ATTRACTION IN THE CASE OF THE ENGIRSH ARC. 
37 
the attracting- mass to lie wholly on a plane — a supposition which will lead to no error 
in the calculation of attraction on stations in the British Isles. 
13. Let the horizontal plane through the station be taken as the plane of xy, the 
axes ofxand 1 / being chosen in such a position (at right angles to each other) as may 
be found in any particular case most convenient for the application of the resulting 
formula. Let « be measured vertically downwards. Suppose the attracting mass 
cut by vertical planes, parallel to the co-ordinate planes, into a number of masses 
on rectangular bases — the bases being of any size, large or small, and if necessary 
all different from each other, the dimensions of each being determined by the contour 
of the upper surface of the mass, so that a fair average height of the mass may be 
easily found. The smaller the parallelograms are the more accurate will be the 
result, but then they will be more numerous and the calculation more tedious. The 
nearer the parallelograms are to the station, and also the more irregular the contour 
of the country, the more attention will be required to make a judicious dissection of 
the mass. By supposing the several divisions thus made to be levelled down to their 
average height above the level of the sea, 1 conceive the whole attracting body to 
consist of a number of tabular masses of various dimensions. 
Let A be the station on which the attraction 
is to be found ; Kx, hy the axes of x and y ; 
BCDE the projection on the plane of xy of a 
tabular mass, lying in contact with the ver- 
tical plane zx, and very near to the plane %y ; 
AB = m; X, Y coordinates to the furthest angle 
D ; xyz coordinates to any point in the mass; 
H the height of the mass, its base being on the 
sea-level ; h he height of A above the sea ; both 
H and h I suppose to be small compared with X and Y, so that the squares of ^ and y 
may be neglected ; § the density of the mass. 
Then §dxdydz is the mass of an element, 
qdxdydz 
y 
E 1 
" ZJ 
a 
c 
1 
A 
B 
H 
c 
a' 
a' 
y 
E' 
D 
qxdxdydz 
its attraction on A, 
its attraction on A parallel to x. 
{x^-i-y^ + z^\^ 
Hence, whole attraction of the tabular mass on A parallel to x 
xdxdydz 
{x^ 
from x—m to x=X, 
3/ = 0 t0 3/=Y, 
z=ih — H to z — h. 
Integrating first with respect toy, then 2 , and then x, 
xdxdz 
attraction 
