154 KOTE ON EIXSTEIX'S PLANETAKY EQUATIOX. 



§ 5. r=;2m raises points of interest. We have then in 



equation fi) R=0, and - =00 (c cannot be zero; see below). 

 ^ ds 



ITT u 2 d^ h T> A J du du d^ n . dr n 



We have r^ = — R = 0; and -— = tts ir- =0 .-. --= 0. 



dt c dt d^ dt dt 



.'. apparently, when r=2m, every particle comes to rest, and 

 since the higher differential coefficients with regard to t also 

 vanish, all motion ceases on r=2m. 



Thus the " inner orbits " considered above reduce to spirals 

 originating on the sphere r=2m, or to curves from points on r=2m 

 to apses on r=l/a and then back to r=2m. 



§ 6. — To prove that these peculiarities of r^=2m are not ruled 

 out by c===0, the equations connecting a, /3, 7 in § 3, viz., 



a + /3 + 7 = l/2m, f3y + ya + a(3=l/h-, afty = {l-c")/2 mh" 



give (I -C-) (o+yS + y) (/3y + ya + ci/3) = a/3y 



.-. if c = 0, (;8 + y)(y + a)(a+^) = 0, 



which is inconsistent with ^y + yo + a/3 = a square (]/h^). 



