Newton- Einstein Planetary Orbit. 143 



two observers a few metres or kilometres apart and an un- 

 disturbed particle passes between them in a direction perpen- 

 dicular to PQ. They remain fixed at their observation 

 posts watching the particle, for many million years, and at 

 last face one another, and P reports to Q that the particle is 

 now for the first time directl}^ behind him. It seems difficult 

 to believe or to disbelieve that what Q has to report to P at 

 the same instant is that it is directly behind himself also. 



The relative rest of Q with regard to P is a case of con- 

 strained motion, for if Q has the minutest velocity relative 

 to P, and is undisturbed, he will rotate (in the belief of P) 

 about P in the same period as does the particle. In such 

 case our paradox disappears, but it still remains if we 

 suppose that a constraint bringing about relative rest is so 

 much as possible. 



Yours, &c, 

 Alex. M c Aulay. 



University of Tasmania, 

 June 1st, 1920. 



X. Newton-Einstein Planetary Orbit. 

 By Sir George Gkeenhill *. 



npHE modification in the Problem of Two Bodies of the 

 JL elliptic planetary orbit under ordinary Newtonian 

 gravity to the Sun, due to the additional term introduced 

 by Einstein into the attraction, varying as the square of the 

 angular velocity round the Sun, has engaged the attention 

 of many writers in the recent numbers of the Philosophical 

 Magazine. 



The result may be stated as a change of Newton's ellipse 

 into Einstein^s orbit, from 



1 _cos'itf sin 2 10 cn'jpfl Mi'j;>fl m 



and so provides an Elliptic Function application to a Central 

 Orbit. It is proposed hero to standardize this problem in 

 the Dynamics of a Particle, and to examine closely the units 

 employed. 



According to Einstein, a term he denotes by 3>?ig> 2 must 



be added to Newton's term jxu 2 to give the total central 



acceleration to the Sun ; here w denotes the angular velocity 



of the planet, and in a central orbit to = lm 2 , with u the 



* Communicated bv the Author. 



