( 50 ) 



parameter of the problem. The parameter which I will use here is 

 T' = 2 T. This change from C to T' can also he conceived as no 

 more than a simplification of language. Instead of saying: "the 

 solution corresponding to the value of C for which the period of the 

 solution of the first sort is-Vs 7" ", I say : "the solution corresponding 

 to the value T"\ 



In Darwin's work ft has the constant value 0.1. If now we 

 choose a convenient element g, we can conceive the curves <P (T', §) 

 to be drawn. Next imagine the same thing to be done for other 

 values of n, and take ft, T' and s as rectangular coordinates. The 

 curves <P (T' , g) belonging to the various values of ;i then produce a 

 surface, every point of which represents a periodic solution. 



If, on the other hand, we take for 7" a fixed value 7",, 

 considering fi as the variable parameter, then we have another 

 problem, also admitting families of periodic solutions, which can be 

 represented by curves if ? (ft §) = 0. If 7", varies these curves describe 

 again the same surface. The form of this surface will now be 

 investigated. Its section by the plane \i = 0.1 then gives all periodic 

 solutions of Darwin's problem. 



The element which I will use is £ = e cos «j , where e is the 

 excentricity and a> the longitude of (lie perihelion of the undisturbed 

 orbit, which is the limit of the orbit of P for lim. ft = 0. The 

 longitude c3 is counted from a fixed axis which at the beginning 

 of the period co-incidcs with SJ. The orbit of P is not periodic 

 unless tu has one of the two values or .t. Moreover at the begin- 

 ning of the period P must be on the line SJ, i.e. there must be 

 either opposition or conjunction. 



Solutions of the first sort are characterised b\ £ = 0. These solu- 

 tions can have any period, therefore the whole plane $ = is a 

 part of our surface. The line § = 0, fi = 0.1 represents Darwin's 

 family A. For a value of T' = 27', which lies between 330° and 

 354°, i.e. between 1.83 jt and 1.97 jr, this family loses its stability 

 and becomes unevenly unstable. So there must be another family 

 which at this point has a member in common with the family .1. 

 This new family must have the period 7", and is therefore of the 

 second sort. If for the sake of argument we assume the change of 

 stability to take place at the value T' = 1 .9 Jt, then we know of 

 the branch of the curve $ = 0, which represents this family, that 

 for 7" <^ 1.9 jt it is evenly unstable and for 7"^>1.9.t it is stable. 



Now there are only four possible periodic solutions of the second 

 sort, distinguished by the following positions of /' at the beginning 

 of the period : 



