( 49 ) 



same thing is true of t lie branches on which ■/. decreases. The two 

 branches between which lies the part of the line x = x on which 

 P<Cft>> are either both stable, or both unstable, and similarly the 

 two branches enclosing the other half of the line y. = •/.„. If T is 

 the period of one of the branches and T' of another, and if 7', and 

 T\ are the values of these periods in the point (x /3 ), then 7', and 

 T\ are mutually commensurable. It' 7"'„ is their least common 

 multiple, then «, 1 7 , " = 0. It' e.g. T\ = 2 7',, then the instability is 

 even with reference to the period 7". 



As an illustration of these general rules I may be allowed to 

 mention a few of the simplest cases. 



1. The curve $ = () is tangent to the line y. = y.„. There are 

 two families, springing from a common member, which come into 

 existence at this value of the parameter. One of them is stable, and 

 the other is evenly unstable. An example of this is presented by 

 Darwin's families B and C of satellites. 



'2. The curve has a double point. Two families are "crossing" 

 each other, at the same lime exchanging their stability. 



'A. The curve consists of one branch tangent to the line x = x e 

 and another branch intersecting the first in the point of contact. 

 The two families which come into existence at this value of the 

 parameter arc both stable or both unstable. The third family, which 

 y.xists boih for y. ^> x a and for x<^x 0J becomes stable if it was 

 uitstalde and unstable if it was stable. 



The cases 2 and 3 are the only ones occurring in the present 

 investigation. 



The proof of the above supposes that the problem can be reduced 



to the second order, so that there are only two characteristic exponents 



(-)- «. and — «)■ The choice of the parameter is determined by the 



way in which this reduction is effected, or is conceived to be effected. 



Darwin uses the iutegral of Jacobi for this reduction. Consequently 



his parameter is the constant C to which this integral isequal. This 



constant C is a function of die two elements u and e. The first of 



these can be replaced by the mean motion //, or by the period 



2n 

 ï'= — In consequence ot the reduction ol die problem by 



>i — 1 



means of the integral of Jacobi one of these elements, say 7', is 



eliminated. This therefore appears no longer as an arbitrary constant 



of integration, but is entirelj determined by C and e. < hi the other 



hand C is entirely determined by 7'and r. Now Darwin's calculations 



show that 7' continually increases if C decreases. Il is therefore 



irrevelant for our purpose whether we consider 6' or 7' as the 



4 



Proceedings Royal Acad. Amsterdam. Vol. X. 



