516 Prof. W. B. Morton on the Forms of 



As 6 increases u approaches the limiting value 



the path continually approximates to a circle of radius 

 4m/(l — 2m) (fig. 5). This form may be called the "asym- 

 ptotic circular orbit." 



Fig. 5. 



It is not at first sight obvious how this form of orbit can 

 be reached as the limit of class (i.). On examination it will 

 be seen that, as the number of convolutions increases, the 

 path lingers in the neighbourhood of w = l + H. This is 

 owing to the rapid change in the elliptic function F (</>), 

 when cj> and the modular angle are both nearly 90°. As a 

 consequence, a large change in 6 is accompanied by only a 

 smali change in r until the case of asymptotic approach to 

 the critical distance is reached. 



When m is very small the critical velocity h t may be taken 

 as 4??i. For the Sun and Earth m is about 10 ~ 8 , the Sun's 

 mass being 1*47 km. and the radius of the orbit 1*49 x 10 8 km. 

 The critical velocity for capture is 4x 10" 8 x speed of light 

 = 12 metres per sec. or about the 2500th part of the orbital 

 velocity. 



