characteristic Quantities occurring in Central Motions. 17 

 Further, equations (13) and (24) hold good — 

 I 2tt 



2 C r 2 ~ P 2 P ' 

 from which, by elimination of -g- results 



k pp n+1 

 If herein for i we put its value from (28), we obtain 



J=a/?-^ < 35 ) 



and the combination of this with equation (33) gives 



q=pj. ........ (36) 



Consequently q stands in a very simple relation to p. 



Its behaviour is also very similar to that of p. It can like- 

 wise vary only between the limits and 1, and takes these 

 boundary values simultaneously with jo. When c = 0, then, ac- 

 cording to (32), is q also =0; and consequently the vis viva of 

 the rotation-motion also, and therefore the quantity p, has the 

 value zero. If, now, c is increasing, p and q increase at the same 

 time. When jo has reached the value 1 the path has become 

 circular. For this case we have, in the differential equation 



d^r 1 



m -=-3- = — kr n + tnc 2, -a- 



dr ir 



d 2 r 

 for the oscillatory motion, to put the differential coefficient -=g 



equal to zero ; and the equation thereby arising can be brought 

 into the following form : — 



m c 2 



~k r n+3 ~ ' 



Further, where r is constant, we may regard r and p as sig- 

 nifying the same thing, and hence the preceding equation can 

 also be written 



m c 2 



and this gives, in accordance with (33), for q the value 1. 



Since, according to (36), q is represented by the product joJ, 

 in which J is a function of p only, q itself is likewise a function of 

 p alone, and accordingly^ can reciprocally be regarded as a func- 

 tion of q only. Thence it follows, further, that J and J x , which 



Phil. Mag. S. 4. Vol. 46. No. 303. July 1873. C 



