702 AEGHDEACON PEATT OX THE DEELECTIOX 
be carried to any degree of exactness by the formnlffi. The measni-es of the SuTA cy 
give almost exact results. The lengths of the arcs measured are known to a wonderful 
degree of exactness. But the smallest disturbing causes affecting the plumb-lme mtro- 
duce an element of doubt and uncertainty, through the astronomical amphtudes, which 
vitiates the whole. This is unhappily the case with the Great Indian Arc. The eiTors 
brought to light in Colonel Everest’s volume (of 1847), ^iz. 5"-236 m one portion and 
-3"-789 in another, are too important not to be strictly and numerically accounted for. 
The cause must lie in the determination of the vertical. The deflection of the plumb- 
line caused by the attraction of the vast Mountain Eegion on the north wiU not account 
for these errors: it would make them very much larger (and would not explam the 
negative sign), as I showed in my Paper in 1855. No hypothesis of deficiency of matter 
below, which we can conceive, will remove the anomaly. The distuihmg cause must he 
elsewhere ; perhaps in the immediate neighbourhood of Kalianpui’ and Damaigi a, 
which should be most carefully surveyed for the pui-pose of detecting it; or it may be 
hidden beneath the surface, and arise from sources which we cannot possi% examne 
and calculate, because of our ignorance of their position and magnitude. [This will be 
illustrated in the following paragraph.] Some would recommend that we ignore the 
above errors altogether, and put them to the account of these uncalculated causes. 
They would reverse the problem, and make the errors the measui-e of the distohmg 
cause. But unfortunately this will not do; for it proceeds upon the assumption that 
the meridian is elliptical, and moreover that its ellipticity is known and is equal to the 
mean ellipticity of the whole earth; which is, in short, begging the question at issue*. 
* The degree of influence which errors in the amplitude and ellipticity may have in the mapping of the 
country may be learnt from the following approximate calculation. . , 
Let A be the length of the arc, A the amplitude, a the senihaxis major, fi, the latitude of the middle point, 
a the ellipticity of the arc, m the mean radius yector, then m=a (^1-- sj : also 
_a|l„|=(l + 3cos2ju,)| = OT|l-|£Cos2^j • • • ) 
Suppose that the quantities used in this formula are incorrect, and that they ought to be A + rfA, X+</X, 
B + dB', the corrections are connected by the following equation : 
^ = ^ + ^'-^COS 2a.dB. 
A X in 2 
How iu the arc between Kaliana and Kalianpur, A it 371 miles 0alci.lated on the supposition that the 
ellipticity = X is 19117. cos 2p=0-69293. I shall suppose that the radius m is correct, and 
^ 300/ 
therefore (hn—0 ; 
• -0-889J£ ; 
A X 
JA=-GX-17270*) = ^Ga- 17270*) miles. (/3.) 
X o2o 
In this formula d\ must be put = to the deflection in seconds, with the neffafive sign, for the following 
