1880.] 



History of Planet and Single Satellite. 



267 



._ .(2-539 ocx) 553 exp. [-062 arc tan (1740a; + '665) ] 

 3 (a? co -826 ) 289 (« 2 + 765® + -47 7) " 1814 



x-> . 7 



4* (x*-2-6x 6 + l) 

 n — 2'6—x. 



(22), 



2^7=(2 i 6-.a;)3-i-, 



The maximum and minimum values of i are given by the roots of 

 the equation a; 4 — 2'6x z + 2 = 0, viz., a;=2'467 and aj=l*103. The max- 

 imum and minimum values of e are given by the roots of the equation 

 a? 4 — 2'6a; + if =0, viz., a;=2'495 and a;=l'0095. The horizontal asymp- 

 totes for if A and jjA are at distances from the axis of x equal to 

 exp. ('062x^7r) and exp. ( — *062 X ^-w), which are equal to 1*102 and 

 *908 respectively. 



Fig. 2 shows the curve illustrating the changes of i, the obliquity of 

 the equator to the invariable plane. 



The asymptotes are indicated by broken lines ; that at A is given 

 by a;='826, and is the ordinate of maximum energy ; that at B is given 

 by a?=2*6, and gives the configuration of the system for which the 

 planet has no rotation. The point C is given by x= 2*539, and lies on 

 the ordinate of minimum energy. Geometrically the curve is divided 

 into three parts by the vertical asymptotes, but it is further divided 

 physically. 



The curve of energy has four slopes, and since the energy must 

 degrade, there are four methods in which the system may change, 

 according to the way in which it was started. The arrows marked on 

 the curve of obliquity show the direction in which the curve must be 

 read. 



Since none of these four methods can ever pass into another, this 

 figure really contains four figures, and the following parts of the figure 

 are quite independent of one another, viz. : (i) from — oo to O ; (ii) 

 from A to ; (iii) from A to C ; (iv) from + oo to C. The figures 

 3 and 4 are similarly in reality four figures combined. For each of 

 these parts the constant A must be chosen with appropriate sign ; but 

 in order to permit the curves in fig. 2 to be geometrically continuous 

 the obliquity is allowed to change sign. 



The actual numerical interpretation of this figure depends on the 

 value of A. Thus if for any value of x in any of the four fields the 

 obliquity has an assigned value, then the ordinate corresponding to 

 that value of x will give a, scale of obliquity from which all the other 

 ordinates within that field may be estimated. 



