitious body which is supposed to start from the position of the star at time T, and to continue in motion during the time (t — T) with a velocity equal and parallel to that of the Earth at the time t.

Now the velocity of the Earth is compounded of its velocity v relative to the centre of mass of the solar system and the velocity V (supposed constant) of the system. If V' is the resultant and U the velocity of light, we have

$ES=U(t-T),\ SS'=V'(t-T),$

E (fig. 3) being the position of the Earth at time t, S the position of the star at time T, ES' the observed direction, and SS' the virtual displacement of the star. But this displacement is compounded of two, SS₀ and S₀S', such that

$SS_{0}=V(t-T),\ S_{0}S'=v(t-T).$

Let ES₀ = Uτ. If then we ignore the velocity V, as in practice we do, and assume a virtual star in the permanently displaced position S_O, we shall infer from observation a fictitious velocity of the Earth v' such that

$S_{0}S'=v(t-T)=v'\tau$

or

$v'/v=(t-T)/\tau =ES/ES_{0}=U/{\sqrt {U^{2}+V^{2}+2UV\ \cos \phi }},$

where φ is the angle between the true direction of the star and the direction of the secular motion of the solar system. Otherwise expressed, the result is as if the constant of aberration for the given star is changed in the ratio of $U:{\sqrt {U^{2}+V^{2}+2UV\ \cos \phi }}$. This is an elementary deduction of Villarceau’s result, and on the assumed premises no other effect is to be expected.

The subject of secular aberration was discussed by Seeliger,^[5] who arrived at results which are quite inadmissible. His expressions for the effects of aberration in R.A. and declination become infinite at the pole, whereas it is obvious that there is no singularity in the neighbourhood of this point in the sky. There is no special connection between the pole and the direction of the operative vectors considered above, and such results can only be due to faulty analysis. It would have been unnecessary to refer to this paper, had not the same errors been reproduced by Weinek^[6] in an elaborate memoir of recent date. A correct treatment of the question has been given by Schwarzschild^[7] on the lines of the present paragraph. But of course the whole question has now entered on a new phase.

11. It is easy to see that the old and the new laws of aberration agree to the first order of small quantities. Beyond the first order, the present accuracy of astronomical observations does not encourage us to look for any means of discriminating between them. Nevertheless it is interesting to ask whether the principle of relativity, as outlined above, has robbed us of the theoretical possibility of detecting the effect of the motion of the solar system through the ether of space. This has been asserted, and we have now to find a satisfactory justification for the assertion. It may be thought that a sufficient argument is contained in the statement italicised in § 6. But this does not appear to cover the case of aberration without a more critical examination. The motion of the observer does affect his observations, even when expressed in terms of the transformed variables, and we cannot dispense with the necessity of explaining in detail how the expected compensation operates when the observations are corrected in a definite, specified manner. And the question is subtle enough to leave ample room for misapprehension. The most obvious instance of this seems worthy of attention. Let us imagine observations made simultaneously by an observer E on the Earth and an observer S on the Sun (or rather the centre of gravity of the solar system, supposed to be moving with uniform velocity through space). Referred in the usual way to a sphere (fig. 4), let S be the apex of the Sun’s ether-velocity V₀, and E the apex of the Earth’s ether-velocity V₁. Let P represent the true direction of a star, and P₀, P₁ the positions

which it appears to the observers S and E respectively to occupy. Now we should have complete agreement between the two observers, and total compensation for E, if the correction applied by E had the effect of changing P₁ into P₀. The observer E must suppose that he is moving relatively to S in the direction of some point E' on the great circle SE; and if he is to deduce the position P₀ in space from the observed position of the star P₁, the point E' must lie also on the great circle P₀P₁. Since P₀, P₁, E' are points on the sides of the triangle SPE, and S, E, E' are points on the sides of the triangle PP₀P₁, we have

