February 17, 192 1] 



NATURE 



795 



«/;*/( - ^v - L ) (^ , - ^;;) Wf</,,/c=o, 



(7) 





(system S), and \ their distance in the natural 

 state (system S^), these distances being both 

 determined in the manner specified in what pre- 

 cedes. Then 



Again, if P" is the particle f, >; + </>;, f, and if 

 the angle P'PP", calculated as stated before, has 

 in the two cases the values 5 and 5o( = 4ir), we 

 shall have 



The six deformations i, . . . will be considered | 

 as infinitely small. In the problem we have in 

 view, they are of the order of magnitude v^jc^, so 

 that our final result will be correct to that order. 



If we put 



tA(|,« + ,f> + ff'), . (6) 

 the well-known expression for the potential energy 

 of an isotropic elastic body, U will be invariant 

 for any change of co-ordinates. 



.\s to the kinetic energy T, it is to be replaced 



by an expression containing p — . Finally, we 



must write, instead of (3), 



We have still to add the formula; that are 

 found by working out the above definitions of 

 (n (^ etc., viz. :— 



at on ^"Oj^'abVavt 



,, t'j, X', are the components of the velocity, 

 ul .., = ,). 



In our problem the body is supposed to move 

 in a normal system of co-ordinates. By this our 

 formula; simplify to ' 



1 If in (7) wc replace (i - e'/f*)' by i - v^/k^, omilting the constant ittm 

 - 1*0 and negledinf U.r*/>(*, we at* led back M ike ordinary formala (3), 



a^ a,. 2(a)v.3j- -2(0)^.-5- 



d$ dn 



c»-v» 



When applied to a revolving body, these equa- 

 tions will enable us to determine the deformation 

 that is produced, wholly independently of the 

 theory of relativity, by centrifugal force, a de- 

 formation that will in reality far surpass the 

 changes we want to consider. To get free from 

 it we can consider the ideal case of a " rigid " 

 body — i.e. a body for which the moduli of elas- 

 ticity A and B in (6) are infinitely great. The 

 centrifugal force will then have no effect on the 

 dimensions, but the changes required by the theory 

 of relativity will subsist. The assumption has 

 also the advantage of simplifying the calculations ; 

 indeed, since U becomes infinitely great, the term 

 — c^p in {7) may be omitted. 



I have worked out the case of a thin circular 

 disc rotating with constant speed about an axis 

 passing through its centre, at right angles to its 

 plane. The result is that, if v is the velocity at 

 the rim, the radius will be shortened in the ratio of 



I to I— ii- . The circumference changing to the 



same extent, its decrease is seen to be exactly 

 one-fourth of that of a rod moving with the same 

 velocity in the direction of its length. That there 

 would be a smaller contraction was to be ex- 

 pected ; indeed, the case can be compared to what 

 takes place when a hot metal band is fitted tightly 

 around a wheel and then left to cool. 



At first sight our problem seems to lead to a 

 paradox. Let there be two equal discs A and B, 

 mounted on the same axis, A revolving and B at 

 rest. Then A will be smaller than B, and it must 

 certainly appear so (the discs being supposed to 

 be quite near each other) to any observer, what- 

 ever be the system of co-ordinates he chooses to 

 use. However, we can introduce a system of 

 co-ordinates S' revolving with the disc A ; with 

 respect to these it will be B that rotates, and so 

 one might think that now this latter disc would 

 be the smaller of the two. The conclusion would 

 be wrong because the system S' would not be a 

 normal one. If we leave S for it, we must at 

 the same time change the potentials /jas- a"d if 

 this is done the fundamental equation will cer- 

 tainly again lead to the result that .'\ is smaller 

 than' B. 



The Geometrisation of Physics, and its Supposed Basis on the Michelson-Morley 



Experiment. 



By Sir Oliver Lodce, K.R.S. 



SO much has been written about the Michelson- 

 Morley experiment that it would be needless 

 to refer to it here, had it not been interpreted by 

 philosophic writers in an interesting but over- 

 violent and, as some think, illegitimate manner. 

 Historically it really docs lie at the root of the 

 remarkable attempt which is being made to gco- 



NO. 2677, VOL. 106] 



mctrise physics, and to reduce sensible things 

 like weight and inertia to a modification of space 

 and time. The work of ^reat Geometers has been 

 pressed into the service, and a differential- 

 invariant scheme of expression has been utilised 

 to do for physics in general, and especially for 

 gravitation, what Maxwell's equations did for 



