FIGURE AND STABILITY OF A LIQUID SATELLITE. 
185 
As already remarked, we shall put M — f 77 -pa 3 -also, since the three axes of 
the ellipsoid are k cos y cosec /3, k cos /3 cosec /3, k cosec /3, we have 
k 3 cos fi cos y _ \a 3 
Hence 
sin 3 /3 
l+X' 
k 3 cos 2 y _ a 3 cos y tan /3 
3\r 3 sin 2 /3 3 (1 + X) r 3 
and the coefficient of the series in the expression for g does not become infinite when 
X vanishes; however, it is perhaps more convenient to leave it in the form as written 
above. 
This expression for gravity is the result required, but it is to be noted that it is 
determined on the hypothesis that the distant body is a particle or sphere instead of 
being an ellipsoid. 
The development ceases with terms of the seventh order, and the harmonic terms 
of third and higher orders have been neglected. Now the harmonic deformation of 
Roche’s ellipsoid of the third order of harmonics is of order F/E in inverse powers 
of r. This deformation is treated as surface density. If we were to proceed to closer 
approximation, we should have to take account of the square of the thickness of the 
layer; such terms would be of order k H /r s . Since, then, we are avowedly neglecting 
terms of this order, it is no use to carry the development higher than terms of the 
seventh order. 
§ 6. Form of the Expression for the Gravitational Lost Energy of the System. 
The system consists of two ellipsoids, say e and E, with their longest axes co-linear, 
and each of them is distorted by deformations expressible by ellipsoidal harmonics of 
orders higher than the second. To the order of approximation to be adopted these 
deformations may be replaced by layers of surface density, which may be denoted by 
l and L respectively. 
The lost energy of the system may be represented symbolically by 
V = i (e + If + i ( E+ If + (e +1) (E + L). 
Let s, S denote two spheres of masses equal to e and E and concentric with them 
respectively. 
Then the whole may be written 
U= i ee+ i EE+eE+%ll + ±LL+ (e + S)l + (E+s) L 
+ (e-s) L + (E-S)l + IL. 
In the term SI I divide S into two parts, namely S 1} which is to contain all the 
VOL. CCVI. — A. 2 B 
