of experiments on the pendulum. 89 
mean as 1 to q, q = 1 whence a = — ; and if q = 
1.55, a — — .58, affording an expression which is, in all pro- 
bability, accurate enough for every astronomical purpose. 
If the variation of density were supposed to proceed equably 
with the variation of quantity, it would obviously be as the 
square of the distance from the centre, and the density would 
be as 1 — ax 3 , the mean density being found at the surface of 
a sphere containing half as much as the whole earth; and 
this might be considered as the most natural hypothesis, if we 
disregarded the effects of compression : but the arithmetical, 
progression of densities, from the centre to the surface, seems 
in every way improbable. 
3. On the irregularities of the earth’s surface. 
A. If we suppose the plumb line to deviate from its general 
direction on account of the attraction of a circumscribed mass, 
situated at a moderate depth below the earth's surface, the 
distance of the two points of greatest deviation from each 
other will be to the depth of the attracting point as 2 to \/ 2. 
Let the magnitude of the additional mass be to that of the 
earth as a to 1, and let its distance from 
the centre be b; then supposing the 
earth a sphere, and its radius unity, and 
calling the angular distance of any point 
from the semidiameter passing through 
the mass x y the linear distance from the 
mass will be v/ ( Px + (qx — b ) 2 ) = 
v' ( f *x -f q'x — 2 b$x &*) = ( 1 4. 
b 2 — zbgx); consequently the disturbing attraction will be 
i+bb—2b<;x : k ut *h e s i ne of the angle subtended by the two 
MDCCCXIX. N 
