AECHDEACO]^’ PEATT ON THE INDIAN AEG OE MEEIDIAN.' 
585' 
gg_|_0-05) are 33"-42, 21"'42, and 16"’30, which are very nearly equal to the calculated 
deflection. Also the values of u and v give the following results for the semi-axes and 
ellipticity : — 
■«=20919988, 5=20846981, £= 
1 
5. The mean ellipse, as determined in the British Ordnance Survey Volume, gives 
1 
«=20926500, 5=20855400, £= 
294 * 
If §5 be the excess (or, in case of a negative sign, the defect) of the semi-axes of any 
of the four ellipses described above, compared -with the mean ellipse, then the following. 
are true : — 
Arc I. 
'ha = 11 T 3 miles, 
'hh = 3-85 miles. 
Arc II. Arc III. Arc IV. 
10'90 miles, — 3‘73 miles, — 1’23 miles. 
3’93 miles, — 2’20 miles, — -I'GO miles. 
§ 3. The deviation of the local elliptic core from the form of the mean ellipse. 
6. The fom’ several ellipses enumerated in the last section, representing the form of 
the arc between Kaliana and Damargida under different data and methods of calcula- 
tion, are not necessarily concentric with the mean ellipse ; but they must have their 
axes parallel to those of the mean elhpse, because the latitudes are measured from the 
same or parallel lines. 
Suppose one of these four ellipses drawn through the extremities of the arc, Kaliana 
— Damargida, and an ellipse equal to the mean ellipse also di’awn through those two 
fixed points, with the axes of the ellipses parallel to each other. Let «, 5, g be the semi- 
axes and ellipticity of the first of these, u and /3 the coordinates to its centre measured 
from some fixed point near that centre, and therefore near the centre of the earth. The 
squares and products of a — 5, g, a and j3 may be neglected. Let s be the length of the 
arc, E the distance of the point of the arc in mid-latitude from the origin of coordinates, 
I and I' the observed latitudes of the extremities (viz. 29° 30' 48" and 18° 3' 15"), X and 
7)1 the amplitude and middle-latitude of the arc. I proceed to find the difference in 
length, and the distance at the mid-latitude of the local and mean arcs lying between 
the two stations, and also the distance of the centre of the local ellipse from that of the 
ellipse equal in dimensions to the mean ellipse, but drawn through the two stations at 
the extremities of the. arc, as described above. 
7. First. The difference in length of the arcs. 
1 3 
s=-(a-j-l/)A—^(a—h) sin cos 2m. 
Let c be the chord, r and >•' and d the polar coordinates from the centre of the 
ellipse to the extremities of the arc ; 
c^=F-jrr'^—2rr'cos (^—d)=2rr'{l — cos 
r=a(l — ssiiFl), r'=a(l—ssm^l'). 
