February 17. 1921] 



NATURE 



793 



of Fizeau's water-tube experiment, the prediction 

 of the law connecting electronic mass with velo- 

 city, and the prediction of ponderomotive electro- 

 magnetic forces in moving media. 



One final, and therefore crucial, test remains : 

 gravitation. It was soon noticed that the hypo- 

 thesis was inconsistent with the exact truth of 

 Newton's gravitational law of force mm' jr^- Thus 

 the hypothesis of relativity predicts that a freely 

 moving planet cannot describe a perfect ellipse 

 about the sun as focus. This prediction is made 

 on quite general grounds, just as the conservation 

 of energy predicts that a stream of water cannot 

 flow uphill. But the conservation of energy by 

 itself is powerless to predict what will be the 

 actual course of a stream of water, and in pre- 

 cisely the same way the hypothesis of relativity 

 atone is powerless to predict what will be the orbit 



of a planet. Before this or any other positive 

 gravitational predictions can be made, additional 

 hypotheses must be introduced. The main trunk 

 of the tree is the relativity hypothesis already men- 

 tioned ; these additional hyfxjtheses form the 

 branches. The trunk can exist without its 

 branches, but not the branches without the trunk. 

 Whether the branches have been placed on the 

 trunk with complete accuracy is admittedly still 

 an open question- — it must of necessity remain so 

 until the difficult questions associated with the 

 gravitational shift of spectral lines have been 

 finally settled — but the main' trunk of the tree can 

 be disturbed by nothing short of a direct experi- 

 mental determination of the absolute velocity of 

 the earth, and the only means which can possibly 

 remain available for such a determination now are 

 gravitational. 



The Michelson-Morley Experiment and the Dimensions of Moving Bodies. 

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



AS doubts have sometimes been expressed con- 

 cerning the interpretation of Prof. Michel- 

 son's celebrated experiment, some remarks on the 

 subject will perhaps not be out of place here. 1 

 shall try to show, by what seems to me an unim- 

 peachable mode of reasoning, that, if we adopt 

 Kresnel's theory of a stationary aether, supposing 

 also that a material .system can have a uniform 

 translation with constant velocity v without 

 changing its dimensions, we must surely expect 

 the result that was predicted by Maxwell. 



Let us introduce a system of rectangular axes 

 of co-ordinates fixed to the material system, the 

 axis of X being in the direction of the motion. 

 Then, with respect to these axes, the a;ther will 

 tlow with the velocity —v. The progress of waves 

 of light, relatively to them, may be traced 

 by means of Huygens's principle; for this 

 purpose it suffices to know the form and (josition 

 of the elementary waves. F"or the sake of gene- 

 rality I shall suppose the propagation to take 

 place in a material medium of refractive index 

 ji, so that, if c is the velocity of light in the 

 jcther, the velocity in the medium when at rest 

 would be cjfi. The elementary wave formed in the 

 time dt around a point P will be a sphere of radius 

 ic/fi.)dt, of which the centre P' does not, however, 

 coincide with P, the line PF" being in the direction 

 opposite to that of C)X. and having the length 

 {v/ii.*)dt (Kresnel's rfH-fTicicnt). 



If f} is any point on the surface of the sphere, 

 VQ can be considered as an clement of a ray of 

 light, and ic— PQ 'dt will be the velocity of the ray. 

 Confining ourselves to terms of the second order, 

 i.e. of the order t'/c*. and denoting by 8 the 

 angle between thr ray and OX, wc have 



' -'*-H'lcot«-t- "Vl+co»««) 



w 



^>' 



fl) 



ray of light passing from .\ to B will be deter- 

 mined by the condition that the integral 



Now, let A and B be points having fixed posi- 

 tions in the material system. The course .« of a 



NO. 2677, VOL. 106] 



J to. 



(«) 



is a minimum. Using the above value of ijio, it is 

 easily shown that, if quantities of the second 

 order are neglected, the course of the ray is not 

 affected by the translation v, so that, if Lq is the 

 path of the ray in the case t' = o, and L the real 

 path, these lines will be distant from each other 

 to an amount of the second order only. Hence, 

 if in the case of a translation v we calculate by 

 means of (i) the integral (2), both for L and L^, 

 the two values will differ by no more than a quan- 

 tity of the fourth order ; indeed, since the integral 

 is a minimum for L, the difference must be of 

 the second order with respect to the distances 

 between L and L,,, and these distances are already 

 of the second order of magnitude. 



It is seen in this way that, so long as we 

 neglect terms of an order higher than the second, 

 we may replace 



K' by l''\ 



.' W J It' 



an argument that must not be overlooked in the 

 theory of the experiment. On the ground of it 

 we shall commit no error if, in determining the 

 paths L, and Lj of two rays that start .from a 

 point A, and are made to interfere at a pf)int B, 

 we take no account of the motion of the apparatus. 

 The change in the difference of phase prtKluced by 

 the translation will be given by the difference 

 between the values which the integral 





I + CO*' 4) t/s 



takes for the lines L, and L, so determined. If, 

 along the first of them, ros*J=i, and along the 



