July 21, 1923] 



NATURE 



103 



The Rotation of the Earth and its Influence on Optical Phenomena.^ 



By Prof. H. A. Lorentz, For. Mem. R.S. 



THERE are different ways in which, by means of 

 optical phenomena, the motion of a system can 

 be detected. I shall speak of them successively, with 

 a view especially to the rotation of the earth, briefly 

 considering also the optical effects that are due to 

 the annual motion, which can be taken to be a trans- 

 lation. 



1. Doppler's Principle. — In the first place there 

 is Doppler's principle. If r is the distance from a 

 luminous source to an observer (or to the slit of the 

 spectroscope), Vr = drjdt the relative velocity in the 

 direction of the hne r, and n the real frequency of the 

 light emitted by the source, the observed frequency 

 will be w + Sw, where 



. Vr 



bn= ——n, 

 c ' 



c being the velocity of light. The corresponding 

 change of the wave-length X is given by 



c 



The velocity of the earth's translational motion is 

 30 km./sec, i.e. y^y^^ c. It can give rise for yellow 

 light to a change in wave-length of about half an 

 Angstrom unit. The displacement of spectral lines 

 produced by it is perfectly observable ; in fact, star 

 velocities of some 50 km./sec. are measured with a 

 considerable precision. 



If the observed shift of the spectral lines of a star 

 is corrected for the motion of the earth, one finds 

 the velocity of the star with respect to the sun. In 

 the case of many spectroscopic binaries, the determina- 

 tion of the elements of their orbits would be wholly 

 impossible if the motion of the earth were not taken 

 into account. 



The velocity of a point of the earth's surface due 

 to the rotation is much smaller than the translational 

 velocity. Even for a point on the equator, it amounts 

 to no more than 0*46 km./sec. The displacement of 

 a spectral line corresponding to this is, for yellow light, 

 about 0*009 A.U., 1/660 part 

 of the distance between the 

 D-lines. This can scarcely 

 be observed. If it were some- 

 what greater, one would see 

 that the lines in the solar 

 spectrum lie somewhat more 

 towards the violet at sunrise 

 than at sunset. It must be 

 remarked that the conse- 

 quences which one draws from 

 Doppler's principle would re- 

 main true whatever might be 

 the state of motion existing in a medium surrounding 

 the earth. The question only is whether two successive 

 vibrations emitted by the source take equal or unequal 

 times to reach the slit of the spectroscope. 



2. Huygens's Construction. — In the second place, 

 the propagation of waves and rays of light may be 

 modified by a motion of the system, a modification 



' Lecture delivered at University College, University of London, on 

 May 17. 



NO. 2803, VOL. I 12] 



Fig. I. 



that can be found by means of Huygens's construction. 



Let Sj (Fig. i) be the wave-front, i.e. the surface that 



is reached at a certain time ^ by a vibration emitted 



by the source at some previous instant. Then, around 



each point A, A', A" ... of 



Sj one can describe the 



elementary wave formed in 



a time dt. The surface Sg 



tangential to them all will 



be the new position of 



the wave-front. The lines 



AC, A'C, . . ., joining the 



centres of the elementary 



waves to the points where 



they are touched by Sg, 



are elements of rays, i.e. of 



the lines which determine the lateral limitation of 



beams of light. The velocity of a ray is given by 



Fig. 



AC 



(I) 



and the course of a ray of light 5 between two given 

 points A and B is determined by the condition that 



Cds 

 I u 



(2) 



is a minimum (Fermat's principle). 



This general method can be applied to the case of 

 ether moving through the diagram with respect to 

 which one wants to know the propagation of light. 

 The elementary wave around a point A (Fig. 2) is a 

 sphere with radius cdt (c velocity of light in ether), 

 but drifting along with the ether. The centre of the 

 sphere will be at B, if AB is in the direction of the 

 velocity v with which the ether moves across the 

 diagram and has the length vdt. From the triangle 

 ABC one finds, if 6 is the angle BAC between the 

 velocity of the ether and the ray AC, and if terms of 

 the order (v/cY are neglected. 



ds 

 u 



ds 

 c 



cos Qds. 



(3) 



The figure also shows to what extent the ray AC 

 deviates from the normal BC to the wave-front. 



3. Stokes's Theory of Aberration. — In this 

 theory it is supposed that the ether is set in motion 

 by the earth, like an incompressible fluid, the velocity 

 of the ether at any point of the surface being equal 

 to the velocity of the earth. At some point P just 

 outside the region where there is an appreciable 

 velocity of the ether, the light coming from some 

 star S will have its wave-front at right angles to PS. 

 The above construction gives the direction of the ray, 

 i.e. the direction in which the star is observed ; the 

 result agrees exactly with that of the well-known 

 elementary theory of aberration. Stokes further 

 supposes that the motion of the ether is irrotational, 

 so that V depends on a velocity potential. In this 

 case (3) shows that (2) may be replaced by (i/<")/^^ 

 plus a term that is independent of the path ; the ray 

 of light is therefore a straight line, and the ordinary 



