ol-P articles with Hydrogen Nuclei. 507 



sech f ; the exact shape will depend on the values of k 

 and 7. 



Now consider I , the integral on the right. Here two 

 cases arise which exhibit an important difference. Ify>l 

 (that is, i£ p>a) the integrand has no turning points, as its 



denominator never vanishes. If we plot 1^ against — — rj 7 

 we have a curve like EF in fig. 8. But if y<l the de- 

 nominator vanishes as soon as sin ( — — 77 ) = 7, and the curve 



IT 



bends back like l t . - — rj then shrinks through zero to the 



value given by sin I — — rj j= — 7, where there is another 



turning point. So the curve is a sinuous line like GH in 

 fig. 8. The three curves are all drawn with the same hori- 

 zontal base line in the diagram ; k :md 7 will determine the 

 size and shape of I,, and 7 of 1^, and also whether the latter 

 is of the type EF or GrH. To see the form of any orbit it is 

 only necessary to select the proper k and y curves and then 

 draw a number of horizontal lines. These will give simul- 

 taneous values of f and rj. In particular the line through C 

 in fig. 8 will give the value of rj on the second asymptote. 

 As 97 is the ordinary vectorial angle when f is large, the 



value of 6 is simply ^l — —rj J. Notice that when y< 1, 6 is- 



almost as likely to be negative as positive. 



It is unnecessary to describe in detail the rather tedious 

 processes involved in the computation. I. has to be trans- 

 formed to reduce it to a standard elliptic integral of the 

 first kind, and two cases arise, differing only analytically,, 

 according as the second or fourth root of (9*3) is the larger. 

 In both its cases I needs little transformation, but the 

 calculation is slightly complicated by the fact that the value 

 of L. to which it is equated must always be diminished by 

 some multiple of the ' complete elliptic integral/ (which 

 determines the number of oscillations in the orbit) before 

 recourse can be had to the tables. 



If we wish to obtain complete curves between p and 6, 

 we are limited by the fact that y 2 must be greater than 1 — 2k 

 for all values of 7, and therefore k must not be less than 1/2. 

 Calculations were therefore made taking £ = 0'5, 0*6, 0*7, 

 0'8, 1-0, 2*0, and in each case the value of was computed 

 for a succession of values of />. The curve 0*5 has an 

 infinite number of oscillations as 6 approaches zero. This is 



