764 
AECHDEACON PEATT ON THE DEELECTION 
surface, so as to raise no considerable mountain mass. The effect on the plumb-line at 
a station over the place where the expanded and compressed portions meet would never- 
theless be considerable. 
I will proceed to calculate the effect on the plumb-line of a large given space (the 
space I use is 4 millions of cubic miles, L e. half a cube of 200 miles each way, the 
thickness being 100 miles and always vertical), situated at different depths and distances 
from the station where the plumb-line is, and of a density equal to x^th part of the 
density of the part of the earth where its middle point lies. 
Let A/ (fig. 3) be the meridian through A. Draw upon it five four-sided figures 
bounded by parts of great circles diverging from A, and also by circles of which A is 
the centre : and let their dimensions be so chosen, that the meridian length of each is 
200 miles ( = 2° 52' 40") and the area of each equals a square of 200 miles each way. It 
will result from this, that the angular distances of the sides of the spaces from A will be 
4°, 6° 53', 9° 46', 12° 38', 15° 31', 18° 23'. By pig. 3, 
comparing these with the angular distances of 
the “compartments” by which I have before 
divided the meridian, it will be seen that the 
five “ spaces ” comprise the following whole 
and fractional parts of the “ compartments 
Space ah includes i of 18th, whole of 19th to 23rd, J of 24th compartment. 
Space he includes f of 24th, whole of 26th and 26th, ^ of 27th compartment. 
Space cd includes ^ of 27th, whole of 28th and 29th, ^ of 30th compai-tment. 
Space de includes 1% of 30th, whole of 31st , f of 32nd compartment. 
Space e/ includes f of 32nd f of 33rd compartment. 
The angular distances of the middle points of the five spaces from A are 5 26 , 8 19 . 
11° 12', 14° 4', 16° 57'; and therefore the chords of these angles (rad. = 4000 miles), or 
the values of c for the middles of the five spaces, are 
379, 681, 781, 980, 1173 miles. 
The angles of the lunes of which the five spaces are portions are found fr'om the 
expression ^ 
area o f space x 180 
sin 1° 26' 20" sin (angular distance of middle point from A) 
The results are |3=30° 7', 19° 42', 14° 41', 11° 46', 9 47'. 
18. Suppose now that each of the five spaces is covered by a mass one mile in imifoi in 
height ; and that in each case this mass is then distributed uniformly to the three depths 
in succession, 100, 300, and 500 miles, as represented in fig. 4, which is a vertical sec- 
tion. I propose to find the deflection caused at A by the mass under these several 
circumstances. For this purpose we must use formula (3.) of paragraph 10. The calcula- 
tion is much simplified, as h is always the same, and =1, and Az=0. The values of 
