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SIR C4. H. DARWIN ON THE 
the satellite, that one is in limiting stability, in which the two solutions coalesce with 
minimum angular momentum. 
A rotating liquid planet will continue to repel its satellite so long as it has any 
rotational momentum to transfer to the orbital momentum of the satellite. Hence an 
infinitesimal satellite will be repelled to infinity, and the configuration of limiting 
stability for an infinitesimal satellite attending a planet, which always presents the 
same face to it, is one with infinite radius vector. 
Very nearly the same conditions hold good when both planet and satellite are 
subject to frictional tides. In § 2 it is proved that when each body is constrainedly 
spherical, the radius vector of limiting stability is infinite when the ratio of the 
masses is infinitely small. The radius vector decreases with great rapidity as'the 
ratio of the masses increases, and when the masses are equal, the radius vector of 
limiting stability is 1738 times the radius of a sphere whose mass is equal to the sum 
of the masses, or is 279 times the radius of either of the two spheres. Thus, when 
the ratio ot the masses falls from zero to unity .(and this embraces all possible cases), 
limiting stability occurs with a radius vector which falls from infinity until the two 
spheres are only just clear of one another. 
V hen we pass from the case ot the two spheres to that of two masses, each of 
which is a figure of equilibrium under the attraction of itself and its companion, and 
subject to centrifugal force, the calculation becomes exceedingly complicated. Since 
the radius vector of limiting stability in every case must be greater than that of 
Hoche’s ellipsoid in limiting stability, and since in the latter case instability sets in 
through the deformation of the smaller body, it follows that in every case of true 
limiting secular stability of the system, instability supervenes through tidal friction. 
When the ratio of the masses is small, we have seen that limiting stability occurs 
when the two masses are far apart. In this case the deformations of figure are small, 
and could easily be computed by spherical harmonic analysis. 
Foi finite values of the ratio of masses, when spherical harmonic analysis would 
fail, a fair degree of exactness in the result may be obtained from the approximate 
foimula of § 19. there would be no serious error from this formula when the ratio 
of masses is less than a half, but for greater values of the ratio it seems necessary to 
have recourse to the laborious processes employed in determining Roche’s ellipsoids. 
I thought, then, that it might suffice to compute the configuration of true secular 
limiting stability in the case of equal masses. It is illustrated in fig. 5, and we see 
that the radius vector is 2*638. We found that for a pair of equal spheres, instability 
only set m when the radius vector, measured in the same unit, was 1738. Thus the 
defoimations of the two masses forbid them to approach as near to one another as if 
thej veie spheies. It should be noted that instability in this case must arise from 
tidal friction, because Roche’s ellipsoid in limiting stability was found to have a 
radius vector of 2 -514. 
V hen PoiNCARh announced that there is a figure of equilibrium bearing some 
