210 
SIR G. H. DARWIN ON THE 
the condition determining the ratio of the masses for given radius vector is expressed 
by a certain equation which was written /(a/r)' = 0 . Further, it was proved that if 
/(a/r) is positive the two spheres of liquid are too far apart to admit of junction, and 
il it is negative they are too near. Finally we found that all these solutions were 
unstable. 
The solutions for the two spheres showed them to be always very close together, and 
as all the solutions for two ellipsoids, when they are in limiting stability, made them 
much further apart than were the two spheres, it seemed somewhat improbable that 
two ellipsoids could be similarly joined by a pipe, and certain that they would be 
unstable if such junction were possible. Nevertheless, it seemed conceivable that the 
additional terms, which must appear in /(a/r) when the constraint to spherical form 
is lemoved, might alter the conditions so that the junction of ellipsoids by a pipe 
should become possible. It thus became expedient to solve a problem analogous to 
that of § 3 when the two masses of liquid are ellipsoidal. 
I he conditions of equilibrium of two ellipsoids unjoined by a pipe are given in 5 10 , 
and the additional condition corresponding to junction by a pipe is 
clV x 2 dl n 
+W-TT = 0. 
dk 2 dk 
In the piesent investigation I shall neglect the higher ellipticities, denoted / s and 
d", and terms of higher order than those in 1 /r 5 . 
With this degree of approximation we have 
v = (in P dy 
+- 
\ 2 
L( 1 + A ) 2 r 10 (1+A) 
1 o 
¥ 
+ { e E) 2 , 
where (eE) a is given in (14) (with omission of terms in 1 /r 7 ), 
Also 
, _ 2 r dv 
* k) Vo {y 2 —\ ) 1/2 (v 2 — 1 /k 2 ) 112 ’ 
and ' V E has a symmetrical form in // and K. 
a 3 
We have besides or = - 3 np (1 + £), where £ is given in (24); and I is given in ( 22 ). 
I he differentiation with respect to k and the subsequent re-arrangement of the 
equation are rather tedious, and I will not give the details of the operations. It may 
however be well to note that k, are functions of A, and that 
ckjj _ 1_ , d'V _ 1 
dk 3A(1+A/’ ~dk ~ 3 (1 +k) X ' 
