342 
ARCHDEACON PRATT ON THE CONSTITUTION 
final integral. Making these changes, and still estimating, as I always shall do, the 
attraction positive downwards, 
Vertical Attraction of the Cap above the station-level 
r3w 7« 3 3w 5 /3 w 
""i - c 2c 3 4c 5 y c 
3m 6 
4? 
m 3 \£ w/ 2_ | /t 2 + u 2 t 
2c 3 )c'c c 2 _|V u 2 ' c 
-3 
u 4 u 2 \ 
4c 4 c 2 J 
lo g, 
u t ft 2 -f u 2 t 
2c \/ u 2 c 
u 
2c 
+ 1 
As these formulae are to be applied to find the vertical attraction of the superficial 
portions of the earth, it may be here stated that, as the attractions will be always small 
quantities, the earth may be regarded as a sphere, and c taken equal to the mean 
radius 3956 miles, as the height or depth of any Cap above or below the sea-level will 
be comparatively small. 
10. These formulae may be much reduced for use by approximation. The square of 
t-^c will be neglected ; for the greatest value t a c will have in this paper will be 1 4 - 13 ; 
and therefore its square will be 1 4- 169. Expanding, then, in powers of 1 4-c and neg- 
lecting its square, and observing that as & occurs in the denominator of every term of 
the coefficient of the log., we may neglect f everywhere in the log. itself, we have 
Vertical Attraction of the Cap below the station-level 
=2^+*~g+S-x/?+S+ V 
6c 3 
7 u 3 t 
6c 3 4 c 5 
+ 
u A t 
'6c 3 
, ut 7v?t t u b t | 
~ 4 ~2c~Y2? Jr S?^ r 
( — — 1 
\2c 2 1 
4c 4 ' 
-=*Uog, (1-- 
=2vg(u+t — 
If we take the density of the surface to be half the mean density of the earth, and g 
be gravity, then 
Vertical Attraction of a Cap below the station-level 
— 4c( M- ^ — ^ \Ar+£ 2 + 2e ^ (1) 
The second formula in like manner gives 
Vertical Attraction of a Cap above the station-level 
= ~4c( m +^~V / u 2 +f- (2) 
11. The cap may be divided into Zones and a Central Portion in the following way. 
Let u and w be the chords of the angular distances from the station of the bounding 
