604 
ON THE DEFLECTION OF THE PLUMB-LINE AT ARTHUR’S SEAT, 
If now we put the angles DCP=ip, ABP=ip', we shall get, since 
V V 
-=COt® =7= cot®, 
a ^ a 
Attraction = 
We may thus determine a sufficiently close approximation to the effect of the hollow 
of the Forth. An examination of a map of Scotland, on a sufficiently large scale, 
will show that a rectangle of eighteen miles by twelve, having its longest side inclined 
40° to the meridian, may be placed so as to cover the greater part of the Forth with 
some exactness, having also Edinburgh opposite to the middle point of the south side 
and Arthur’s Seat nearly two miles from this side, as in the accompanying diagram. 
The angles (p and <p' will be found to be 73° and 18°, and therefore the 
attraction of a rectangular stratum of these dimensions with thickness 
h and density § will be, (2’302.'> being the reciprocal of the modulus 
of the common system of logarithms) 
=2f 
h 
5280 
(2-3025) log 
cot 9° 
cot (36° 30') 
2-3025 X -6695 
5280 
and therefore the corresponding or resulting deflection is 
2 
2-3025 X -6695 x 12-44-7 
5280 
|A=0"-00727|/i 
in seconds, h to be expressed in feet, ^ the mean density of the earth. 
An inspection of a chart of the Forth will show that the depth may be taken at a 
very even average of 30 feet below mean water-level ; so that the attraction of the 
water (^=1) upon Arthur’s Seat causes a deflection =0"-04 to the north-east at mean 
water ; the latitude of points in the neighbourhood is consequently variable to the 
amount of about 0"-02, depending on the tide. 
We may now suppose the water to be removed and the hollow filled up with rock 
to a mean level of 70 feet. Then taking 2-5 for the mean density of the rock, the 
attraction of this stratum would be 0"-36, or resolved in the direction of the meridian, 
the deflection north would be 0"-28. If the hollow were filled up to a mean level of 
150 feet, the deflection north would be 0"-50. 
From this we conclude that the existence of the hollow of the Forth will account 
for but a small portion of the deflection of 5". 
To the south of Edinburgh the country gradually rises, until at the southern 
boundary of the country the mean level is about 1000 feet with peaks rising to 1750 
feet. The contours for the county of Peebles are not yet sufficiently advanced to 
permit the calculation of the attraction of the hills in the north of that county. We 
may however extend the calculation to the southern borders of Edinburghshire. 
