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MR. J. II. JEANS ON THE CONFIGURATIONS 
figure proves to be unstable also for all compressible masses, then the only possible 
figures of stable equilibrium for a compressible mass will be pseudo-spheroids and 
pseudo-ellipsoids. 
4. When the mass is compressible a complication can occur which, as it happens, 
does not arise in the incompressible problem. The resultant normal force at any 
point on the surface of rotating mass is —cQ/dn, this being the resultant of gravity 
and centrifugal force. In the incompressible problem, cQ/cn does not vanish, except 
in the unstable configurations at the far ends of the spheroidal and ellipsoidal series, but 
this is not necessarily the case in the compressible problem. We shall find that 012/3n 
can vanish either on the series of pseudo-spheroids or on the series of pseudo-ellipsoids, 
and when this happens matter will necessarily be thrown off at the points at which 
dQfcn vanishes. The series of figures of equilibrium may accordingly be abruptly 
terminated at any stage by the vanishing of ciljcn. 
5. Thus it appears that the series of figures of equilibrium for a compressible mass, 
until the stage at which fission begins, will consist of pseudo-spheroids and pseudo¬ 
ellipsoids, these series possibly being abruptly terminated by the vanishing of dQ/dn 
at any point. Our problem is to study these series of configurations; our method 
will be as follows : — 
For an incompressible mass the density at the centre, which we shall denote by p m 
is identical with the density at the boundary, which we denote by <r. For a 
compressible mass, will be different from <r, and a rough measure of the extent to 
which the density differs from uniformity will be given by a quantity e defined by 
c _ A >~' 7 
Po 
We know the solution of the problem when e = 0 ; we require to obtain it for all 
values of e. Our method is to adopt the known solution when e — 0 as generating 
solution and to obtain, by what amounts to a method of successive approximations, an 
expansion for 12 in powers of 6. The success of the method will depend on the extent 
to which the series so obtained is convergent. 
the value of 12 obtained in this way will be a function of x, y, z, e and of the 
constants which enter into the law of compressibility. When x, y , 0 are small the 
value of 12 will be found to be convergent for all values of e, but as x , y, z increase, 
the range of convergence of the series contracts. But in the most important case we 
shall consider, it is found that the series is convergent, even for points furthest 
removed from the origin, for values of e up to some value between two and three. 
The largest value of e which is of physical interest is e = 1, corresponding to a- — 0, 
and for this value of e the series converges with considerable rapidity, so that the first 
few terms will give a fair approximation to the truth. 
Our method, then, is to obtain the series of pseudo-spheroids and pseudo-ellipsoids 
as deformations of the already known series of spheroids and ellipsoids, expanding in 
(3) 
