598 
ON THE DEFLECTION OF THE PLUMB-LINE AT ARTHUR’S SEAT, 
The ang-Ie of deflection produced by any horizontal attracting force acting on the 
pliiinb-line is measured by the ratio of the attracting force to the force of gravity or 
the attraction of the earth. 
The attraction of the earth upon any point on its surface in latitude X is* 
62 V 
1 — e— 
where b is the polar semiaxis, e the ellipticity of the surface, and m the ratio of the 
centrifugal force at the equator to the equatorial gravity ; if we put a for the radius 
of the equator, the attraction may also be expressed thus : 
here m=7j;Q5 sin^ nearly; whence it will follow that the term within 
289 ' 
the bracket will only influence the attraction by less than a six-hundredth part of its 
amount, and will therefore only affect the calculated deflection in that ratio. There- 
fore it is sufficiently exact to assume the attraction equal to that of a sphere whose 
radius is equal to the mean of the principal semidiameters of the earth, or 3956 1 
miles: hence the attraction on any point on its surface 7r.^(3956'l), taking the 
mile as the unit of measure linear. The defection, therefore (expressed in seconds), 
caused by any attracting force A on the surface of the earth may be taken as 
Itt.S. ( 3956-1) sin 1"— 8 ^ 
h being the mean density of the earth. Consequently the deflection caused 3y such 
a mass as we have been considering at the origin of coordinates or attracted point, is 
D=|(r 2 — ri).f sin g X 12''-447, 
or log — X I2"-447, 
where s is put for the difference of the sines. 
The calculation of the attraction or deflection of the plumb-line at any point of the 
hill is easily effected by means of these formulae. If through any one of the stations 
observed from, we draw on a contoured plan of the hill and surrounding country, a 
1 2 1112 
number of lines the sines of whose azimuth are successively 9? 12 ’ T 2 ’ T^’ 12 ’ 
counting from the south meridian in either direction, and from the north meridian in 
either direction ; and draw also a number of circles whose radii are 500, 1000, 1500, 
2000 .... feet, being in arithmetical progression with a common difference of 500 feet ; 
the hill will be thus divided into a number of prisms, the deflection caused by any 
* Airy’s Mathematical Tracts, pp. 167, 173. 
