242 
SIR G. H. DARWIN ON THE 
two ellipsoids also remains wonderfully nearly constant throughout a wide range of 
change in the value of X; for when X - <A4 it is (R653, and when X = 1 it is 0-660. 
only falling to 0'633 at its minimum. 
thus far I have been speaking of the modified problem of PiOCHE in which the 
planet assumes the appropriate figure of equilibrium, but I have also obtained the 
solution of Roche’s problem for an infinitely small satellite and a spherical planet. 
As stated in the Preface, the radius vector of limiting stability, which has been called 
“Koche’s limit,” is found to be 2A553, and the axes of the critical ellipsoid are 
proportional to the numbers 10000, 5114, 4827. These maybe compared with the 
2'44 and 1000, 496, 469 determined by Koche himself. When we consider the 
methods which he employed, we must he struck with the closeness to accuracy to 
which he attained. 
For the infinitely small satellite “the modification” of Koche’s problem hardly 
introduces any sensible change in the results, but for satellites of finite mass stability 
will continue to subsist for a slightly smaller radius vector for the spherical than for the 
ellipsoidal planet. In other words, the ellipticity of the planet induces instability, 
earlier than would be otherwise the case. 
Roche did not attempt to investigate how closely his equations were capable of 
giving the ellipsoid most 1 nearly representative of the truth, nor did he estimate how 
far the ellipsoid is an accurate solution. These points are considered above, and it 
was the desirability of making the investigation with a closer degree of accuracy which 
occasioned many of the difficulties encountered. 
h or the infinitely small satellite the ellipsoidal solution is exact, and with a 
spherical planet, but not for an ellipsoidal one, Koche’s equations give that ellipsoid 
exactly. In this case, however, the change introduced by the modification of Roche’s 
problem is quite unimportant. 
For finite satellites Roche’s equations require sensible modification, and the solution 
of the " modified problem is different from that of the unmodified one, although not to 
an important extent. But the ellipsoid derived from the corrected equations is deformed 
by an infinite series of ellipsoidal harmonic deformations, beginning with terms of the 
third order. Of these, the only ones which have any sensible effect are those which 
may be described as zonal with respect to the satellite’s radius vector. 
By far the most important of these is the third zonal harmonic, whereby the 
satellite assumes a somewhat pear-shaped figure, being sharpened towards the stalk 
end of the pear pointing towards the planet, and bluntened at the other end. In 
consequence of this deformation the shape is slightly flattened between the stalk 
and the middle. 
The fifth and successive odd zonal harmonics accentuate the sharpening of the 
stalk and the bluntening of the remote end. The fourth, sixth, and successive even 
harmonics also accentuate the protrusion of the stalk, but tend to fill up the deficiency 
at the remote end. 
