240 
SIR G. H. DARWIN ON THE 
Two problems are solved here simultaneously; for the analysis required for their 
solutions is almost identical, although the principles involved are very distinct. 
We conceive that there are two detached masses of liquid in space which revolve 
about one another in a circular orbit without relative motion—-just as the moon 
revolves about the earth ; the determination of the shapes assumed by each mass, 
when in equilibrium, is common to both our problems. It is in the conditions which 
determine secular stability that the problem divides itself into two. 
One cause of instability in the system resides in the effect on each body of the 
reaction on it of the frictionally resisted tides raised by it in the other. If now the 
larger of the two masses were rigid, while still possessing the same shape which it 
would have had if formed of liquid, the only effect on the orbital stability of-the 
system would be due to the friction of the tides of the smaller mass generated by the 
attraction of the larger one. Investigation shows that in this case, as the two masses 
are brought nearer and nearer together, instability would not supervene from tidal 
friction until the two masses were almost in contact; but it is clear that the 
deformation of the figure of the liquid mass presents another possible cause of 
instability. In fact, instability, as due to the deformation of figure, will set in when 
the masses are still at a considerable distance apart. It amounts to exactly the same 
whether we consider the larger mass to be rigid, or whether we treat it as liquid and 
agree to disregard the instability which arises from the friction of the tides raised in 
it by the smaller body. Accordingly we may describe the stability just considered as 
“ partial, whilst full secular stability of both bodies will depend on the tidal friction 
of the larger mass also. 
The determination of the figure and partial stability of a liquid satellite (he., apart 
from the effects of the tidal friction of the planet) is the problem of Roche. He, 
however, virtually regarded the planet as constrainedly a sphere, whilst in general I 
have treated it as an ellipsoid with the form of equilibrium. 
It has ah eady been remarked that, as the radius vector of the satellite diminishes, 
pai tial instability first supervenes from the deformation of the smaller body. It 
therefoie hardly seems worth while to consider the partial stability of a system in 
which the liquid satellite (hitherto described as the smaller body) is greater than the 
planet. We may merely remark that in this case the problem conies to differ very 
little from that involved in the determination of the full secular stability of two liquid 
masses ; for if we consider the case of a large liquid mass (the satellite) attended by a 
small body (the planet), it clearly makes very little difference in the result whether 
01 not the tidal friction of the small body is included amongst the causes of instability. 
This being so, I have not thought it worth while to continue the solutions of 
Roche s problem (modified by allowing the planet to be deformed) to the cases in 
which the satellite is larger than the planet. The ratio of the masses of satellite to 
planet is denoted above by X, and the field examined by means of numerical solutions 
extends from X = 0 to X = 1, while the part omitted extends from X = 1 to X = oc. 
