516 
MR. HENNESSY’S RESEARCHES IN TERRESTRIAL PHYSICS. 
the former will not be the same as if in solidifying all the particles retained the same 
positions which they had when constituting the entirely fluid mass. All investiga- 
tions heretofore made of the variation of gravity at the earth’s surface being grounded 
on this very untenable hypothesis, it seems desirable that a more general solution of 
the question should be obtained, in which such a supposition would not be involved. 
The forces acting on a particle at the shell’s outer surface which we have here to 
examine, are — 1st, the shell’s attraction; 2nd, the attraction of the nucleus; 3rd, 
centrifugal force. If the laws of arrangement of the matter composing the shell 
be continuous, the first of these forces will be equal to the difference between the 
attraction of the entirely solid spheroid and that of the spheroid produced by the 
complete solidification of the nucleus, the particles in both spheroids arranging them- 
selves according to the influence of the forces acting on them. 
5. Attraction of the Shell . — In a spheroid differing but little from a sphere, and com- 
posed of homogeneous strata varying in form and density, the expression upon which 
its attraction on an exterior point depends, is thus written*, — 
V=77^V<^a+^r/‘V(‘'‘W.+|w.+&c.), (6.) 
the radius of each stratum being of the form «(1 -f/Sio), a being the radius of the 
sphere equal in volume to the mass included within the surface of that stratum, and 
tt;=Wo+Wi-l-W 2 -f &c.. Wo, Wi, &c. being functions satisfying the equation of 
Laplace’s coefficients, fa the density of any stratum, a' the value of a at the surface, 
and r' the radius of the attracted point. 
If the above expression be supposed to refer to the spheroid included within the 
shell’s outer surface, it is evident that a corresponding expression may be obtained 
for the spheroid included v/ithin the shell’s inner surface by merely changing |8 into 
(3', and a! into ; |S' being a constant depending on the ellipticity of that surface, 
and the value of a corresponding to it, we shall have therefore 
( 7 .) 
If (6.) and (/.) be differentiated with respect to r, and the first then subtracted from 
the second, we shall obtain for Gj, the attraction of the shell, the expression 
-|3'7^V(a*W.+^V.+^W.+....)} (8.) 
Let us conceive the surfaces of both spheroids to be covered with a homogeneous 
fluid to a small depth compared with their radii. The bounding surfaces of these 
fluid strata will depend on the attractions of the spheroids, and also on the attractions 
of the fluid particles. If the density of each of these strata be nearly the same as that 
* See PoNTEcouLANT, Theorie Analytique, &.C., Livre V. No. 23. 
