236-238] 
The Adiabatic Model 
261 
From these formulae I have proved* that the value of V{ is 
where 4 , (q) = P - if DP + ~ (J/)> D'P - ... - Je [...] + (237 5), 
being formally similar to the <p defined by our previous equation (99*7) 
except that f and D are now defined by 
the lower limit in the integration with respect to q 2 is determined from the 
equation 
We obtain the various equilibrium configurations by inserting the value 
for Vi just obtained, and the value for p given by equation (236‘6) into the 
single equation of equilibrium, namely, 
Since this equation must be satisfied throughout the mass, we may equate 
the coefficients of all powers and products of x, y, z to zero separately, thus 
obtaining equations giving the coefficients in P 0 . 
The values of Vi and p are each equal to their values for the configura- 
(p 0 — a). As a consequence the equations of the boundaries of the configura- 
where F 0 ( x , y, z) = 0 is the equation of a configuration of equilibrium for a 
mass of uniform density p 0 , and so of a Maclaurin spheroid or a Jacobian 
ellipsoid. The function F 1 (x, y, z) is found to consist of terms of degrees 4, 2 
and 0 in x, y, z , the function F 2 (x, y, z) consists of terms of degrees 6, 4, 2 
and 0, and so on. 
238. A configuration of special interest is that at the point of bifurcation 
between the pseudo-spheroidal series and the pseudo-ellipsoidal series. This 
is derived by distortion from the point of bifurcation at which the Maclaurin 
spheroids join the Jacobian ellipsoids, so that we take F 0 ( x , y, z) = 0 to be 
the equation of the boundary in this latter configuration. 
a 2 + A, o* + A, c‘ + \ 
+ ecf> ( q ) -q 2 = 0. 
Vi + | to 2 (¿c 2 + y 2 ) = cons. 
.(237-6). 
tions of rotating incompressible masses plus terms in ( p 0 - a) and powers of 
tions of equilibrium for the rotating compressible mass are obtained in the 
form 
Phil. Trans. 218 (1919), p. 157, and Problems of Cosmogony and Stellar Dynamics , p. 166.