178 



Mr. G. H. Darwin. 



[June 19, 



the equilibrium breaks down by the satellite receding, the recession 

 will go oh until the system ha>s reached the state corresponding to B. 



The point P, in fig. 2, shows approximately the present state of the 

 earth and moon, viz., when x=3'2, y = 'S. 



It is clear that, if the point I, which indicates that the satellite is 

 just touching the planet, be identical with the point A, then the two 

 bodies are in effect part of a single body in an unstable configuration. 

 If, therefore, the moon was originally part of the earth, we should 

 expect to find A and I identical. The figure 2, which is drawn to 

 represent the earth and moon, shows that there is so close an approach 

 between the edge of the shaded band and the intersection of the line of 

 momentum and curve of rigidity, that it would be scarcely possible to 

 distinguish them on the figure. Hence, there seems a considerable 

 probability that the two bodies once formed parts of a single one, 

 which broke up in consequence of some kind of instability. This view is 

 confirmed by the more detailed consideration of the case in the paper 

 on the " Precession of a Viscous Spheroid," of which only an abstract 

 has as yet been printed. 



Hitherto the satellite has been treated as an attractive particle, but 

 the graphical method may be extended to the case where both the 

 satellite and planet are spheroids rotating about axes perpendicular to 

 the plane of the orbit. 



Suppose, then, that h is the ratio of the moment of inertia of the 

 satellite to that of the planet, and that z is equal to the angular velocity 

 of the satellite round its axis, then hz is the moment of momentum of 

 the satellite's rotation, and we have 



h=x + y-\-hz for the equation to the plane of momentum, 

 2e=y 2 -\-hz 2 — ~ for the equation of energy, 



fly™ 1 



and x s y=l, xh=l for the equation to the line of rigidity. 



The most convenient form in which to put the equation to the 

 surface of energy is 



where E, y, z are the three ordinates. 



The best way of understanding the surface is to draw the contour- 

 lines of energy parallel to the plane of yz, as shown in fig. 4. 



The case which I have considered may be called a double-star 

 system, where the planet and satellite are equal and Any other 



case may be easily conceived by a stretching or contraction of the 

 surface parallel to z. 



It will be found that, if the whole moment of momentum h has 

 less than a certain critical value (found by the consideration that 



