Planetary Orbits on the Theory of Relativity. 521 



As the parameter h is increased a, /3 approach limiting 

 'allies, viz., the roots o£ the quadratic 



"-(i-^Msr 1 )- 



and so the orbits approach a definite limiting " hyperbolic" 

 shape. On fig. 9 (7/) 5 ( 6 ») ? (d) are shown three particular 

 cases of orbits in this class : (b) is the orbit which looks like 

 a parabola, having the angular range between apse and 

 infinity —it, (c) is that for which h = l, and (d) is the 

 limiting form for /i=c© , all drawn for m=J. The scale of 

 fig. 9 is t J q that of the others. 



Modifications for larger values of m. 



As m approaches ^ the radius of the asymptotic circle 

 becomes unity and the orbit becomes identical with the 

 ordinary circular orbit ; the " elliptic orbits from aphelion " 

 are squeezed out. The later forms follow the same sequence 

 as before. 



When ^<m<\ for h = h 1 the quadratic has a double root 



( -. — — - J, which is now less than unity. Therefore, it runs 



\-irn 2 J J 



from 1 to oo in stages (ii.) and (iii.) as well as in stage (i.) ; 

 the captured orbits extend over the first three classes. The 



/3 1 \ 

 positive quantity H now approaches (^""j— ) and k 2 ap- 

 proaches zero, so the elliptic functions become circular 

 instead of hyperbolic. The equation for (ii.) is 



M = l + Htan s ±p0, 



so the centre is reached at = 7r/p, where p 2 = 2mH, 



In case (iii.) the arrangement of the roots of the cubic is 

 1, a } /3. The integral is 



M = a+(l-*)/cn s ip0, P = (a-/3)/(l-/3), p 2 = 2m{l-{3). 



Thus u = 'X) for = 2K/p. When « = 1 the circular orbit 

 is reached and the succeeeding forms are as before. 



For ^<m<^ there is a further extension of the range of 

 captured orbits. The critical values h. 2 and h s now change 

 places, Ji 2 >h 3 . All paths get to the centre until h—-h. 2 , the 

 orbit is then circular and succeeding paths are hyperbolic. 



For m>\ the roots of the quadratic are always imaginary, 

 so that all orbits are captured. 



These results may be summarised in tabular Eorm, the 

 Phil. Mag. S. 6. Vol. 42. No. 250. Oct. 1921. 2 N 



