441 



is taken equal to the average height of the surface above the sea- 

 level. The dimensions of the bases may differ from each other, and 

 are determined by the contour of the surface in such a way that the 

 average height in each mass may not depart materially from the 

 height of any part of it. The investigation leads to the following 

 Rule for determining the horizontal attraction deflection of the plumb- 

 line caused by any one of these Tabular Masses (as the author calls 

 them) : 



RULE. Take the origin of coordinates at the station where the 

 plumb-line is. Let the plane of xy be horizontal, and the axis of x in 

 the vertical plane in which the amount of deflection 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 be considered 

 + ve , and y + ve or ve 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. Y y^ Y y 



x' X' X' x' 



Look in a Table of Tangents for the four angles of which the tan- 

 gents equal the above ratios. 



Form four more angles by adding (subtracting if they be negative) 

 half of each of these angles just found to (or from) 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. 



1" 



This result, multiplied by Hfeet and by - , will give the required 



oo9 



deflection of the plumb-line in seconds of a degree H being the height 

 of the Tabular Mass above the sea-level, and its density being taken 

 equal to half the mean density of the earth, which is that of 

 granite. 



The only restriction to be attended to in the application of this 

 Rule is, that the ratio of the height of the attracted station above 

 the sea to each of the horizontal coordinates of the nearest angle 

 of the Tabular Mass, must be so small that its square may be 

 neglected. 



If any part of the attracting mass is nearer to the station than 

 this, it must be divided into vertical prisms and the attraction of each 

 found ; for which the author gives a formula in a note. 



