10 PRESIDKNt's address — SECTION A. 



It is found that the ])ath is a plane curve given by the differential 

 equations 



"fs-" ■. H" 



where h and k are constants of integration. 



A comparison with the ordinary Newtonian Equations 



'ty ^r'l'^Y = _!!! + !!!!, 



dtl \dt a r ^ 



dt ''' 

 shows that the paths approximately correspond if ^-^ _ j _ ^^y^ 

 and if m is the gravitational mass of the attracting particle and h the 

 angular momentum. ' 



The only outstanding discrepancy arises from the term 2mh''/r^ 

 in equation A (1), and it is found that in this term lies the explanation 

 Df the motion of the perihelion of Mercury. 



The ratios ni/a, ni/r are very small in practical applications. For 

 example, if we take the kilometre as our unit of length, for the earth's 

 orbit a = 1 ■ 49 . 10* and co = 6-64.10-1'. Hence the mass of the 

 sun, expressed in kilometres by absorbing the imit dimensional 

 gravitation constant, is 1 • 47. 



Thus in the solar system ni/r, h'^/r'^ and/ 1 — — jare all of order 10 *, 



By eliminating s from equations A we find that the differential 

 equation of the orbit is 



(V'u , m 



^:?'^+^ = ^ + '■'"""' 



where u = r~^. 



Forsyth has given an exact solution of this equation, and an approxi- 

 mate solution is contained in Eddington's Report to the Physical 

 Society. 



This solution is 



II = 

 h 



n = Yz [^ +ecos(0 — a — ha)} 



where a is the longitude of perihelion and ha = —jr 4*- 



Hence in a complete revolution of the planet, perihelion advances 



through a fraction of a revolution equal to —p. 



9 

 Measurable perturbations depend on the ])roduct eha- 



