AND ON THE REMOTE HISTORY OF THE EARTH. 
463 
In these equations P and Q stand for the cosine and sine of the obliquity of the 
ecliptic. 
Several conclusions may be drawn from this result. 
If e, e, e" are zero the obliquity remains constant. 
Now if the spheroid be perfectly elastic, the tides do not lag, and therefore the 
obliquity remains unchanged; it would also be easy to find the correction to the 
precession to be applied in the case of elasticity. 
It is possible that the investigation is not, strictly speaking, applicable to the case 
of a perfect fluid; I shall, however, show to what results it leads if we make the appli¬ 
cation to that case. Sir William Thomson has shown that the period of free vibration 
of a fluid sphere of the density of the earth would be about 1 hour 34 minutes.And 
as this free period is pretty small compared to the forced period of the tidal oscillation, 
it follows that E, E ', E", will not differ much from unity. Then putting them equal 
to unity, and putting e, e', e" zero, since the tides do not lag, we find that the obliquity 
remains constant, and 
. m 
This equation gives the correction to be applied to the precession as derived from the 
assumption that the rotating spheroid of fluid is rigid. This result is equally true if 
all the seven tides are kept distinct. Now if the spheroid were rigid its precession 
TC 
would be — cos i, where e is the ellipticity of the spheroid. 
The ellipticity of a fluid spheroid rotating with an angular velocity n is -§■ — 1 or \ —; 
0 "5 
but besides this, there is ellipticity due to the non-periodic part of the tide-generating 
potential. 
By (3) § 1 the non-periodic part of V is \wtv z (^— cos~ 6)(\ — Gqrq ~); such a disturb- 
T 
ing potential will clearly produce an ellipticity -( I — Gp~q 2 ). 
5 
If therefore we put e (J =l—, and remember that Gp> 2 q 2 —^ sin 2 i, we have, 
"5 
"I sin 3 f) 
Hence if the spheroid were rigid, and had its actual ellipticity, we should have 
TC 0 • I I T " 7 I 3 • o A 
-r-— — cos t- 4-A — cos l —4 snr i) 
dt n 12 tyi, v 2 ' 
(32') 
* Phil. Trans., 1863, p. 608. 
3 o 
MDCCCLXXIX. 
