548 Mr. C. G. Darwin on the 



and of energy is 



■&j 



1 Mm 9 3 Mm(M 2 -Mm + m 2 ) E<? 



2M + m 8C 2 (M + w.) 3 r 



Ee Mm 2r* + r*0 a 



The integration constant is taken as — W, so that W may 

 be positive for elliptic orbits. Following the usual pro- 

 cedure we eliminate the time between (19) and (20), and 

 express the orbit in terms of 6 and n, where v=l/r. The 

 result is an equation of the form 



(l-p)ifi + 2g«-h, . . (21) 



/duy 



U) =* u ' 



Eg EVMmW 



where « - 02(M + ^ , p - (J2 ^ 2 (M + ^ 



_^e Mm / W M 2 -Mm + m 2 \ 



9 ~ p 2 M + m\ C 2 Mm(M-f m) /' 



2W Mm A W M 2 -Mm + m 2 - 



/l '~ f M 



1m / VV Mf.— JSLm + m'\ 



+ m\ 2G a - Mm(Mi-m) J 



The solution of this equation to the same order of approxi- 

 mation as before is 



u = q •+- s cos \0 + 1 cos 2k6, .... (22) 



where \= 1 — |a(/ — J/3, 



s 2 =g 2 —k + a#(4/ _ 3/,) 4- /3(2</ 2 - /•), 



Z = -iaQ/ 2 -*). 

 The last three expressions cannot be much simplified, but 



x-i- EV 



2cy, 



which is independent of the masses and depends only on the 

 angular momentum. It is the same as Sommerf eld's result 

 and implies an advance of perigee by 7r 1 Et 2 e 2 /G 2 p 2 each revo- 

 lution. The term in / makes a slight increase in the radius 

 at the apses, and a decrease at the ends of the latus rectum. 

 The solution of the relative orbit is completed by finding the 

 time from (19). The formula is complicated and of no special 

 interest. 



