Newton-Einstein Planetary Orbit. 145 



the expression of Kepler's Law III, and implying that Gr is 

 a constant for all matter throughout the Universe. 



In these C.G.S. units, Einstein's m must denote a length, 

 in centimetres. It is mysterious then that Einstein is quoted 

 as calling m the mass of the Sun, as if a mass could be 

 measured in centimetres, by a metre rule, and not in 

 grammes; some mysterious unexplained astronomical units 

 must .have been employed, and writers should enlighten us 

 on this point of the theory. 



The differential equation in Particle Dynamics of the 

 Central Orbit, analytical expression of the normal component 

 of the acceleration, becomes when Einstein's term is added, 



d 2 U P Lb o 9 , kn 



d& +w= hs? = v +Smu ' ■ ■ ■ ■ (5) 



and integrating, with li 2 ~/jd, 



U) +» 2 = C+ T +-W, . . . . . (6) 



\dd) =2m ( u ~ a - u ~~ fi- u — 7)i • • ( 7 ) 



and a, /3, 7 are the roots of a cubic in u, where 



a+/ 3+ 7 =i-, fr + '>*+afi=±, «fh=-±, (8) 



In a closed orbit, a, ft, 7 are positive ; and taken in the 

 sequence so that 



« > /3 > u > y, 



x /(ot-y)du 



ipdd = 



^/(l.CL-U.$-U.U-i)\ 



a — 7 



^ = M-7)=-+^y, 



(9) 



bringing in an elliptic integral, which is to be reversed in 

 Abel's manner ; and then, expressed by the inverse elliptic 

 functions, 



kp e = zn-\/ U fV =cn -: /£=? =dn -x /fr? ] 



2/ V /3-T V £-7 V «-7 ! (W) 



h — , 



a — y J 



equivalent in the direct notation, as in (1), to 



w= 7 cn 2 ip0 + /3sn 3 ip0 (11) 



Phil Mag. S. 6, Vol. 41. No. 241. Jan. 1921. L 



