MEAN DENSITY OF THE EARTH. 
373 
ished in the ratio of the cosine of the angle to unity or in 
the ratio ^ • Multiplying these, we have the attrac¬ 
tion at J in the line formed by the intersection of a vertical 
plane through 0 J and a horizontal plane through J. In 
order now to reduce this effect to the direction N J we must 
multiply it by the cosine of the angle a. Thus we obtain 
for the attraction of the elementary mass, 0, in the direc¬ 
tion N J and in a horizontal plane the expression 
x h d r 2 cos a da dr dz 
(r 2 + z 2 )§ W 
For the total effect this expression must be integrated, be¬ 
tween limits, for the three variables, azimuth, distance, and 
height, and therefore the total attraction is 
The result of this integration is, when the vertical heights 
are small compared with the horizontal distances, 
T 
F—hdh {sin a 2 — sin a x ) Nap. log y' 
( 7 ) 
The above integration is not difficult and is given in Clarke’s 
Geodesy, page 295. Its explanation, however, is not given, 
and it is here stated in order that we may have a clear con¬ 
ception of its application. 
From formula (7) it is evident that if the mountain be 
divided into rectangular prisms bounded by radial planes 
and concentric cylinders in such way that the sines of the 
azimuths of the planes are in arithmetical progression and 
the radii of the cylinders are in geometrical progression the 
attractions for compartments will vary directly as the heights 
of those compartments. With such division, therefore, the 
total attraction may be found by simply summing the heights 
