128 General Relativity without Equivalence Hypothesis. 



The equations of motion of the planet, which can easily he written down 

 according to (41 a), yield a plane orbit. If this plane is taken to be 



= const. = 7r/2 ; then, with e=l, and w=— cot ^, the differential 



equation of the orbit is 



R R' 

 g.+fl+yju-p (47) 



where 



The solution is 



A = _R 2 sin 2 - . -=- = const., 

 R at 



7=M 2 /#. 

 M 



l+*cos(0\/l+y)}, .... (48) 



where e=const. With appropriate "initial" data this is an ellipse, with 

 eccentricity e and with moving perihelion. The motion of the perihelion 

 is, per period T of revolution, 



i. e. rejecting y 2 and higher powers, and replacing c, 



h 2 c 2 ~ c 2 T 2 {l-e 2 Y 



(49) 



where a is the major semiaxis. Thus, the proposed equations give a 

 negative (or retrograde) secular* motion of the perihelion, equal to minus 

 one sixth of that yielded by Einstein's gravitation theory. 



By (49) we should have, per century, the perihelion motion multiplied 

 by the eccentricity, 



e8w= — 1"-48 for Mercury, 



and — 0"-010, -,0"-011, -0"'021 for Venus, Earth, and Mars respec- 

 tively. According to Newcomb ('Astronomical Constants ') the secular 

 excess for Mercury, not accountable for by the perturbation due to all 

 the other planets of our system, is +8" , 48±0 , 43. Thus, if the above 

 equations of motion are accepted, the true excess for Mercury would be 

 still greater, viz. 



e«-or = 8"-48+l""48=9' / -96 (±0-43), 



or Sto- = 48"'44. I understand from a conversation with Mr. Harold 

 Jeffreys, who has already found a satisfactory representation of Mer- 

 cury's 8"-48 and of the motion of the node of Venus by means of a 

 modification of Seeliger's zodiacal-light matter (M.N., R.A.S., December 

 1916, p. 112), that the above, increased, excess of about 10" could 

 equally well, and possibly " more easily," be accounted for by an 

 appropriate distribution of "the said disturbing matter. 



