WITHOUT THE USE OF THE SOUNDING-LINE. 
673 
Mountain Limestone 
Granite 
Basalt 
Bed Sandstone . . 
Slate 
2-86 
2- 63 to 2-76 
3- 0 
2-3 to 2-52 
2-8 to 2-9 
Average density of above 2*763 nearly. 
It is dependent, in the second place, upon the total gravitation of the earth in 
reference to a point on its surface, and upon the influence exercised in that general 
result hy the strata of matter in the immediate vicinity of that point. 
Mathematical Investigation . — In Plate 63. fig. 1 the circle represents the circum- 
ference of the earth, which I propose to consider for the present irrespectively of its 
rotation, and as being spherical and of uniform density. 
Let P be the point upon the surface of the globe where the attraction is to be 
measured ; then, in order to calculate the amount of variation that will be produced in 
the total attraction of the earth, supposing it to be of uniform density, by a given depth 
of water below the attracted point P, a line is drawn from that point to the centre of 
the earth, and the same is divided into an unlimited number of indefinitely thin slices, 
by planes perpendicular to that line. 
In taking one of these slices at the distance h from the attracted point, an expression 
is obtained representing its aggregate attraction, thus — 
The slice is composed of concentric rings of sectional area 
and of the capacity 
which gives 
dh . dx=dh . z . da : cos a, 
2tt . z . sin a . dh . z . da : cos a, 
2ir . z . sin « . dh . z . da. 
2 ? 
as the differential of the attraction, where z and z in the numerator and z 2 in the deno 
minator, although variable quantities, always vary together, or 
ddA 1 =2or .dh . sin a . dot,. 
This expression has to be integrated between the limits of h and 0, and a and 0 ; 
thus 
dh. sin a . da=: f 2'xdh i a . dot,=2ft ^ dh{\ — cos a). 
Jo Jo Jo 
f sin a . doi=l — cos a, 
5 a 
Since 
MDCCCLXXVI. 
