FIGURE AND STABILITY OF A LIQUID SATELLITE. 
209 
On introducing this in the above formula for a 3 /r 3 , I find 
T 3 = ¥ 0 + M X [! -(2-M x-(109 + 67X + 0X 2 ) X 2 
r 
(157712 + 261395X+97656X 2 +1568X 3 ) x 3 -..]. 
On writing 
a = 14 
1 a 3 
1 + Xr 3 ’ 
and inverting the series we find 
X = « + (2 —X) a 2 + (l 17 + 59X + 2X 2 ) a 3 
(306872+ 269975X+ 5739X 2 +908X 3 ) a 4 ... . 
(43). 
This series expresses a function of y in series proceeding by powers of a 3 /r 3 , and a 
similar series must also connect a function of V with the radius vector, so as to 
determine the figure of the second ellipsoid appropriately. This second series may 
be written down by symmetry. 
A. 
Since X must now be replaced by 1/X, the function corresponding to a is 
or \a, and the function corresponding to y is ^ Sm ^ 
If then we write 
7 [(3 +cos 2 r) X+l] 
X = 
sin 2 T 
7 [(3 +cos 2 r) X+l] ’ 
the symmetrical series for the other ellipsoid is 
X = a + (2X—1) a 2 +(l 17X 2 + 59X+ 2X 2 ) a 3 
+ (306872X 3 +269975X 2 + 5739X 3 + 908)a 4 ... 
Now a is easily computed for the first ellipsoid, and then is computed by the 
series. Thus we have 
; s (4X+1)X 
& " XX+} ' 
We obtain in this way a fairly accurate value of F corresponding to the value of y 
which determines the first ellipsoid. We can then compute K' by the method of the 
last section. We may thus obtain a good idea of the values of F and K with which 
it is necessary to work in order to obtain the final solution. 
§ 13. The Equilibrium of Two Ellipsoids joined by a Weightless Pipe. 
In § 3 the problem is considered of the equilibrium of two masses of liquid, each 
constrainedly spherical, when joined by a pipe without weight. It was shown that 
VOL. CCVI.- A. 2 E 
