Planetary Orbits on,the Theory of Relativity. 513 



h is the product of apsidal distance and velocity of projection 

 (measured by unit of proper time), it follows that a path of 

 the same shape is got by projecting with the same velocity 

 at increased distance in presence of a centre of force whose 

 strength is increased in the same proportion. Accordingly, 

 in what follows m will mean the ratio of the strength of the 

 centre to the initial apsidal distance (both measured, as 

 alread}' explained, in km.), and h will be the ratio of the 

 velocity of projection to the velocity of light. 



The first step is to adjust the value of the arbitrary con- 

 stant, so tliat {u — 1) is a factor of the expression, which then 

 takes the form 



M«-i){« 2 -(^-i>+i-(^-i)} 5 



and the question turns on the nature of the roots of the 

 quadratic factor. 



The magnitude of m being fixed, suppose li to increase 

 from zero. There are three critical values of h — say, h lf h 2 , 7i 3 . 



(1) When the roots of the quadratic are equal, 



*H(i+>)'-'- 



(2) When one root is unity, 



J- =1-3 



k 2 m 



(3) When one root is zero, 



V 2m 



The values (1) and (2) coincide when m = }, (2) and (3) 

 when m=i, (1) and (3) when m — \\ these are therefore 

 critical values of m. The greatest variety in types of orbits 

 is found when m is less than the smallest or! these values, so 

 it is convenient to discuss this case first and afterwards to 

 indicate how the results are modified for larger values of m. 

 The polar angle 9 will be measured from the initial apse-line 

 u = l. In the integrated equation u is expressed in terms of 

 elliptic functions of the argument pO, where p is a number 

 for which different expressions are found in the sep irate 

 cases. To illustrate the results obtained I have drawn a 

 series of curves for the case »i = £, for which the critical 



