180 



On the Secular Effects of Tidal Friction. [June 19 y 



that point must always slide down hill. It does not, however, neces- 

 sarily follow that it will always slide down the steepest path. The fall 

 of the guiding point into the ravine indicates the falling together of 

 the two stars. 



Thns, if the two bodies be started with less than a certain moment 

 of momentum, they must ultimately fall together. 



Next, suppose the whole m. of m. of the system to be greater than 

 the critical value. Now, the less steep of the two valleys of the 

 former case (viz., the one in which the origin lies) has become more 

 like a semicircular amphitheatre of hills, with a nearly circular lake at 

 the bottom; and the valley facing the amphitheatre has become merely 

 a falling back of the cliffs which bound the ravine. The energy curve 

 in fig. 2 would show a section perpendicular to the ravine through the 

 middle of the lake. 



The origin is nearly in the centre of the lake, but slightly more 

 remote from the ravine than the centre. 



In this figure li was taken as 4, and h as unity, so that it represents 

 a system of equal double stars. The numbers written on each con- 

 tour give the value of E corresponding to that contour. 



Now, the guiding point of the system, if on the same side of the 

 ravine as the origin, may either slide down into the lake or into the 

 ravine. If it falls into the ravine, the two stars fall together, and if to 

 the bottom of the lake, the whole system moves round slowly, like a 

 rigid body. 



If the point be on the lip of the lake, with the ravine on one side 

 and the lake on the other, this corresponds to the motion of the two 

 bodies rapidly round one another, moving as a rigid body ; and this 

 state is clearly dynamically unstable. 



If the point be on the other side of the ravine, it must faJl into it, 

 and the two stars fall together. 



Tt has been remarked that the guiding point does not necessarily 

 slide down the steepest gradient, and of such a mode of descent 

 illustrations will be given hereafter. 



Hence it is possible that, if the guiding point be started somewhere 

 on the amphitheatre of hills, it may slide down until it comes to the 

 lip of the lake. As far as one can see, however, such a descent 

 would require a peculiar relationship of the viscosities of the two stars, 

 probably varying from time to time. It is therefore possible, though 

 improbable, that the unstable condition where the two bodies move 

 rapidly round one another, always showing the same faces to one 

 another, may be a degradation of a previous condition. If this state 

 corresponds with a distance between the stars less than the sum of the 

 radii of their masses, it clearly cannot be the result of such a degrada- 

 tion. 



If, therefore, we can trace back a planet and satellite to this state, 



