FIGURE AND STABILITY OF A LIQUID SATELLITE. 239 
cases the convergence breaks down. It appears, however, justifiable to argue from 
these results that the unstable body continually elongates until its end coalesces with 
the other elongated body. I have no doubt but that the same holds true when the 
masses are unequal, and that we should always find r—(c+C) diminishing until the 
two meet. The poorness of the approximation of course would prevent us from 
making good drawings when coalescence is approaching. 
§ 26. On the Possibility of Joining the Two Masses by a Weightless Pipe. 
This subject is considered in § 13, and it is there shown that if a certain function 
written f (a/r), for a given solution of the figures of equilibrium of two detached 
masses of liquid, is positive, the two masses are too far apart to admit of equilibrium 
when joined by a pipe without weight—and conversely. 
Now I have computed /"(a/r) in a number of cases of Roches ellipsoids in limiting 
stable equilibrium, and have found it always to be decisively positive. 
The corresponding function for two spheres is given in (3) of § 3, and its first term 
is + a 3 /r 3 . When we compute it for two ellipsoids, we find the corresponding term to 
have become negative, and the additional terms, which are given in (44), § 13, are 
also negative. Hence f (a/r) is decidedly less for two ellipsoids than it is foi two 
spheres of the same masses with the same radius vector. Thus the deformation of the 
two bodies tends in the direction of making it possible to join them by a pipe without 
weight, but it seems certain that in the cases of the Roche’s ellipsoids in limiting 
stability such junction remains impossible. 
I also computed f (a/r) for the much elongated ellipsoids which are roughly 
computed in the last section and finally overlap, and always found /(a/r) to be 
positive, as far as the approximate formula went. The additional terms tend, 
however, more and more to cause f (a/r) to vanish, and the approximation becomes 
very imperfect. Now I believe, although I cannot prove it rigorously, that if we 
could obtain a more exact evaluation of the forms of these elongated ellipsoids, and if 
further a more exact value of f (a/r) were calculable, we should find J (a/r) vanishing 
near the stage when the computations would show the two ellipsoids to overlap. It 
therefore seems probable that there is a figure of equilibrium consisting of two 
elongated masses joined by a narrow neck. These ellipsoids are very unstable when 
detached, and, according to the principles of § 2, it seems inconceivable that junction 
by a neck of fluid could render them stable. 
Part III.— Summary. 
Since the foregoing investigation may be read by mathematicians, while astronomers 
and physicists will perhaps wish to learn the nature of the conclusions arrived at, I 
shall devote this part of the paper to a general discussion of the subject, without 
reference to the mathematical processes used. 
