( 55 ) 



By the perturbing influence of J this is changed to T = 2n -\- t. 

 The conditions that the perturbed orbit shall be periodic are: 



T T 



ƒ da. rdX , 



dt = x — - dt r=z Q n -\- x, 



dt J dt 







where ). is the mean longitude of P. For the computation of the 

 integrals we must use the mean motion affected by perturbations, 

 i.e. n = 3 -f o. The left-hand members of these equations of condition 

 are therefore functions of t and a, and these two unknowns can 

 be determined from them. 



If in these equations of condition we neglect the square and higher 

 powers of e, they become 



t = ™ (2 .-T + t) (i [13 ' ± {21 AW + 10 A* + 2 A, C« j] I 



tJ.T -f t = (3 + ö) (2jt + t) — «a (2,t -f t) [i AJS> ] 



The upper sign in these equations must lie used for the family 

 CC', the lower sign for BB'. The sum within the { j being larger 

 than /•>'", we find that for the family B1V r is negative, while for 

 CC' it is positive, as has also been found above. Further the first equa- 



i/r 



Hon shows that the numerical value of the differential coefficient 



d(i 



for the first family (BB') decreases if ;i increases, while for the 



other family it increases. Thus the left-hand branch of % ((i, T') = 



has its concave side towards the line T' = 2.t, and the right-hand 



branch its convex side, as is shown in fig. 5. 



In the numerical computation we must not forget thai the formulas 



(1) can only be considered as approximatively true. The solution of 



the equations is easily effected by means of the tables of Runkle, the 



argument for the determination of the differenl functions A^ beine 



° p 



computed by 



, 10 



I find in this manner for the two families: 



B: — = — 0.085 T' = 1.83ar 



C: — = + 0.29 T = 2.58 n 



These are the periods of those orbits of the two families, which 

 have % = 0, and which therefore co-incide with a member of the 



