822 Prof. J. W. Nicholson on Uniform Rotation, Principle 



light. The principle of relativity must therefore be valid as 

 a first order principle, as will be shown analytically below. 

 The exact extent of the divergence between the cases of 

 rotation and translation is of course important in the theory 

 of the Michelson-Morley experiment. The statement is 

 frequently found that such an experiment would give a null 

 result if the experimental accuracy were more refined. This 

 is incorrect if the circuital relations are a valid foundation 

 for electromagnetic theory, for it will be shown that they 

 must apparently lead to an effect, although of a higher order 

 than any experiment yet devised can detect. The effect may 

 be of the second order in the velocity ratio, but actually of 

 higher order than this on account of the large radius of the 

 earth. The assumption that the earth's motion may be treated 

 as translation for a point on its surface has always been 

 tacitly made, with of course sufficient justification from first 

 principles when terms beyond the square of the velocity 

 ratio are neglected. The explicit statement of the ground of 

 the assumption, or of its exact order of validity, does not 

 appear to have been made, nevertheless, although there is no 

 necessity for invoking the subsequent analysis of this paper 

 in order to obtain it. 



Swann * has drawn attention to the normal acceleration 

 as a factor destroying the principle of relativity for rotation, 

 but his analysis is vitiated by the use of an incorrect form 

 of the solenoidal relations of the vectors, which leads to the 

 conclusion that even to the first order, the correspondence 

 between the fixed and moving systems will not, in general, 

 be found. This, as we have seen, cannot be correct. The 

 analysis in other respects proceeds on the natural lines 

 initiated by Larmor and Lorentz in their treatment of 

 uniform translation in one direction. 



In this paper, a formal examination is made of the limits 

 within which a correlation can be obtained on the ordinary 

 analytical lines. It will appear that a second order cor- 

 respondence cannot be found, so that the geometry of uniform 

 rotation cannot be expressed in a simple general way. This 

 is not an unexpected result, since the case of uniformly 

 accelerated motion in one direction, which is probably of 

 simpler character, has not been so expressed. 



Let a set of cylindrical coordinates (r, 0, z) be chosen, 

 where z is the axis of rotation of a system. If (/ g Ji), 

 (a b c) are the components of polarization and magnetic 

 induction along r, 0, and z respectively increasing, then- 

 time rates with respect to axes in which the initial line of 

 * Phil. Maff. March 1911. 



