1881.] On the Tidal Friction of a Planet, fyc. 



323 



have to be solved in order to trace the changes in the system of 

 satellites. 



Expressions are also found for the rotation of the planet, and for the 

 energy E, in terms of the resultant moment of momentum of the 

 system and of the £'s. 



It is then shown how these equations may be solved by series, pro- 

 ceeding by powers of the time. As, however, the series are not rapidly 

 convergent, they are not appropriate for tracing extensive changes of 

 configuration. 



The case where there are only two satellites is then considered in 

 detail, and it is shown that, if a surface be constructed, the points on 

 which have E and the two £'s as their three rectangular co-ordinates 

 (E being drawn vertically upwards and the gf's being horizontal), then 

 the solution of the problem is expressed by the statement that the 

 point, representing on the surface the configuration of the system, 

 travels down the steepest path. 



The contour-lines on this " surface of energy " are illustrated by 

 figures, and the graphical solution found therefrom is interpreted and 

 discussed. 



The second part of the paper contains a discussion of the part played 

 by tidal friction in the evolution of the solar system. 



It is proved that the rate of expansion of the planetary orbits which 

 arises from the friction of the tides raised by the planets in the sun 

 must be exceedingly small compared with that which arises from the 

 friction of the tides raised by the sun in the planets. Thus the investi- 

 gation in the first part of the paper, where the satellites are treated as 

 particles, is not applicable to the solar system. 



Although the problem of finding the changes in a system, formed by 

 a rigid or perfectly fluid sun attended by tidally disturbed planets, is 

 easy of solution, yet it seemed inexpedient to attempt a numerical 

 solution which should be applicable to the solar system. 



It appeared, however, likely that a knowledge of certain numerical 

 values would throw light on the question. Accordingly the moments 

 of momentum of the orbital motion of the planets round the sun, of the 

 sun's rotation round his axis, of the orbital motion of the satellites 

 round their planets, and of the rotation of the planets about their axes 

 are evaluated with such degree of accuracy as the data permit. 



From a comparison between the orbital momenta of the planets and 

 their rotational momenta, it is concluded that tidal friction can scarcely 

 sensibly have enlarged the planetary orbits since the planets had a 

 separate existence. 



By parallel reasoning (although the argument has much less force) 

 it also seemed improbable that the orbits of the satellites of Mars, 

 Jupiter, and Saturn have undergone very large extensions since the 

 satellites had separate existences, and it seemed nearly certain that 



