158 
MR. J. H. JEANS ON THE CONFIGURATIONS 
Here V is the total gravitational potential, including that of tidal forces from 
neighbouring stars, if any are present. From equation (l) it follows that the surfaces 
p — cons., Q = cons., and p — cons., all coincide, so that the free surface, being a 
surface of constant pressure, must also be a surface of constant density, say p = a, 
and must also.be one of the equipotentials Q ~ cons., say Q = C. The value of the 
gravitational potential over the surface must accordingly be 
V = C ~^(x»+y 2 ), 
this being the potential of the rotating mass itself and of certain tide-generating 
masses outside. We may suppose the positions and structure of these tide-generating 
masses to be given, so that V 2 Y is given at all points outside the surface of the 
rotating mass, while Y is given at all points on the surface by the above equation, 
and vanishes at infinity. By a fundamental theorem in potential theory, it follows 
that Y is determined uniquely at every point outside the mass in terms of a>, C and 
the shape of the boundary, whence dV/dn must be determined uniquely in terms of 
these quantities at every point of the boundary. 
If M is the total mass of the rotating body, 
4ttM = jj ^ dS, 
where the integral is taken over the surface of the body, and on substituting the 
value of 3Y /dn, this becomes a linear equation, which may be regarded as determining 
C uniquely in terms of M. 
Thus when M, « and the equation of the boundary are given, it appears that Y and 
dY/dn are uniquely determined over the boundary. If the value of p at the boundary, 
say <x, is given, and if we also know the law of compressibility at the boundary, 
then p and dp/dn are uniquely determined over the boundary. Thus not one, but 
two, surfaces of constant density are fixed, namely the boundary S and the surface 
iust inside it, say S'. 
The mass, say M', inside S' is in equilibrium under the rotation w and gravitational 
forces which originate from the external tide-generating masses and also from the layer 
of matter between S and S'. It now follows from the preceding argument that the 
surface of constant density next inside S', say S", is also determined. There are 
now three surfaces of constant density fixed, and by a continual repetition of the 
foregoing process, we can fix all such surfaces in turn. # Thus when the external 
boundary is given, and also M, w and the tide-generating masses, which may be 
* This cannot be defended as a piece of rigorous mathematical reasoning, but there can, I think, be no 
doubt that it is true for practical purposes. I have discussed the mathematical complications elsewhere 
(‘ Roy. Soc. Proc.,’ A, vol, 98, p. 413). A more formal proof of the theorem will be found in the ‘ Monthly 
Notices of the R.A.S.,’ vol. 77, p. 187. 
