450 Royal Society : — 



out a method for calculating the deflection of the plumb-line at sta- 

 tions on the Indian arc, caused by the attraction of the Himalayas 

 and of the vast regions beyond, with a view to the correction of the 

 astronomical amplitudes of the measured subdivisions of the arc 

 before they are applied to the determination of the ellipticity of the 

 earth. 



The same subject is taken up in the present paper, but in refer- 

 ence to the English arc between Dunnose and Burleigh iVIoor ; and 

 a different method of calculating the attraction is given. 



1 he paper consists of three parts. In the first, the ellipticity of 

 the English arc is calculated without taking account of attraction. 

 The arc is divided into five parts, and the lengths and amplitudes 

 assigned to them in Mudge's Trigonometrical Survey of England, 

 vols. ii. and iii., are made the basis of the calculation. These por- 

 tions of the arc are compared two and two, and ten values of the 



ellipticitv thence deduced : the mean of which is — — The 



^ ' 43-8687 



ten values, of which this is the mean, diflfer considerably among each 

 other, indicating that there is some disturbing cause, like local at- 

 traction, affecting the plumb-line, and therefore the apparent lati- 

 tudes. The variations of the observed amplitudes are then dis- 

 cussed ; and the necessity of calculating the local attraction pointed 

 out. 



In the second part a formula is obtained for calculating the at- 

 traction. The method is different from that given by the author in his 

 first communication. The curvature of the earth is neglected, as 

 this would have no sensible effect on the results in the British Isles. 

 The attracting mass is divided into a number of smaller masses 

 standing on rectangular bases at the sea-level, and the height of each 

 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 amoimt 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 

 -f ™, and y +^'= or — ™ 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_ X. 1 



x' X' X' x' 



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

 gents equal the above ratios. 



