228 The Configurations of Rotating Liquid Masses [ch. yiii 
distances between the two masses. Since the equations determine p uniquely 
in terms of a and b, none of the curves p = cons, can ever intersect. 
The special case of p = oo has already been fully investigated; when p— oo, 
M' — 0 and the problem reduces to that of a single mass rotating freely in 
space under its own gravitation alone. Thus when p = oo . the solution of the 
equations represents the two linear series of Maclaurin spneroids (a = b) and 
Jacobian ellipsoids. In fig. 33, let S represent the spherical configuration 
a = b = r 0 , and let SM represent the series of Maclaurin spheroids (a — b > c ). 
Let B represent the point of bifurcation with the series of Jacobian ellipsoids, 
and let JBJ' represent this latter series. 
When p = — 1 equations (2067) shew that b — c. Thus the curve p= — 1 
is represented by the curve ab 2 = r 0 s ; let this be the line T'ST" in fig. 33. 
All points which lie to the left of the median line OSM represent con 
figurations for which b> a, and therefore configurations in which the primary 
is broadside on to the secondary. These configurations are unstable, for they 
would be unstable even if the primary were constrained to remain rigid. 
They need not trouble us further and we may confine our attention to the 
right-hand half of the diagram. 
Linear series for all values of p pass through S. The series for p = + oo 
is the broken line SBJ, that for p = — 1 is the line ST, while that for p = — oc 
is easily seen to be the line SO, along which a—b < c. Since two linear series 
cannot cross, it is clear that the series for a very large positive value of p 
must be asymptotic to the line SBJ. All the series from p = + oo to p = - 1 
accordingly lie within the small area bounded by the lines JB, BS, ST. 
The series in the area OST are of course series for which p is negative 
and numerically greater than 1, while those in the area MBJ are again 
series for which p is negative, a second series for p = —oo coinciding with the 
line MBJ. 
Let us now confine our attention to the series which lie inside the area 
JBST, these being as we have seen the only ones of physical interest. Each 
series starts at S and ends at the point in which the lines BJ and ST ultimately 
meet at infinity. Thus each series begins with a sphere and ends with an 
infinitely long prolate spheroid. As we pass along any one of these series /z 
changes while p remains constant. The value of oo 2 which is given by equations 
(206‘6) accordingly changes also, this giving the value of a real angular velocity 
when p is positive, and being regarded simply as an algebraic quantity when 
p is negative. The value of a> 2 vanishes only when p = — 1 or when p = 0 ; 
consequently it vanishes at S, along the line ST, and at the points at which a 
is infinite, but nowhere else. It follows that &> 2 is positive everywhere inside 
the area SBJT. 
Since on 2 vanishes at both ends of every series, it follows that on passing 
along each series eo 2 at first increases, and then after passing a maximum