( 51 ) 



II ■. P in opposition in aphelion (<5 = 0, 5 = -(- < \ 



B' : „ .. ,, ,. perihelion {ta = 3t, 5 = — e 



C •■ „ „conjunction ,, perihelion i» =:0 ) |^-|-g, 



C' : „ ,, ,, .. aphelion (tö„ = .-r, S = — e„ 



With reference to rotatin»' axes, of which the axis of x co-incides 



with SJ, the orbits B and A" are identical, and similarly C 



->J 



Orbit of family B or 5' 

 Fis. 1. 



and (". The orbits A' and I!' are of t lie form represented in fig. 1. 

 The orbits C and <" are of the same form, rotated through 180', 

 i.e. with the double point a\va\ from •/. 



The families 11 and IV arc stable Tand <" are unstable. This is 

 easily found by considering the equation which determines the 

 exponent a. This equation is (see Poini \kk. Acta Math. XIII, p. 134) : 



./-' !(• 



« ' « 2 = 



d(SJ 



(V C„-2 n, n, C„ + V^u) 



Now using the variables employed by Poincaré I.e. pages 128 

 and 171. we find easily 



If further in if' (i.e. the average value of the perturbing function 

 over one period we neglect the terms which contain a higher power 

 of e than the second, we find 



i|< = /< AV- cos e s = w, -|- 3 co, 



where e is the mean longitude of I' at the beginning of the period, 

 and K is a positive constant. 

 We find thus 



4* 