${\begin{array}{ccccc}{\frac {\sin SE'}{\sin EE'}}&\cdot &{\frac {\sin EP_{1}}{\sin P_{1}P}}&\cdot &{\frac {\sin PP_{0}}{\sin P_{0}S}}=1\\\\{\frac {\sin P_{0}E'}{\sin P_{1}E'}}&\cdot &{\frac {\sin P_{1}E}{\sin PE}}&\cdot &{\frac {\sin PS}{\sin P_{0}S}}=1.\end{array}}$

Let

$PS=\phi _{0},\ P_{0}S=\phi '_{0},\ PP_{0}=\phi _{0}\ \phi '_{0}$

$PE=\phi _{1},\ P_{1}E=\phi '_{1},\ PP_{1}=\phi _{1}-\phi '_{1}.$

Hence

${\frac {\sin \ SE'}{\sin \ EE'}}={\frac {\sin \left(\phi _{1}-\phi '_{1}\right)}{\sin \phi '_{1}}}\cdot {\frac {\sin \phi '_{0}}{\sin \left(\phi _{0}-\phi '_{0}\right)}}$

${\frac {\sin P_{0}E'}{\sin P_{1}E'}}={\frac {\sin \phi _{1}}{\sin \phi '_{1}}}\cdot {\frac {\sin \phi '_{0}}{\sin \phi {}_{0}}}.$

The first of these shows that

${\frac {\sin \left(\phi _{1}-\phi '_{1}\right)}{\sin \phi '_{1}}}\div {\frac {\sin \left(\phi _{0}-\phi '_{0}\right)}{\sin \phi '_{0}}}=const.$

for all stars, which happens to be true according to the ordinary theory of aberration. But it is not true according to the "relativity" law of aberration, which gives

${\frac {\sin \left(\phi -\phi '\right)}{\sin \phi '}}={\frac {\sin \beta +(1-\cos \beta )\cos \phi }{\cos \beta }}.$

In fact it appears that a law of aberration, which would explain the absence of a visible effect arising from the secular motion by supposing that the corrected observation coincides in space with the standard observation, would require simultaneously

$\sin(\phi -\phi ')/\sin \phi '=f_{1}(V),\ \sin(\phi -\phi ')/\sin \phi =f_{2}(V),$

and this is not possible. We have then to look in another direction for the explanation of the way in which the "relativity" law effects a compensation, and for the necessary hint I am indebted to Mr. Eddington, who very kindly supplied me with a particular case which was both simple and illuminating.

12. A general proof will now be given. Let the ether-velocity of the observer S be $V_{0}=U\ \sin \beta _{0}$, and of the observer E be $V_{1}=U\ \sin \beta _{1}$, and let the angle between them be α, being measured from V₀ towards V₁. The components of V₀ along and perpendicular to V₁ are $V_{0}\cos \alpha ,\ -V_{0}\sin \alpha$. Hence for E the components of the apparent velocity of S are (by § 6)

${\frac {V_{0}\cos \alpha -U\ \sin \beta _{1}}{1-V_{0}\cos \alpha \sin \beta _{1}/U}},\ {\frac {-V_{0}\sin \alpha \ \cos \beta _{1}}{1-V_{0}\cos \alpha \sin \beta _{1}/U}}$

or

${\frac {U\left(\cos \alpha \ \sin \beta _{0}-\sin \beta _{1}\right)}{1-\cos \alpha \sin \beta _{0}\sin \beta _{1}}},\ {\frac {-U\ \sin \alpha \ \sin \beta _{0}\sin \beta _{1}}{1-\cos \alpha \sin \beta _{0}\sin \beta _{1}}}.$

Now E must infer that his own velocity relative to S, $V_{10}=U\ \sin \beta _{10}$, has these components reversed in sign; and if θ be the angle between V₁ and the resultant we have

$\sin \beta _{10}\cos \theta =\left(\sin \beta _{1}-\cos \alpha \ \sin \beta _{0}\right)/\left(1-\cos \alpha \sin \beta _{0}\sin \beta _{1}\right)$

