152 NOTE OX EIXSTEIX'S rLAXETARY EQUATIOX. 



the desired Mercury result. Whether this discrepancy can be at 

 all accounted for by the fact that a ray of light is there treated 

 as a travelling particle, I cannot say. 



2. In Eddington's extension of the formula to the " gravita- 

 tion of a continuous distrilnition of matter "' (chap, vi, p. 59) an 

 approximate result is obtained: — 



_ (ds)2=^ - [1 + 2V) (dx- + dy^ + dz-) + ( 1 - 2V) (dt)^ 

 where V=2(ni/r), the Newtonian potential. 



Apparently this result does not include the previous formula 

 for a single particle as a special case — a somewhat surprising 

 incident. 



§ 3.— The main object of this Note is to call attention to a 

 consequence of the second equation in § 1. 



;'|)^^2.u.-u«.2--i-= („) 



The corresponding Newtonian equation is, of course, 



(du/d^)^= -u^H — ^-^ -const., giving the elliptic orbits. 



On the fundamental principle that at a distance from the 



gravitating centre Einstein's equations must agree witli Newton's, 

 we identify, in the case of the solar system, Einstein's integra 

 tion constant m with Newton's " mass " of the sun (in gravitation 

 units). 



Eddington solves equation (ii) by a rather unpleasing series 

 of approximations. The straightforward and accurate elliptic- 

 function solution is pointed out by Forsyth in Nature for 8th 

 April, 1920, p. 186, 



Ee-writing (ii) in the form ij^j =2m (u — a) (u — P) (u — y) 



where a>>/3>»7, and at first assuming a, (3, y to be all real and 

 ])ositive, we see that u may either lie between (3 and y, giving 

 the ordinary two-apse orbit which, presumably, is indistinguish- 

 able by observers from Newton's ellipse, 



or u may be greater than a, i.e., r less than -. 



This ease gives ^a/'^= h f 

 ^ 2 V, 



dt 



(iii) 

 v/(t-a)(t— /3)(t— yj 



Thus u-a, U-/3, u-y are the (semi-axes)^ of that confocal 

 ellipsoid which has internal potential ^v/J^/^. Since, by elemen- 

 tary Elliptic Function Theory, these (semi-axes)- are periodic 

 functions of the potential, the period being 2oj, 



'd^ 

 V(t-a)(r^fl)(t-y) 



— —-^ — —j^ 7- , we obtain a periodic 



orbit, the period of being mJ \/2m. And since equation (iii) 

 makes this orbit go through the origin when ^=0, we are reduced 

 to a single-apse flight from the origin out to u=a and back in 

 l^eriod w/ V^m. 



