344 
ARCHDEACON PRATT ON THE CONSTITUTION 
which is precisely the same formula as before, — k being put tor k because it is measured 
down below the sea-level, and the density being that of the deficiency of attracting 
matter. 
Hence for all cases in which the thickness of the mass on the zone is such that the 
fourth power of its ratio to the mid-distance of the zone may be neglected, 
Vertical Attraction of the mass on the zone 
3qvo — u 1 f^ . 2h — k c 2 \ 
12. In order to simplify this as much as possible, I shall so divide the sea-level 
into zones that w—u is the same for the central portion and each zone, equal cl. Sup- 
pose n is the number of divisions — that is, a central portion, n— 2 zones, and a central 
portion at the antipodes, all following the above law. Let u } , u 2 , u 3 , . . . u n be the chords 
to the successive bounding circles. Then 
u l == d, u 2 — 2 cZ, rid, also =i 2c , 
M,_l U % 2 Ur, 11 ^ 
‘ ‘ 2 c n 2c n ’ ' * 2c n ' ' ' 2c n ’ 
and the formula becomes 
Vertical Attraction of the mass on the rth zone after the central portion 
_ 3 9 k ( i\ 2h ~ k n * \ ni 
4c 4c r(r + 1) J ' ' 
In the calculations of this paper I shall take n— 160, which makes the radius of the 
central portion and the width of each zone = 49 '45 miles, nearly ==50. 
I would here observe that matter may be always transferred in imagination in azi- 
muth round the station without altering its effect on the vertical attraction. An appli- 
cation of this principle to an actual case in the earth may often assist in getting a better 
average for the mass. Any zone may be subdivided into smaller zones, if necessary, 
according to the same law. Also any zone may be divided into four-sided compart- 
ments by great circles so drawn through the station as to divide it into portions, the 
average heights of which may represent the mass, if it be very irregular, better than the 
mean height of the whole would. 
If the fourth power of the thickness may not be neglected, as is done above, the for- 
mulae (1) and (2) must be used without expanding the radical. This I shall revert to 
in the latter part of the next Section. 
§ 4. Formula* for the “ Resultant Vertical Attraction" of the Central Portion 
of the Cap, and of the Zones. 
13. By the expression “ Resultant Vertical Attraction” of a mass I mean the vertical 
attraction of the mass, diminished by the effect of the attenuation spread uniformly below 
the sea-level according to the hypothesis. In the case of the ocean, the resultant ver- 
tical attraction will be the (negative) attraction of the deficiency of matter in the ocean, 
