( 52 ) 



« 3 = 3 ft Ke* ,v„— 4 cos f. 

 Thus, for positive values of' jj, a" is negative, and therefore the orbit 

 is stable, when (here is opposition at the beginning of the period. 



For positive values off/ therefore BB' is stable and CC' is unstable, 

 for negative values 1 ) of fi BB' is unstable and CC' is stable. It is 

 evident that, lor | = 0, B and B' co-incide, and similarly 6' and ( ". The 

 branch of <P = which intersects g = in the point T' = 1.9 n 

 therefore represents either the family BB' or the family CC' . In 

 the first ease it is stable, and therefore it must on both sides of the 

 point of intersection bend round towards the right. In the other case 

 it is unstable and encloses the stable part of the line § = 0. 



Now Darwin has, for C= 39.0, i.e. I 1 ' = 1.97 n, actually com- 

 puted and drawn an orbit, which shows the form of lig'. 1, viz.: 

 the orbit x = — . 337 which has already been quoted. This orbit 

 thus belongs tc die family B, but il also belongs to B'. It belongs 

 to B if P is in aphelion at the beginning of the period and in 

 perihelion in fne middle of the period (being at both times in opposi- 

 tion to ./), a. id to />" in the opposite case. The branch of the curve 

 cp =z which passes through the point T' = 1.9 n therefore represents 

 the family JiJi' , and nol CC'. Consequently it is stable, and that 

 part of the section of our surface by the plane ft — 0.1, which lies 

 to the left of the line T' == 2 st, is thereby completely determined. 

 This section is represented in Fig. 2. Stable families are there, and 

 in the following figures, represented by heavy full lines, unevenly 

 unstable families by broken lines, and evenly unstable ones by 

 dotted lines. 



\e' 



Fie. 2. 



We next consider the section of our surface by the plane (i = 0. 



') The meaning of a negative value "I fe is thai the force emanating from J 

 is repulsive, the force from S remaining attractive. 



