of Relativity, and Michel son- Mori ey Experiment. 825 



but the vectors and the time differ from the stationary values 

 by first order terms, as in the theory of translation. 



But we cannot proceed in the ordinary way to a more 

 complete correspondence, mainly on account of the term 

 'd/'dr'Qr'h') in the last equation. It is evident that a new 

 time variable ^ for which dt x is a perfect differential cannot 

 be employed without introducing time differentiations on the 

 right, and it may be shown that, in particular, the trans- 

 formation required by either the principle of relativity or 

 the buckling theory of the rotating body cannot reduce the 

 equations to the old type, even with any readjustment of 

 the vectors concerned. 



The principle of relativity thus cannot be derived analyti- 

 cally for the case of rotation, except as a first order principle, 

 and as we have seen, its extension beyond the problems con- 

 templated by Einstein is not warranted. We must therefore, 

 since the motion of the earth is not translation, expect second 

 order velocity effects due to the motion of the earth in ex- 

 periments like those of Michelson and Morley. But these 

 effects are, by virtue of another factor, incapable of detection 

 by present experimental means. Let a be the radius of the 

 earth, and consider an experiment performed at its surface. 

 The value of: e _1 is 



€ -i = l-a) 2 (a+£)7C 2 , 



where f is the height of a point above the surface. If 

 B~ 1 = l — a) 2 a 2 /C 2 , make the transformation 



dt" = &dt l9 (a^B &!,<?!) = («',&', c'), 1 ^ 



Then the equations (12) are unaltered in type except for a 

 substitution of ej8 for e. But e/8 = l-\-2co 2 a^/C 2 , and the 

 equation of this quantity to zero can only affect the result 

 of any experiment, as prophesied by the equations, to a re- 

 lative order &) 2 af/C 2 or v 2 %jaC 2 , where f is the greatest 

 length concerned in the apparatus, and v is the surface 

 velocity. 



The value of f in the Michelson-Morley experiment was 

 about 20 metres, so that f/aisof order 10 " 6 . But v 2 /C 2 is of 

 order 10~ 14 approximately, so that the final effect is actually 

 of the third order in the velocity ratio, by virtue of the other 

 factor. Morley and Miller * found it possible to conclude 

 that if there is any effect of this nature it is less than one 



* Phil. Mag. [6] ix. p. 680 (1905). 



