582 
AECHDEACON PEATT ON THE INDIAN AEG OF MEEIDIAN. 
Indian Arc being curved differently to the mean meridian of the earth. As each new 
disturbing cause — first the mountains, secondly the possible variation in density below, 
thirdly the ocean — was thought of and the efiect calculated, the resulting curvature, of 
course, came out differently. 
In the present communication, however, I shall demonstrate that no change in the 
curvature of the arc, within reasonable and indeed wide limits, can possibly have any 
appreciable effect on the calculated amplitude. It is this fact which leads to the result 
I have announced in the first paragraph, I will explain how this result did not flow 
from my former calculations. The length of the arc s between two stations is 
a(l — sin X cos 2??i, 
X and m being the amplitude and middle latitude, and a, s the semi-major-axis and 
ellipticity. In order to find the effect produced on the dimensions of the ellipse passing 
through the two stations by increasing or decreasing the amplitude, this was differen- 
tiated, 5 and m being considered constant. This gives an equation connecting da and ds 
with dX, the change of the amplitude. A relation was then assumed (in the absence of 
a better metliod) betAveen da and db, viz. that the mean value of a and b is the same in 
the two ellipses *. The calculation which I now give rests upon the fact, that the length 
of the chord of the arc must be the same in both the ellipses, the local and the mean, 
draAATi through the stations at the extremities of the arc. There was a difficulty in 
following this course before, Avhich I have now overcome, I find the length of the arc 
in terms of the unknown chord and semi-axes, and then differentiate with respect to the 
semi-axes, remembering that the chord is constant. All the terms now being small, an 
approximate value may be used in them for the chord in terms of either semi-axis and 
the observed latitudes of the extremities of the arc, 
§ 2. Statement of the several calculations which have been made of the form of the 
northern 'portion of the Indian Arc. 
4. I will bring together the results of the various measures and calculations which 
have been made of the arc between Kaliana and Damargida, divided near the middle 
by Kalianpur. 
I. By a comparison of the two portions of the arc. Colonel Eveeest, taking the ob- 
served amplitudes, got the following results {a and b being the semi-axes of the ellipse) 
«=20985260 feet, 5=20875737, g=JL; 
and he found that the amplitudes, calculated on the supposition of the arc being part of 
the mean ellipse, are 5"‘236 in excess and 3"-791 in defect of the observed amplitudes f, 
II. Captain Claeke, in his Chapter on the Figure of the Earth, in the volume of the 
British Ordnance Survey lately published, gives formulae which enable us to calculate 
* Philosophical Transactions, 1859, p. 762, Note. 
t See Colonel Eveeest’s Volume, 1847, p. 428 ; also p. clxxvii of the Preface. 
