522 MR. GL H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID. 
The enormous height of the lunar mountains compared to those in the earth seems, 
however, to give some indications that a cooling celestial orb must contract by a 
perceptible fraction of its radius after it has consolidated."' Perhaps some- of the 
contraction might be due to chemical combinations in the interior, when the heat had 
departed, so that the contraction might be deep-seated as well as superficial. 
It will be well, therefore, to point out how this contraction will influence the initial 
condition to which we have traced back the earth and moon, when they were found 
rotating as parts of a rigid body in a little more than 5 hours. 
Let C, C 0 be the moment of inertia of the earth at any time, and initially. Then 
the equation of conservation of moment of momentum becomes 
C n 
Co%o 
1 
And the biquadratic of Section 18 which gives the initial configuration becomes 
C n n, 
a 
x' 1 —(l-j-/x) . -~x+ =0 
The required root of this equation is very nearly equal to 
Cq%o 
C 
Now 
. To w o 
x 3 =I2 ; hence tt is nearly equal to (l+/r)~A But in Section 18, when C was equal 
to C 0 , it was nearly equal to (l+/r)r/ 0 . Therefore on the present hypothesis, the value 
* Suppose a sphere of radius a to contract until its radius is a + Sa, but that, its surface being incom¬ 
pressible, in doing so it throws up n conical mountains, the radius of whose bases is b, and their height h, 
and let b be large compared with h. The surface of such a cone is nb+ 7j-(& 2 + Hence the 
excess of the surface of the cone above the area of the base is and 4iTra 2 =4nr{a+Sa)' 2 -\-^mrh :i . 
Sa n //A 2 
a lb \aj 
Then suppose we have a second sphere of primitive radius a', which contracts and throws up the same 
number of mountains; then similarly — — =-d/ \ and-:— = (-—. Now let these two spheres be 
J a' 16 \a / a a \ha / 
the earth and moon. The height of the highest lunar mountain is 23,000 feet (Grant’s ‘Physical Astron.,’ 
p. 229), and the height of the highest terrestrial mountain is 29,000 feet; therefore we may take 
^=ff. Also -='2729 (Herschel’s ‘Astron.,’ Section 404). Therefore =-§-§■ of -2729 = '344, and 
(—^ =T183 or (-—d =8"45. Hence — -j-— =84 ; whence it appears that, if both lunar and terrestrial 
\h'a/ Xha/ a 1 a 2 
mountains are due to the crumpling of the surfaces of those globes in contraction, the moon’s radius has 
been diminished by about eight times as large a fraction as the earth’s. 
This is, no doubt, a very crude way of looking at the subject, because it entirely omits volcanic action 
from consideration, but it seems to justify the assertion that the moon has contracted much more than 
the earth, since both bodies solidified. 
