51 4 



Prof. W. B. Morton on the Forms of 



values of j^ are §\, 5, and 3. Details of these orbits are 



given m the table at the end of the paper. The special 



values chosen for j^ are such as to give unbroken values for 



the modular angle of the elliptic functions in order to facili- 

 tate the computations. 



Case m < -g-. 



(i.) If 7t</i 1 , then unity is the only real root of the cubic, 

 u runs from 1 to oo , r from 1 to 0, and we have a captured 

 orbit." 



Let the result of substituting u = l in the quadratic factor 



/r m 



the integral is 



M=l+H. 



1 — en 





where p 



1 + QYipQ en 2 %pQ 



2 = 2mH and P = (k + -^—~)/2T1, u = ao for 



V 4 m 2 J 



= 2K/p, giving the angular range of the motion from the 

 apse to capture. 



/ 1- 3\ 



As h approaches h u H approaches \~ ""9)5 ^ approaches 



Fiff. 1. 



Fig. 2. 



unity, and tlie particle makes a continually increasing 

 number of circuits round the centre before being drawn in 

 (figs. 1-4). 



