014 ON THE FIGURE, DIMENSIONS, AND MEAN SPECIFIC GRAVITY OF THE EARTH, 
The average number of equations in each figure is about 44 ; the greatest number 
of equations in any one figure is 77. Each figure was worked by two independent 
computers. This of itself alone would have been insufficient to secure freedom from 
error, but the final working of every possible triangle, after the corrections were 
applied to the observed angles, secured perfect accuracy. 
Corrections to all the observed bearings having been obtained in the manner 
explained, it is clear that since all the geometrical relations of the figure are satisfied, 
no discrepancies can present theniselves between the calculated values of the distance 
between any two points by whatever series of angles it may. be obtained. The 
triangles are calculated by Legendre’s theorem. This theorem may be applied to 
triangles of any magnitude, up to two or three hundred miles, without fear of error. 
The greatest errors that can result in the values of the sides a, 6 of a spheroidal triangle 
as calculated from the side c by spherical trigonometry, using the geometric mean of 
the principal radii of curvature of the surface for the mean latitude of the triangle as 
the radius of the sphere, are (the position of the triangle in being the variable 
quantity with respect to whicli the errors are the greatest possible), 
— -{■ ni^ Arf 
cos^ 
^“12R2 
where R is the radius of the sphere, e the eccentricity of the earth’s surface, Imn the 
cosines of the angles of the triangle opposite respectively, and K its mean latitude. 
Comparison of Bases. 
The absolute length of any side, or the linear scale of the triangulation, is made to 
depend on the bases measured with the compensation bars at Lough Foyle and on 
Salisbury Plain. The discrepancy between the measured and calculated length of 
these bases is about 5 inches ; this discrepancy is divided so that each of the two 
bases shall exhibit an error proportional to the square root of its length : the com- 
parison of all the bases is then as follows; — 
i 
Date. 
Bases. 
Length in terms 
of Ramsden’s 
S cale. 
Length in terms 
of Ordnance 
Standard. 
Length in 
Triangulation. 
Difference. 
1791. 
1794. 
1801. 
1806. 
1817. 
1827. 
! 1849. 
Hounslow Heath 
Salisbury Plain 
Misterton Carr 
Rhuddlan Marsh 
Belhelvie 
Lough Foyle 
Salisbury Plain 
ft. 
27404-24 
36574-23 
26342-19 
24514-26 
26515-65 
ft. 
27406-190 
36576-830 
26344-060 
24516-000 
26517-530 
41640-887 
36577-858 
ft. 
27406-363 
36577-656 
26343-869 
24517-596 
26517-770 
41641-103 
36577-656 
ft. 
+ 0-173 
+ 0-826 
-0-191 
+ 1-596 
+ 0-240 
+ 0-216 
— 0-202 
