1313 
We may remark that’ / cannot exceed unity. A value of f larget 
than 1 would mean that the moment “of inertia about the axis 
pointing to the earth was larger than about the axis which is tangent 
to the orbit, and this would be an unstable state: 
4. The theory of Crarraur would lead to values of J', 8, fand qe 
which are absolutely in contradiction with those found above from 
the observations. 
Although the development of the theory is well known, and also 
its application to an ellipsoid with three unequal axes introduces no 
new principles, it is perhaps not devoid of interest to collect the 
different formulas into a concise summary. 
The forces acting on the moon are: its own gravitational attract- 
ion, the attraction of the earth, and the centrifugal force. Take a 
system of coordinate axes, with its origin in the centre of gravity 
of the moon and the axis of Z along the axis of rotation. We can 
with sufficient approximation suppose the earth to be situated on the 
axis of Y at a constant distance A from the origin. 
The equipotential surfaces are approximately ellipsoids of which 
the principal axes are situated along the coordinate axes, and have 
the lengths 
Br NE (i)? MEEO) 
Further the equipotential surfaces are also surfaces of equal den- 
sity. The density at any point is denoted by A and the mean density 
within any equipotential surface by D. We have thus 
1 ted 
D= | A —[p* (1 —o) (1 — v)] dp. 
: 0 
As we will only develop the theory to the first order of r ando 
inclusive, we require D only to the order zero, thus 
5 
Ê 
3 7. 
D=; farde 
pe 
Further we introduce the ‘integrals 
Ê b 
1 d lo 
S= [AZ ermee, T= [a de, 
8) dp reeds 
0 2 
£ b 
