62 



Ellipsoidal Configurations of Equilibrium [OH. in 



For the configuration R of closest approach, Darwin gives the value of r 

 as 2'343, but he does not compute the axes. 



For the contact configuration C, Darwin finds* 

 r = 2-372, a = a' = 1-186. 



In these solutions the figures are assumed to be ellipsoidal ; the harmonic 

 deformations which have to be superposed will of course bring the vertices 

 closer, so that actual contact will occur before the vertices of the ellipsoids 

 touch. Darwin gives the following figure, which he describes as " highly 

 conjectural " for such a case. 



Fig. 10. 



Darwin calculates the value of R in the configurations of limiting stability 

 and of closest approach (i.e. the minimum possible value of .R) in some other 

 cases ; from his results we can compile the following table : 



p= 1, 0-8, 0-5, 0-4, 0. 

 R (limiting stability) = 2'638, 2*574, 2'59, oc . 



# (closest approach) = 2'343, 2'36, 2'457. 



Partial Stability 



65. The entry p = 0, R = oo means, as has already been noticed, that 

 there cannot be secular stability for an infinitesimal planet until it has 

 been driven off to infinity. The agency by which this driving off is ac- 

 complished is, of course, tidal friction; the satellite M raises tides in the 

 primary M' ; the dissipation of energy in the tides provides the dissipation 

 necessary for secular stability to have any meaning, and the tidal forces 

 result in an acceleration of the small body at the expense of the energy of 

 rotation of the large, this process continuing until the bodies are infinitely 

 far apart. 



* Approximately : I have extrapolated to get initial contact in Darwin's table on p. 514 of 

 Coll. Works. Vol. in. Stress should not be laid on exact values, as Darwin specially draws 

 attention to the bad convergence of the series used in this and similar calculations. 



