OF THE PLUMB-LINE IN INDIA. 
740 
the antipodes of A, dividing the superficial mass into so many lunes of attracting matter. 
Circles are then draum with their centres in the vertical line through A, at distances 
increasing according to a peculiar law, which, for a name, I call the Law of Dissection, 
dividing the whole surface into a number of four-sided “compartments*,” which have 
this property, that the attractions of the masses standing on the several compartments 
of a lime thus formed are exactly the same when the heights of the masses are the 
same ; or, which amounts to the same thing, since the heights are all small compared 
ufith the distances from the station, the attractions of the masses standing upon the 
several compartments are in proportion to their average heights. 
7. It is then proved (p. 65) that if (3 be the angular width of the compartment, and 
h the average height above the level of A, the deflection of the plumb-line at A, in the 
direction of the meridian, caused by the mass standing on this compartment, 
= 1"'1392 h sin ^j3 cos Az. 
Hence the deflection caused by the whole mass 
= 1"-1392 2. A sin-|/3 cos Az, Cl.) 
Az being the azimuth, reckoned from the north, of the middle line of the lime. 
8. By an examination of the physical geography of the earth’s surface, I next show 
that the only mass which can affect the plumb-line lies within a space DEFGHIJKLD, 
which I call the Enclosed Space, and then describe : the extreme • diagonal lengths arc 
each about 2000 miles. 
Tables are next formed — of which a summary is given in the following page — con- 
taining all the data regarding the average heights of the masses on the several com- 
partments; and the formula (1.) brings out the following results: — 
At A. At B. At C. 
Deflection of the plumb-line in the ilieridian . . 27"-853 ll"-968 6''-909 
Hence the differences of deflections, or the errors in the astronomical amplitudes, are 
15"-885 and 5"-059. 
After applying these corrections, I proceed to calculate the ellipticity of the Indian 
Arc. With a view to make the results available for any other calculated attractions — a 
* The horizontal attraction of the mass standing on any one of these compartments is shown to equal 
(see p. 62 of former Paper) 
where a and a + ^ are the angular distances of the nearer and further sides of the compartment from A. 
The Law of Dissection, connecting (p wdth a, is this, 
p cos* 
(2^ + 4^ 
^ 21 
sin 1 
1 
<p being assumed equal to i a when these angles are very small. 
