96 
ARCHDEACON PRATT ON THE ATTRACTION OF THE 
The fortimla for thus calculating’ the amplitude is derived from the usual relation, 
arc n _ 
• ^ Ml { I — * 
1 
^ axis major 
where I and L are the latitudes of the extremities of the arc, and e is the eccentricity 
of the ellipse. 
Let a be the semi-axis major; 
£ the ellipticity ; 
A the amplitude of the arc ; 
the latitude of the middle point of the arc. 
Then — X=/— L, 2|a/=/4'L; and the above form leads to the following, 
arc 
= M 1 
(l+3 
sin A ^ \ 1 o/, , r sin A 
-y— COS 2|a-)— — gll — 15 cos A. cos 
arc f 1 / . sin A ^ \ 
.-. A=— |l+-£(^l+3-^-cos2^Aj 
sin A 
2^ +- 1 — 15 
sin A 
COS A COS 4|M- 
1 
/■ 
arc 
If the values for arcs I., II., III. and a are put in the expression — , we obtain the 
three first approximate values of the computed amplitudes of the three arcs. They are 
0-0937324, 0-1052876, and 0-1990201 ; 
or the same expressed in angles are 
5° 22' 13"-715, 6° 1' 57"-145, and 11° 24' 10"-869. 
These must be put in the small terms on the right-hand side of the form for A above, 
and we have the second approximation to the three values, viz. — ■ 
A'=5° 23' 41"-796, A" = 6° 3' 53"-286, A'"=ll° 27' 36"-300. 
A third approximation will not affect these results. 
Now the astronomically determined amplitudes are, as I gather from Colonel 
Everest’s work *, 
5° 23' 37"-058, 6° 3' 55"-973, and 1 1° 27' 33"-032. 
And in art. 43. I have found the differences of deflection of the plumb-line at the 
extremities of the arcs 1., 11., III., to be 
15"-885, 5"'059, and 20"-944. 
* These will be found at pp. Ixx, Ixxi. Expressed in seconds they are 19417"'058, 21835"'973, and 
41253"’031. These are the amplitudes, given under the symbol A/ in the Table of final results at the close 
of his work, and which form the data for the comparison of the Indian arc with arcs in other parts of the w'orld, 
for the determination of the mean ellipticity of the earth. It is obvious that these values of A7 are not correct 
as final data, if mountain attraction is to be taken account of. They should be increased by the diflferences in 
deflection in the plumb-line caused by attraction at the extremities of the arcs. 