$\sin \beta _{10}\cos \theta =\sin \alpha -\sin \beta _{0}\ \cos \beta _{1}/\left(1-\cos \alpha \sin \beta _{0}\sin \beta _{1}\right)$

and we deduce

$\cos \beta _{10}=\cos \beta _{0}\ \cos \beta _{1}/\left(1-\cos \alpha \sin \beta _{0}\sin \beta _{1}\right).$

We take the z-axis perpendicular to V₀ and V₁ throughout, and

(i) The x-axis parallel to V₀. If (λ, μ, ν) are the direction cosines of a star in its true position, S observes this star in the direction (by § 7)

$\lambda _{0}={\frac {\lambda +\sin \beta _{0}}{1+\lambda \sin \beta _{0}}},\ \mu _{0}={\frac {\mu \ \cos \beta _{0}}{1+\lambda \sin \beta _{0}}},\ \nu _{0}={\frac {\nu \ \cos \beta _{0}}{1+\lambda \sin \beta _{0}}}.$

(ii) We turn the x-axis through the angle a to bring it into the direction of V₁. The direction cosines of the star in its true position become

$\lambda \ \cos \alpha +\mu \ \sin \alpha ,\ \mu \ \cos \alpha -\lambda \ \sin \alpha ,\ \nu ,$

and E will observe it in the direction

${\begin{array}{lr}\lambda _{1}=&\left(\lambda \ \cos \alpha +\mu \ \sin \alpha +\sin \beta _{1}\right)/\left\{1+\left(\lambda \ \cos \alpha +\mu \ \sin \alpha \right)\sin \beta _{1}\right\}\\\\\mu _{1}=&(\mu \ \cos \alpha -\lambda \ \sin \alpha )\cos \beta _{1}/\left\{1+\left(\lambda \ \cos \alpha +\mu \ \sin \alpha \right)\sin \beta _{1}\right\}\\\\\nu _{1}=&\nu \ \cos \beta _{1}/\left\{1+\left(\lambda \ \cos \alpha +\mu \ \sin \alpha \right)\sin \beta _{1}\right\}\end{array}}$

(iii) We turn the x-axis through the further angle θ to bring it into the direction of V₁₀. The direction cosines of the star in its apparent position become

$\lambda _{1}\cos \theta +\mu _{1}\sin \theta ,\ \mu _{1}\cos \theta -\lambda _{1}\sin \theta ,\ \nu _{1},$

and when E has corrected this position for his observed relative velocity V₁₀ he will infer that the star lies in the direction

${\begin{array}{lr}\lambda _{10}=&\left(\lambda _{1}\ \cos \theta +\mu _{1}\ \sin \theta -\sin \beta _{10}\right)/\left\{1-\left(\lambda _{1}\ \cos \theta +\mu _{1}\ \sin \theta \right)\sin \beta _{10}\right\}\\\\\mu _{10}=&(\mu _{1}\ \cos \theta -\lambda _{1}\ \sin \theta )\cos \beta _{10}/\left\{1-\left(\lambda _{1}\ \cos \theta +\mu _{1}\ \sin \theta \right)\sin \beta _{10}\right\}\\\\\nu _{10}=&\nu _{1}\ \cos \beta _{10}/\left\{1-\left(\lambda _{1}\ \cos \theta +\mu _{1}\ \sin \theta \right)\sin \beta _{10}\right\}.\end{array}}$

These have to be compared with $\lambda _{0},\ \mu _{0},\ \nu _{0}$, remembering that the axes have been turned through an angle α + θ.

