1862.] 
Local Attraction. 
149 
As noticed in these papers, I conceived also that the difference 
between the geodetic and astronomical amplitudes might arise, not 
solely from attraction influencing the plumb-line, but in part from 
the curvature of the Indian Arc being somewhat different from tho 
curvature of the mean figure of the earth. Geology teaches us, that 
the earth’s surface has undergone changes of level. The surface, 
therefore* cannot be now an exact spheroid. In this case the nor¬ 
mals at the extremities of the actual arc would include an angle not 
precisely equal to the amplitude of the mean or undisturbed arc, and 
part of the errors to be accounted for might, it was thought, arise 
from this; the remainder arising from local attraction influencing the 
plumb-line, and therefore affecting the observed or astronomical ampli¬ 
tude. This served to introduce a new element of difficulty. 
4. The ambiguity, however, with which the question was thus beset 
from all these causes is removed in the Fourth Paper, the last of 
the series, published in the Philosophical Transactions of 1861. The 
following theorem is there demonstrated :—That the length of the 
actual arc, altered as its form and position may be by geological 
changes, is nevertheless sensibly equal to the length of the mean or 
undisturbed arc. Hence, if we calculate the amplitude by using the 
measured length of the arc, and the mean axes, as is done in the 
Survey, it will come out the mean or undisturbed amplitude. The 
consequence of this is, that the relative position of places laid down 
on a map from geodetic operations is correct, and free from all sensi¬ 
ble error arising from local attraction, from whatever causes local 
attraction may arise. 
This is a most important practical result, and frees the Survey 
operations from a doubt which has attached to their high scien¬ 
tific accuracy, ever since it has been discovered that the influence 
of the Himalayas and of the ocean is so considerable, and that 
variations in the earth’s crust below may have an important disturb¬ 
ing effect. This theorem, moreover, gives us a direct means of 
estimating at once the difference of local attractions, and of local 
deflections caused by them, at the extremities of an arc. For the 
difference is precisely equal to the quantity by which the astronomi¬ 
cal amplitude differs from the mean or undisturbed amplitude found 
as above described. 
5. There is only one desideratum remaining; but one which I 
i 
