146 Sir George Greenhill on the 



In the degenerate case of pure Newtonian gravity, m = ; 

 and then a=oo 3 /3 and 7 finite, 79 — 0, k = ; and the orbit 

 reduces, as in (1), to 



w = 7 cos 2 -|l9 + /3sin 2 i# (12) 



The apsidal angle w in (1) is changed from it to 2K/p ; 

 and the importance of Einstein's theory lies in the applica- 

 tion to Astronomy, where the term m is considered small, in 

 the endeavour to account for the anomalous amount of the 

 advance of the apse in the orbit of Mercury. 



In that case where m is small, ex. is very large compared 

 with /3, 7; and we take 



m->0, 2ma=l - 2m(/3 + 7)-^ 1 



^i_z *+e=i, ^i + iP,l (13) 

 p \Z0-9 



7T IT}) ^OL ' 



irp 



KU) 



/3 4- 7 _ 1 ^ (By 1 m A-S7 m . 



_ 4^r~4/77ia 2 4a 2_> 4/7^ _> T' V^T' ' ^ ^ 



making the advance of the apse in one revolution 360° (mjl). 

 The addition of a term varying as u z or co* to the central 

 attraction would cause the ^differential equation of Newton's 



orbit to change 



from 















d 2 u 



d¥ +u= 







into 



? + 



?iw, 







(16) 



/du\ 2 



\de) z 



= + 



I 



into 



c+ 



*-<•- 



-n)u 2 , 



• 



(17) 



/du\ 2 

 \dd) z 



= j3 — u . u — 7, 



into 



a- 



-n)(J3-u 



.u-y) 





(18) 



u- 



= 7 cos 



2 £0 + £ sin 2 1<9, into 

 7 cos 2 ^<7# + /3 sin 2 io 



&, <? = A/ 



'(1-n) 



.} 



(19) 



the character of the integral and of the equation of the orbit 

 is not altered essentially, except by an expansion or con- 

 traction, in a fan-like manner, of the vectorial angle 0. 



