FOR DETERMINING THE MEAN DENSITY OF THE EARTH. 
335 
These different slices, it may be remarked, correspond to equal surfaces on the sphere; 
and upon these it is not improbable that the irregularities may mainly depend. 
55. In much of the preceding reasoning, it will be remarked, 1 have tacitly 
assumed that large continental elevations or large marine depressions, as we find 
them on the earth, do not interfere materially with the general law of attraction based 
on the spherical distribution of matter. For the reasons wliich seem to sustain this 
assumption, I would refer to a paper by me (printed in the Philosophical Transactions, 
1855) on the Attractions of Mountain Masses. It will also be remarked that I have 
not introduced the consideration of the earth’s rotation. I conceive its effects to be 
extremely insignificant; but the formulse applying to it are so unmanageable, that I 
have not pursued it to details. 
Considering now that it is sufficiently shown that, on the supposition that the 
surface in the neighbourhood of U is truly spherical, we may use the method of 
article 2, with no other uncertainty than that explained in article 53 : I shall pro- 
ceed with the corrections for the inequalities of the surface near U. 
56. First, I shall investigate the attraction of the matter included between two 
horizontal planes, figure 3, upon points U and L in these planes, whose distance or 
the separation of the planes is equal to the distance UL in figure 2. 
Divide the whole of the matter into cylindrical rings, of which UL is the axis : let 
the internal and external radii of one of these rings be ^ and + Call the azimuth 
of any part of the ring the end-surface of the prism included between d and 
is Let be the vertical ordinate measured upwards from the lower plane; the 
solid content of the part of the prism included between z and is : its 
. , . -r .... 17 * d.plp.M.'^Z , , , , 
attraction on the point L, supposing its density to be a, is — / a — ; and the resolved 
part of this, in the vertical direction, is Integrating with respect to 
between the limits «=0 and ^ = c=UL, we have — rV Integrating 
^ ^ \? ip^ + c^J ^ ® 
Inte- 
with respect to 6 for the whole circumference, we have 2^.(1. (ip _ 
grating with respect to g, we have 27r This is the attraction 
upwards on the point L. The attraction downwards on the point U will be the 
same ; and thus the difference of attractions on U and L, estimated in the downwards 
direction, will be A'tt . 
If the planes be continued without limit, or be infinite, this expression becomes 
Arc.d. Now the attraction of a shell whose thickness is c, computed as in article 2, 
is 0 for the point at the inner surface of the shell, and Azc.d for the point at the outer 
surface, and therefore the difference is A^rcd. Hence it is indifferent whether we 
consider the difference of attractions at the upper and lower stations (independent of 
the change in the attraction of the nucleus caused by the change of distance from it), 
