L 626 j 



LX. Oil the Advance of the Perihelion of a Planet, and the 

 Path of a Ray of Light in the (gravitation Field of the Sun. 

 By Prof. A. Anderson *. 



r |^HE particular integral of Einstein's contracted tensor 

 X equations, which hns been applied to the case of the 

 motion of a planet in the sun's gravitational field is 



neglecting the term in cf>, as the motion may be considered 

 to take place in the plane <£ = 0. The quantity denoted by 

 y is l — 2mjr, where m is a constant. A contra variant vector 

 is found that is suitable for the motion of a planet, and then 

 it appears that 



2 _ 2m m 2mh 2 



r a 



,,y j 



where a is a constant and h = vp. 



Thus m is identified with the astronomical mass of the 



sun, and a with the semi-axis major of the planet's orbit. 



972 

 The term 3 indicates an advance of the perihelion of the 



planet through an angle ]2 in every revolution. The 



advance of the perihelion of Mercury (43" per century) is 

 accounted for. 



In the case of the path of a ray of light ds = 0, and we 

 get, using the same particular integral, 



©'MS)"-*-. 



but this equation is unsuitable. We want an equation which 

 will give us the velocity of light as a function of r. That is, 

 We want an equation of the form 



(drV „/d0\ 2 , .. , 



\dt) +r \jt) = ^me function of ,. 



Now it appears that we may take any function of r instead 

 of r and then call this function r. So that we must get a 

 function of r that, when substituted for r, will give us an 

 equation of the necessary form. If we make r = f(r l ), it is 

 easily proved that w r e must have 



■**(*-^ w 



* Communicated bv the Author. 



