44 
REPORT— 1847- 
the region of the equator. To test the truth of this hypothesis, we may apH 
to the phffiDomena already mentioned (art. 11) of precession and nutation, 
and the corresponding inequality in the moon’s motion. 
If the density of the earth were uniform, the resultant attraction of the 
moon or sun on the portion of the terrestrial n»ass containeil within a sphere 
having its centre coincident with the cenirt* of the .‘•pheroid, and its ladiw 
equal to the earth's yxilar radius, would Juanife»Uy pass through the centre or 
gravity of the earth, and would, ihort'l'ore. havir no cifect in producing pre* 
cession and nutation. In such case these plimnonu-na would be due wilely 
to the attraction on the protuherant equatorial mass wliirh forms the cxc«« 
of the spheroid above the sphere just mention’d. In like niannor the cor¬ 
responding inequality in the lunar motion would also be due entirely to the 
attraction of the same protiibi’rant poriiou of the eartli's Jiiass on the mt'on. 
Supposing the earth’s mass amt cxiernal form to be the same as at jircwnt. 
the annual precession of the pole, as due botli to the action of the moon and 
of the sun, would amount to nearly 5^"; and the coeHicient of the term ex¬ 
pressing the inequality in the moon’s motion would amounl to about 10 . 
Again, if we suppose the removal of the superficial mass from the polar to 
the equatorial regions to take place witliout atlrctitig the density of tlie ri« 
maining portion of the mass, the resultant attraction of the sun or moon on 
that portiun would still pass through the earth's eeutre of gravity, and would 
consequently have no intluence in producing the pliajiioincna in question, 
which would tliereforc depend entirely, us in the former case, on the equato¬ 
rial protuberance. Hence, the mass of the earth atui its cilipticity being tbe 
same in the two cases, the calculated lunar inequalities would bo proportional 
to the densities of the equatorial jirotubcraiices, and the calculated prcces* 
sional motions would be proj)ortional to the deiisiticH directly, and to the mo¬ 
ments of inertia inversely. In i he first case the detisity is the tnean tknfffy 
ol the earth; in the latter, it is the siipcrjwiol tfcnsi(y\ and these densities 
(art. 10) may be taken in tlm ratio of nearly ‘2^ to 1 . The moments of iticrtia 
may be taken in the ratio of about fi; 5, taking the variable density as above 
(art.lO). Consequently, if the spheroidal form of the earth were due to thf 
cause to which it is here assigtiwJ, the annual luni-solar precession would only 
be about 28 , and the coeHicicut of the lunar inequality- about 4". Their 
actual values, as established by observation, arc respetuively about 51’ and 8’, 
presenting the discrepancies of about 23" and d-" respcctivelv. 
In calculating the above amounts of thcprccessum an«i the lunarinequalitj* 
^sumtng the hypotliesis we are considering respecting the earth’s spheroidal 
torm. It has been supposed that the general interior mass would retain iis 
primitive density. 1 Ins would manifestly imt be strictly correct, since the 
density must neces^rily be more or less aO'ected by the removal of so large 
j ‘ deposiiiun about the equator. Let L, denote 
?peeiaaUcity for the solid matter of the earth before being snb- 
that^tfwhi^h"!? "'n considerable pressure than 
denote the m H f «»»>jccted on the surface of the earth; and let E 
inrfirnf^® i^cen subjected for an 
7 f to which all matter at 
probU n k ■? intricacies whicti would be involved in the general 
Skee lia too remote from tim 
earth’s fortoMniri^"’ density uftor the change in the 
orthe rrr r K ®5niilar to the external surface! hut if. 
>vould remain ' S<renter thmi E,„ the surfaces of equal density 
approximately sphencal, notwithstanding the change of pressure 
