112 THE EVOLUTION OF SATELLITES. 



We have uow to examine wbat eifects must follow from tlie attrac- 

 tiou of the satellite on an egg-shaped planet, when the two bodies con- 

 stantly maintain the same attitude relatively to each other. It will 

 make the matter somewhat easier of comprehension if we replace the 

 tidal protuberances by two particles of equal masses, one at P, and the 

 other at P'. If the masses of these particles be properly chosen, so as 

 to represent the amount of matter in tlie protuberances, the proposed 

 change will make no material difference in the result. 



The gravitational attraction of the satellite is greater on bodies 

 which are near than on those which are far, and accordingly it attracts 

 the particle P more strongly than the particle P'. It is obvious from 

 the figure that the pull on P must tend to stop the planet's rotation, 

 while the pull on P' must tend to accelerate it. If a man pushes 

 equally on the two pedals of a bicycle, the crank has no tendency to 

 turn 5 and besides, there are dead i)oints in the revolution of the crank 

 where pushing and pulling have no effect. So also in the astronomical 

 problem, if the two attractions were exactly equal, or if the protuber- 

 ances were at a dead point, there would be no resultant effect on the 

 rotation of the planet. But it is obvious that here the retarding x)ull 

 is stronger than the accelerating ])u.\], and that the set of the protuber- 

 ances is such that we have passed the dead point. It follows from this 

 that the primary eliect of fluid friction is to throw the tidal protuber- 

 ance forward, and the secondary effect is to retard the planet's rotation. 



Action and reaction are equal and opposite, and if the satellite pulls 

 at the protuberances, they pull in return at the satellite. The figure 

 shows that the attraction of the protuberance P tends in some measure 

 to hurry the satellite onward in its orbit, while that of P' tends to 

 retard it. But the attraction of P is stronger than that of P', and 

 therefore the resultant of the two is a force tending to carry the satel- 

 lite forward more rapidly in the direction of the arrow. When the 

 satellite is thus influenced^ it must move in a spiral curve, ever increas- 

 ing its distance from the planet. Besides this, the satellite has a longer 

 path to travel in its circuit, and takes longer to get round the planet, 

 than was the case before tidal friction began to operate.' 



Now, let us apply these ideas to the case of the earth and the moon. 

 A man standing on the planet, as it rotates, is carried past places 

 where the fluid is deeper and shallower alternately; at the deep places 

 he says that it is high tide, and at the shallow places that it is low tide. 

 In the figure it is high tide when the observer is carried })ast P. ISTow, 



I It is somewhat paradoxical that the effect of attempting to liurry the satellite is 

 to malce it actually moveslower. It would be useless to attempt au explanation of 

 this in such au article as the present one, hut the converse case, where a retarding 

 force acts on the body, may be more intelligible. When a meteorite rushes through 

 the atmosphere it moves faster and faster, because it gains more velocity by the 

 direct action of the earth's gravity on it than it loses by the friction of the air. And 

 yet it is the friction of the air which allows gravity to have play; so that we have 

 the paradox of friction accelerating the motion. 