13. The process of eliminating $\lambda _{1},\ \mu _{1},\ \nu _{1}$ and θ is perfectly simple and straightforward, if rather complicated. We find

$1-\left(\lambda _{1}\cos \theta +\mu _{1}\sin \theta \right)\sin \beta _{10}={\frac {\left(1+\lambda \ \sin \beta _{0}\right)\cos ^{2}\beta _{1}}{\left\{1+\left(\lambda \ \cos \alpha +\mu \ \sin \alpha \right)\sin \beta _{1}\right\}\left(1-\cos \alpha \ \sin \beta _{0}\ \sin \beta _{1}\right)}}$

whence

$\nu _{10}=\nu \ \cos \beta _{0}/\left(1+\lambda \sin \beta _{0}\right).$

Also

${\begin{array}{r}\sin \beta _{10}\left(\mu _{1}\cos \theta -\lambda _{1}\sin \theta \right)\left\{1+\left(\lambda \ \cos \alpha +\mu \ \sin \alpha \right)\sin \beta _{1}\right\}\left(1-\cos \alpha \ \sin \beta _{0}\ \sin \beta _{1}\right)\\\\=\cos \beta _{1}\left\{\mu \left(\cos \alpha \ \sin \beta _{1}-\sin \beta _{0}\right)-\left(\lambda +\sin \beta _{0}\right)\sin \alpha \ \sin \beta _{1}\right\}\end{array}}$

whence

$\mu _{10}\sin \beta _{10}={\frac {\cos \beta _{0}\left\{\mu \left(\cos \alpha \ \sin \beta _{1}-\sin \beta _{0}\right)-\left(\lambda +\sin \beta _{0}\right)\sin \alpha \ \sin \beta _{1}\right\}}{\left(1+\lambda \ \sin \beta _{0}\right)\left(1-\cos \alpha \ \sin \beta _{0}\ \sin \beta _{1}\right)}}.$

Similarly a little reduction gives

$\lambda _{10}\sin \beta _{10}={\frac {\left(\lambda +\sin \beta _{0}\right)\left(\cos \alpha \ \sin \beta _{1}-\sin \beta _{0}\right)+\mu \ \sin \alpha \cos ^{2}\beta _{0}\sin \beta _{1}}{\left(1+\lambda \ \sin \beta _{0}\right)\left(1-\cos \alpha \ \sin \beta _{0}\ \sin \beta _{1}\right)}}.$

We come now to the interpretation of these expressions. The components of V₁ along and perpendicular to V₀ are V₁ cos α, V₁ sin α. Hence the apparent components of the relative velocity of E as observed by S are

${\frac {V_{1}\cos \alpha -U\ \sin \beta _{0}}{1-V_{1}\cos \alpha \sin \beta _{0}/U}},\ {\frac {V_{1}\sin \alpha \cos \beta _{0}}{1-V_{1}\cos \alpha \sin \beta _{0}/U}},$

or

${\frac {U\left(\cos \alpha \ \sin \beta _{1}-\sin \beta _{0}\right)}{1-\cos \alpha \sin \beta _{0}\sin \beta _{1}}},\ {\frac {U\ \sin \alpha \cos \beta _{0}\sin \beta _{1}}{1-\cos \alpha \sin \beta _{0}\sin \beta _{1}}}.$

Hence if χ be the angle between V₀ and the resultant $V_{01}=U\ \sin \beta _{01}$, we have

$\sin \beta _{01}\cos \chi =\left(\cos \alpha \ \sin \beta _{1}-\sin \beta _{0}\right)/\left(1-\cos \alpha \ \sin \beta _{0}\ \sin \beta _{1}\right)$

$\sin \beta _{01}\sin \chi =\sin \alpha \ \cos \beta _{0}\sin \beta _{1}/\left(1-\cos \alpha \ \sin \beta _{0}\ \sin \beta _{1}\right)$

and we deduce

$\cos \beta _{01}=\cos \beta _{0}\cos \beta _{1}/\left(1-\cos \alpha \ \sin \beta _{0}\ \sin \beta _{1}\right).$

We see that $\beta _{01}=\beta _{10}$, and when we introduce $\lambda _{0},\ \mu _{0},\ \nu _{0}$ and χ into the last expressions for $\lambda _{10},\ \mu _{10},\ \nu _{10}$ we obtain simply

$\lambda _{10}=\lambda _{0}\cos \chi +\mu _{0}\sin \chi ,\ \mu _{10}=\mu _{0}\cos \chi -\lambda _{0}\sin \chi ,\ \nu _{10}=\nu _{0}.$

