A SHORT NOTE ON EINSTEIN'S PLANETARY EQUATION. 



By W. N. RosFAEAiiE, M.A., ^'<^\Cy47 



Professor of Mathematics, Natal University College, ^"^^ * -^ 



Pietermaritzhurg . '^ "" 



Read July 15, 1920. 



§ 1. — Einstein's Relativity Theory of Gravitation, as given in 

 Eddington's Report to the Physical Society (1918), involves 

 much heavy, but, on the whole, satisfying, mathematical work. 

 The results for the case of a point-centre of gravitation are tha.t 

 ds, the " interval " (including space and time) between two events 

 is given by 



- ids)- ==--!{.' ^ (dv)^ + r-(de)^ + r''sm-0(d<py -Ridty ; . . . . (i) 

 and, hence, for a plane orbit (writing henceforward 6 for 0), 



/du\'' 3 ^ 2mu 1— c^ / 1\ 



(^^^^j=2mu3-u^+ ^^^ - -j^-,-. [n = ~j 



where R =l-2m/r, m being merely a constant of integration, 

 "constant " for all motion in the field of this point-centre; 



1 _ od^ c _ dt 



"=r''-^ ' ~j^= -^', where h, c are constants for any one orbit. 



r. 6, <p are (more or less) identified with ordinary polar co-ordinates. 

 But, " the considerable freedom of choice of co-ordinates allowed 

 by Einstein's equations " is invoked with disturbing frequency in 

 order to sanction " the possibility of using any function of the 

 radius vector instead of r itself." And we are told that we " may 

 meet elsewhere with different expressions for the line-element 

 due to a particle." The one used by Eddington was given by 

 Schwarzschild. By means of it (and of the rather free modifica- 

 tions in the co-ordinate r referred to above) Eddington, following 

 Einstein, obtains clearly enough the two great results by which the 

 new theory claims to have improved on the old Newtonian theory 

 — the theoretical advance of Mercury's perihelion, and the theore- 

 tical deflection of light by a field of gravitation, which seems to 

 have been so triumphantly confirmed by the observation of the 

 May, 1919, eclipse. 



§ 2. — Two discrepancies in the work, arising from the free 

 treatment of the radius vector are worth mentioning : — 



1. As pointed out by Prof. Anderson, of Galway, in the 

 Phil. Mag., May, 1920, the special modification of r necessary 

 to establish the deflection of light (Eddington, p. 53) would, if 

 appHed to the same fundamental equation (i) above, not give 



