104 
ON THE EQUILIBRIUM OF ROTATING LIQUID CYLINDERS. 
icop 
may, to a good approximation, be taken to be ^ , 
centre of gravity of the satellite, this must be equal to 
, and if x, 0 is the 
the factor ‘03 being introduced to allow for the displacement of the centre of gravity 
of the primary. 
Taking = y/(5a), we find the equation 
and putting orjiirp — 
r-^/- = -93 , 
I + w ~7rp 
’43, tliis gives the values 
ax~ = 1'54, . ,r = 1-24. 
Now ya. X is the sum of the semi-axes of primary and satellite divided by the 
semi-axes of the primary. The equation just found is therefore about as true as 
could be expected, the linear diameters of primaiy and satellite being approximately 
in the ratio of 4 to I. 
The ellipse given as curve 4 is stable, and, since the mass of the satellite is small 
compared with that of the primary, we may suppose the combination of primary and 
satellite to be stable. Thus, if our conjecture as to the interpretation of curve 8 
is sound, it appears that the linear series which commences with curve 5 remains 
stable until the mass separates into two masses. 
The motion of a gradually-cooling mass will therefore be through the following 
cycle of changes. Firstly, increase of the ratio [angular momentum -E (area)-] until 
we reach curve 5. Then motion alono’ Poinoaue’s linear series until we reach 
o 
curve 9. At this j)oint separation takes place, and the primary is left as a tidally- 
distorted form of curve 4. As the satellite recedes the tidal distortion decreases, 
and as the value of the ratio [angular momentum -F (area)“j again increases, the 
configuration moves along the Jacobian series of elliptic cylinders imtil curve 5 is 
again reached. Tin’s completes the cycle, and the continual repetiti<m of the cycle 
can only be ended l)y solidification, or some similai’ cause which is outside our present 
considerations. 
I have not attempted to give any discussion of results from the point of view 
of dynamical astronomy. The complications introduced ly the heterogeneity and 
compressibility of natural substances, as well as by the difference ])etween the two- 
dimensional and three-dimensional problems, are so great that any discussion with 
reference to the actual conditions of astronomy would be impo.ssible in the present 
paper. 
I have had the advantage of frequent conversations with Professor Darwtx on 
tlie subject of this j^ajier ; my thanks are also due to Professor Foesytji for advice in 
connection with tlie earlier sections. 
