1880.] History of Planet and Single Satellite. 269 



time, and therefore the infinite value of the obliquity does not indicate 

 an infinite rate of change of obliquity. In fact if we put n=Q in (1) 

 we see that di/idt= — ;j( t2 /2) sin 4/ However, to consider this case 

 adequately we should have to take into account the obliquity in the 

 equations for dnjdt and dxjdt. because the principal semi-diurnal tide 

 vanishes when n—Q. 



Similarly at the minimum of energy the system changes infinitely 

 slowly, and thus the obliquity would take an infinite time to vanish. 



We may now state the physical meaning of fig. 2, and this interpre- 

 tation may be compared with a similar interpretation in the paper on 

 " The secular effects of tidal friction," above referred to. 



A fluid planet of small viscosity is attended by a single satellite, and 

 the system is started with an amount of positive moment of momentum 

 which is greater than 4/3*, with our present units of length, mass and 

 time. 



The part of the figure on the negative side of the origin indicates a 

 negative revolution of the satellite and a positive rotation of the 

 planet, but the m. of m. of planetary rotation is greater (by an amount 

 h) than the m. of m. of orbital motion. Then the satellite approaches 

 the planet and ultimately falls into it, and the obliquity always 

 diminishes slowly. The part from O to A indicates positive rotation 

 of both parts of the system, but the satellite is very close to the planet 

 and revolves round the planet quicker than the planet rotates, as in 

 the case of the inner satellite of Mars. Here again the satellite 

 approaches and ultimately falls in, and the obliquity always diminishes. 



The part from A to C indicates positive rotation of both parts, but 

 the satellite revolves slower than the planet rotates. This is the case 

 which has most interest for application to the solar system. The 

 satellite recedes from the planet, and the system ceases its changes 

 when the satellite and planet revolve slowly as parts of a rigid body — 

 that is to say, when the energy is a minimum. The obliquity first 

 decreases, then increases to a maximum, and ultimately decreases to 

 zero.* 



The part from infinity to C indicates a positive revolution of the 

 satellite, and from infinity to B a negative rotation of the planet, but 

 from B to C a positive rotation of the planet, which is slower than the 

 revolution of the satellite. In either of these cases the satellite 

 approaches the planet, but the changes cease when the satellite and 

 planet move slowly round as parts of a rigid body — that is to say, when 

 the energy is a minimum. If the rotation of the planet be positive, 

 the obliquity diminishes, if negative it increases. If the rotation of 

 the planet be nil, the term obliquity ceases to have any meaning, since 

 there is no longer an equator. 



* According to the present theory, the moon, considered as being attended by the 

 earth as a satellite, has gone through these changes. 



