OF THE PLUMB-LINE IN INDIA. 
773 
27. As I havG not again calculated in this Paper the total deflections of the plumb-line 
at the three stations A, B, C, but only the meridian deflections, I cannot revise the posi- 
tions of the three points where the whole mass must be collected that it may produce 
the same effects as in nature. This is of no importance. The principle devised for 
mterpolating the deflection at any intermediate station between A and C, by means of 
the piopeity of a cui\e, still holds good: and the amount of meridian deflection may 
be represented nearly by tie expression I and L being the latitudes of Kaliana 
and of the intermediate station on the arc between its extremities where the deflection 
IS sought. It makes the deflection at the southern extremity about one-fifteenth too 
laige; but at the northern and middle principal stations it gives it correctly*. 
28. In the last page of my former Paper I compare the curvature of the Indian Arc 
under several hypotheses by means of the formula, 
Height of the middle point of an arc of which the amplitude is X, above the chord of 
the arc. 
being the latitude of the middle point, a the radius of the earth, and sufficiently 
small to allow to be neglected. This formula is correct. But I should not have left 
it in terms of X, the amplitude, but of s, the length of the arc ; since X is not the same, 
whereas 5 is, in the three cases to which the formula is applied. This change will make 
the height above the chord 
which is the same as before, except in the sign of g. The consequence of this is, that the 
arc flatter when attraction is taken account of, and is more curved when it is neglected, 
than the mean curvature. 
* The law of the inverse chord will naturally deviate from the truth, and give too large a value, as we 
recede from the Himmalayas, for the loUowing reason. The Himmalayan Mass has been shown to produce 
the same effect as a comparatively slender uniform prism of great length running nearly east and west. 
Now the attraction of such a prism on a point opposite to its middle, equals its mass divided by the product 
ol the point’s distances from the middle and from either extremity of the prism. Hence when the distance 
irom the middle, compared with the prism’s length, is smaU, the attraction will vary most nearly as the 
inverse distance ; but as the distance increases, the law evidently tends towards that of the inverse sg^ mre, 
which it ultimately attains when the distance is very great compared with the length of the prism. This 
sufficiently accounts for the actual deflection at Damargida being somewhat smaller than that given by the 
formula, and tends therefore to confirm the general calculation. 
Calcutta^ September I, 1858. 
