176 
SIR G. H. DARWIN ON THE 
that is to say, if 
or if 
2^(f G'- + F') + F'^ + ^G' = 0, 
a \ a / a 2 5 
t + v£- + 3 = 0. 
r 3 - G' r 
But this last is true, being the equation (3) determining the figure of equilibrium ; 
hence A = 0 gives minimum angular momentum. 
Since two liquid spheres cannot be joined stably by a pipe, it seems very improbable 
that two tidal ellipsoids could be so joined as to become stable. Indeed, if the 
distortion of the surfaces of the two masses into ellipsoidal forms may be regarded as 
due to the removal of constraints whereby they were previously maintained in a 
spherical form, stability is impossible. 
The question as to whether or not there is an unstable figure with a thin neck will 
be considered later, for the present we are only concerned with the conclusion that 
there is no stable figure of this kind. 
Mr. Jeans has treated an analogous problem in his paper on the equilibrium of 
rotating liquid cylinders, # and has concluded that the cylinder will divide stably into 
two portions. The analogy is so close between his problem and the three-dimensional 
case, that it might have been expected that the analogy would subsist throughout; 
nevertheless, if we are both correct there must be a divergence between them at some 
point. 
§ 4. Notation. 
As the solution given below is effected by means of ellipsoidal harmonic analysis, 
it is well to state the notation employed. It is that used in four previous papers to 
which references are given in the Preface. 
In “ Harmonics ” the squares of the semi-axes of the ellipsoid were 
a 2 = h 2 [v 2 -\^\, b 2 = Jc 2 (v 2 -l), c 2 = JcV. 
The rectangular co-ordinates were connected with 
ellipsoidal co-ordinates v, p, (f> by 
x 
e 
■L = —(*> 2 — l) (p, 2 — t) sin 2 <f), 
z 2 2 2 1 — ft cos 2c/> 
F = ^ i+/s • 
* ‘Phil. Trans. Roy. Soc.,’ Series A, vol. 200, pp. 67-104. 