The interpretation is that the sky as seen by the observer E and referred to the direction in which he observes himself to be moving relative to S, is identical with the sky as seen by the observer S, and referred to the direction in which he observes E to be moving relative to himself. Thus the observations of E, corrected for his observed relative velocity, give the same configuration as the observations of S uncorrected, and the latter is a permanent standard so long as his velocity is uniform. But for E the whole universe is rotated in space through an angle θ + α - χ. This rotation is precisely that which is required to enable the two observers to identify the same star as occupying the position which each observer imagines to represent the apex of the motion of E relative to S, and thus it could not be detected even though the two observers were in communication. The difficulty discussed in § 11 arises from the fact that the observed velocity of E relative to S and the observed velocity of S relative to E are not in the same straight line. We find that

$\tan(\chi -\theta )={\frac {\sin \alpha \left(\cos \beta _{0}+\cos \beta _{1}\right)}{\cos \alpha \left(1+\cos \beta _{0}\cos \beta _{1}\right)-\sin \beta _{0}\sin \beta _{1}}},$

which can be reduced to the form

$\tan {\frac {1}{2}}(\chi -\theta )={\frac {\cos {\frac {1}{2}}\left(\beta _{1}-\beta _{0}\right)}{\cos {\frac {1}{2}}\left(\beta _{1}+\beta _{0}\right)}}\tan {\frac {1}{2}}\alpha$

Thus χ - θ is not equal to α, as it would be in ordinary kinematics.

14. We have thus given a perfectly precise meaning to the application of the principle of relativity to stellar aberration. In doing so we have excluded everything which does not belong to the purely optical aspect of the problem. But stellar aberration is not a purely optical problem. For in practice we do not actually observe the apparent motion of the Sun and use the result to correct our observed positions of the stars. The motion which we do use is derived by calculation from the theory of gravitation. Hence, if we are to be consistent, we must regard Keplerian motion as an appearance, not as a reality. And here we come in contact with the general problem of the dynamics of the electron, which in the historical sense is responsible for the introduction of the principle of relativity. In this wider problem there is a profound modification of the notion of mass, in addition to the transformation which we have admitted in the time and the spatial coordinates. The result of the work of Lorentz and others is to show that these transformations suffice to explain the complete compensation of effects arising from the motion of any system through space over the whole field of electrodynamics as well as of optics. The same will be true of gravitation if gravity can be expressed in terms of electrodynamic entities. If, on the other hand, the nature of gravity were essentially not electrodynamic, it would be possible for some deviation from the accepted laws of dynamics, owing to a motion of translation through space, to be revealed by direct astronomical observations. The possibility of such deviations and of the compensations, partial or complete, which may accompany them, places some astronomical problems in a new position. Thus Poincaré has recently reconsidered^[8] the problem of Laplace concerning the finite propagation of gravitation, and has concluded, contrary to Laplace’s negative result, that the propagation of gravitation with the velocity of light can be brought into consistency with the facts of observation owing to the partial compensations which are now introduced into the theory. Such questions are in the highest degree subtle and intricate as well as interesting. What it is desired to point out is that the developments of modern physical theory concern the astronomer no less than the physicist.

---

1. The fundamental hypothesis concerning the contraction of matter in motion is due to Fitzgerald; the first mention of his suggestion occurs in an abstract of a paper by Sir O. Lodge (Nature, 1892, June 16). This reference I owe to Professor Whittaker.
2. Mon. Not., lxix. p. 504. Professor Whittaker considers that Veltmann’s paper added nothing to Fresnel’s original paper in Annales de Chemie, ix. p. 57 (1818). In attaching importance to Veltmann’s theorem I have followed Jamin (Cours de Physique).
3. Ann. der Physik, 1905, No. 10, p. 891.
4. Conn. des Temps, 1878.
5. A.N., 2610.
6. Denkschr. der Wiener Akad., Bd. 77, p. 204.
7. A.N., 3246.
8. C.R., cxl. p. 1504.