4317 
[6 4-0 —25}=6 pi + q8 +... 
ya-+ b8 He? +... 
and substituting in (12), we can successively determine the constants 
a,b,c... We find that all these coefficients are zero. Consequently 
YOE 
eo. 
This being 50 for small values of 3, it remains true for all 9, 
since 4 and @ satisfy the same differential equation. 
Referring now to the various definitions given above, we conclude 
ew AE OOP Bet bx, 
TR AT oe == be. oe (lS 
o | 8 1 8, CA $x +40, ess a 
Now, since the velocity of rotation equals the mean motion in 
the orbit, we have by KerPrer’s third law, for the average value of £&, 
Ro? = AfM (1+), 
where the factor A is taken from the lunar theory. Therefore 
© = Alu) = 1.0095 
x 
We have thus from (13): 
1 7 = 0.7482, f= 0.2518. 
We found above that for the actual moon the true value of / is 
probably very near unity. We must thus conclude that the distribution 
of mass within the moon is „ot approximately in accordance with 
the theory of CrAIRAUT, 
6. Continuing however to trace the consequences of this theory, 
we now apply Rapav’s transformation of the differential equation 
(10) of Crairaur. Since On, it is sufficient to treat the equation 
for 4. 
Put 
D= DB’ VIA. 
Differentiating, and comparing with (10), we find 
ESS yg rg pa ae 
dg V1 4m 
Os ae i Boma vou 
Now the function “= - 
is nearly constant for small 
BE ke 1] 
values of 4, as will be seen from the following little table 
87* 
