ATTRACTION IN THE CASE OF THE ENGLISH ARC. 
43 
half the mean density of the earth; and put n=20923713 feet. Then the above 
coefficient =^ 6X^7500 ^ being- expressed in feet. Hence also 
Tangent of angle of deflection of the plumb-line at A in the vertical plane through 
X caused by the tabular mass on F6 
= attraction parallel to x-^g 
= 76 i^ 50 Q{*og tan(^45°-f ^ ^i) +log tan(^45°-l-^ ^ 2 ) 
- log tan ( 45 ° -fi ^ 3 ) - log tan (45°+^ ^ 4 ) j ; 
or, angle of deflection of the plumb-line in the vertical plane through x 
=~ njlog tan (^45°+^ -flog tan^45°-f ^ ^ 3 ) 
— log tan (45°-f i ^ 3 ) — log tan (45°+^ ^ 4 ) 
20 . From this easily flows the following Rule for calculating the deviation caused 
in the plumb-line in any plane by the attraction of a tabular mass of which the height 
above the sea-level is H feet. 
Take the origin of coordinates at the station where the plumh-line is. Let the jilane 
of xy be horizontal, and the axis of x in the vertical plane in which the amount of de- 
flection is to be found. 
Write down the coordinates XY xy of the furthest and nearest angles of the tabular 
mass from the origin ; Y is always to he considered positive, and y positive or negative 
accordingly . 
Form four ratios, by first dividing each ordinate by the abscissa not belonging to it, 
and then by dividing each ordinate by its own abscissa, viz. —3 
Look in a Table of Tangents for the four angles of which the tangents equal the 
above ratios. 
Form four more angles by adding (subtracting if they be negative) half of each of - 
these angles just found to 45°. 
From the sum of the log-tangents of the first two of these angles subtract the sum of 
the log-tangents of the second two. 
This result, multiplied by H feet and by will give the required deflection in 
seconds of a degree. 
deflecting the plumb-line ; and V is the only part which has any influence in altering the time of vibration of 
the pendulum, as it is easily proved that a small constant horizontal force, though it affects the arc of vibration, 
has no effect on the time of vibration. Thus the determinations of local attraction by the pendulum cannot 
assist in determining the effect of local attraction on the plumb-line, except in as far as they assist in pointing 
out the relative density of the mass which deranges the normal state of things. 
G 2 
