AECHDEACO]^- PEATT ON THE INDIAN AEG OF MEEIDIAN. 
589 
These, converted into seconds, at the estimate of 69’5 miles to one degree, or 1 mile to 
61"’8, are 
-0"-00394, -0"-01375, 0"-04943, 0"-06204, 
which are absolutely insensible. 
From this it follows, that the length of the arc lying between its two extremities will 
be sensibly the same, whether it coincide with the mean ellipse in curvature, or with 
any of the other ellipses enumerated in § 2. Of course, on determining the geodetic 
amplitude from the formula 
1 3 
sin X cos 2m, 
the amplitude will come out diiferently for these different ellipses, although s is the 
same. But the fact, that the length of the arc is the same whatever the curvature of 
the arc (within the recognized limits), leads to this result — that the geodetic amplitude 
calculated from the length measured by the Survey, by means of the above formula 
applied to the mean ellipse, will be the amplitude corresponding to the mean ellipse, 
however much the actual arc differs from the mean ellipse owing to geological or other 
causes. Hence the deflection of the plumb-line in India from the normal to the mean 
ellipse can in no degree be attributed to the possible or probable circumstance of the 
curvature of the arc differing from the mean ellipse, as it may differ materially from it 
without producing this effect. 
I may illustrate this still further by finding what amount of deviation in the curvature 
there may be without producing even 1" in the calculated length. The formulae of 
paragraph 10 give 
Is =0-0003397?i«-0-0010097^5, 
m=0-0025536$a-0-0075756a5. 
Eliminate 'ha, and these give 
SE= 7 - 517 ^ 5 + 0 - 000015 a 5 . 
Putting ^^=1"=5 y;; 8 mile, and neglecting the second term, 
aE=:0T451 mile = 7 th of a mile. 
The surface of the earth may, therefore, be elevated or depressed through one-seventh 
part of a mile at the middle parts of the arc (about 800 miles long) without producing 
more than 1" difference in the length of the arc. 
The Table in paragraph 10 shows that the length of the Indian Arc, according to no 
one of the four different measures which have been made of it, differs by even -g^th 
of a mile, and in three cases even by much less than this, from the ellipse equal to the 
mean ellipse. 
12. The deviations brought out by the Indian Survey must arise, therefore, altogether 
from local attraction. The effect of the two visible causes — the Mountain-Mass and the 
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