510 MR. G. H. DARWIN ON THE PRECESSION OE A VISCOUS SPHEROID, 
body, and therefore probably a contracting one, and therefore its rotation would tend 
to increase. Of course this increase of rotation is partly counteracted by the solar 
tidal friction, but on the present theory, the mere existence of the moon seems to show 
that it was not more than counteracted, for if it had been so the moon must have been 
drawn into and confounded with the earth. 
This month of 5 hours 36 minutes corresponds to a lunar distance of 2'52 earths 
mean radii, or about 10,000 miles; the month of 5 hours 16 minutes corresponds to 
2 - 39 earth’s mean radii; so that hi the case of the earth’s homogeneity only 
6,000 miles intervene between the moon’s centre and the earth’s surface, and even 
this distance would be reduced if we treated the earth as heterogeneous. This small 
distance seems to me to point to a break-up of the earth-moon mass into two bodies at 
a time when they were rotating in about 5 hours ; for of course the precise figures 
given above cannot claim any great exactitude (see also Section 23). 
It is a material circumstance in the conditions of the breaking-up of the earth into 
two bodies to consider what would have been the ellipticity of the earth’s figure when 
rotating in 5-f hours. Now the reciprocal of the ellipticity of a homogeneous fluid or 
viscous spheroid varies as the square of the period of rotation of the spheroid. The 
reciprocal of the ellipticity for a rotation in 24 hours is 232, and therefore the reciprocal 
of the ellipticity for a rotation in 5ij> hours is (Ai)- of 232 = X 232= L2 2. 
Hence the ellipticity of the earth when rotating in 5T hours is yg-th. 
The conditions of stability of a rotating mass of fluid are as yet unknown, but 
when we look at the planets Jupiter and Saturn, it is not easy to believe that an 
ellipticity of yg-th is sufficiently great to cause the break-up of the spheroid. 
A homogeneous fluid spheroid of the same density as the earth has its greatest 
ellipticity compatible with equilibrium when rotating in 2 hours 24 minutes.* 
The maximum ellipticity of all fluid spheroids of the same density is the same, and 
their periods of rotation multiplied by the square root of their densities is a function of 
the ellipticity only. Hence a spheroid, which rotates in 4 hours 48 minutes, will be in 
limiting equilibrium if its density is (£|) 2 or \ of that of the earth. If this latter 
spheroid had the same mass as the earth, its radius would be (Vi or 1 ‘59 of that of 
the earth. If therefore the earth had a radius of 6,360 miles, and rotated in 4 hours 
48 minutes, it would just have the maximum ellipticity compatible with equilibrium. 
It is, however, by no means certain that instability would not have set in long before 
this limiting ellipticity was reached. 
In Part III. I shall refer to another possible cause of instability, which may perhaps 
be the cause of the break-up of the earth into two bodies. 
It is easy to find the minimum time in which the system can have passed from this 
initial configuration, where the day and month are both 5l> hours, down to the present 
* Pratt’s ‘ Eig. of Earth,’ 2nd edition., Arts. 68 and 7U. 
