1879.] On the Secular Effects of Jiclal Friction. 177 



for this part of the line of momentum, the satellite must approach the 

 planet, and will fall into it when its distance is given by the point h. 



ii. For all points of the line of momentum from D through F to 

 infinity, x is positive and y is negative; therefore the motion of the 

 satellite is clockwise, and that of the planetary rotation anti- clockwise, 

 but the m. of m. of the orbital motion is greater than that of the 

 planetary rotation. The corresponding part of the energy curve 

 slopes down to the minimum b. Hence the satellite must approach 

 the planet until it reaches a certain distance where the two will move 

 round as a rigid body. It will be noticed that as the system passes 

 through the configuration corresponding to D, the planetary rotation 

 is zero, and from D to B the rotation of the planet becomes clockwise. 



If the total moment of momentum had been as shown in fig. 3, then 

 the satellite would have fallen into the planet, because the energy 

 curve would have no minimum. 



From i and ii we learn that if the planet and satellite are set in 

 motion with opposite rotations, the satellite will fall into the planet, if 

 the moment of momentum of orbital motion be less than or equal 

 to or only greater by a certain critical amount,* than the moment 

 of momentum of planetary rotation, but if it be greater by more than 

 a certain critical amount the satellite will approach the planet, the 

 rotation of the planet will stop and reverse, and finally the system will 

 come to equilibrium when the two bodies move round as a rigid body, 

 with a long periodic time. 



iii. We now come to the part of the figure between C and D. For 

 the parts AC and BD of the line AB in fig. 1, the planetary rotation 

 is slower than that of the satellite' s revolution, or the month is shorter 

 than day, as in one of the satellites of Mars. In fig. 3 these parts 

 together embrace the whole. In all cases the satellite approaches the 

 planet. In the case of fig. 3, the satellite must ultimately fall into 

 the planet ; in the case of figs. 1 and 2 the satellite will fall in if its 

 distance from the planet is small, or move round along with the planet 

 as a rigid body if its distance be large. 



For the part of the line of momentum AB, the month is longer than 

 the day, and this is the case of all known satellites except the nearer 

 one of Mars. As this part of the line is non-existent in fig. 3, we see 

 that the case of all existing satellites (except the Martian one) is 

 comprised within this part of figs. 1 and 2. I^ow if a satellite be 

 placed in the condition A, that is to say, moving rapidly round a 

 planet, which always shows the same face to the satellite, the condi- 

 tion is clearly dynamically unstable, for the least disturbance will 

 determine whether the system shall degrade down the slopes ac or db y 

 that is to say, whether it falls into or recedes from the planet. If 



* With the units "which, are here used the excess must be more than 4-f-3j j see- 

 further back. 



