FIGURE AND STABILITY OF A LIQUID SATELLITE. 
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resemblance to a pear, he also conjectured that the constriction between the stalk and 
the middle of the pear might become developed until it became a thin neck of liquid 
joining two bulbs, and that yet further the neck might break and the two masses 
become detached. References to my own papers on this pear-shaped figure and its 
stability are given above in the preface, and the present investigation was undertaken 
in the hope that a revision of Roche’s work would throw some light on the figure 
when the constriction has developed into a thin neck of liquid. 
As a preliminary to greater exactness, I have in § 3 considered the motion of two 
masses of liquid, each constrainedly spherical, and joined to one another by a weight¬ 
less pipe. Through such a pipe liquid can pass from one sphere to the other, and it 
will continue to do so until, for given radius vector, the spheres bear some definite 
ratio to one another; or, to state the matter otherwise, two spherical masses of given 
ratio, revolving in a circular orbit without relative motion, can be started with some 
definite radius vector so that liquid will not flow from one to the other. 
In this system the ratio of the masses and the radius vector are the only 
parameters, and I find that the condition of equilibrium is a cubic equation in 
the radius vector with coefficients which are functions of the ratio of the masses. 
The cubic has three real roots of which only one has a physical meaning, and the 
solution is illustrated graphically in fig. 1. The single circle on the right is the 
larger sphere, and it is maintained of constant size for convenience of illustration. 
The smaller circles on the left represent the solutions for various ratios of masses, 
which are the cubes of the numbers written on the successive circles. 
The solution of this problem seems to me very curious, but it does not possess 
much physical interest, since it is proved in § 3 that all the solutions are unstable. 
The distance between the two masses is much smaller than is the case with any 
of Roche’s ellipsoids, even with minimum radius vector, and accordingly it did not 
seem probable that the parallel problem, when the two masses are liquid and 
deformed, would possess any solution at all ; nevertheless, it was worth while to 
pursue the investigation to the end. 
When the masses are ellipsoidal and are joined by a weightless pipe, the solution 
would become very complicated, but the question may be attacked indirectly. 
When the masses are spherical there is a certain function of the radius vector and 
of the ratio of the masses which must vanish when a channel of communication is 
opened between them. If this function be computed for two given spherical masses 
with given radius vector, we find that it is negative if the two masses are too 
close together to admit of junction by a pipe without disturbance of their relative 
masses, and that it is positive if they are too far apart. 
When the figures of equilibrium of two detached masses of liquid are determined, 
it is possible to form the corresponding function, but part of it consists of an infinite 
series of which it is only practically possible to give the first few terms. Now I 
have computed this function in a number of cases of Roche’s ellipsoids, and have 
