54 
ARCHDEACON PRATT ON THE ATTRACTION OF THE 
northern base attests, it would seem, to the non-existence of any external disturbing 
cause ? For in observing the altitude or depression of one station in the triangulation 
as seen from another, the error on the plumb-line must come into the calculation. 
The answer is, that these small errors occur in the calculation of the horizontal arc 
in very small terms not higher than the second order ; whereas in the expression for 
the inclination of the two verticals at the extremities of the arc they occur in terms 
of the first order. This I will further illustrate. 
5. Suppose the arc divided into n equal 
portions: and let V. the deflec- 
tions of the plumb-line at the w-j-1 stations 
thus chosen. Let A be one of these stations, 
and B the next towards the south ; Kz, 
Ax vertical and horizontal lines through A 
on the supposition that there is no moun- 
tain attraction ; Az', Ax' the vertical and 
horizontal lines as affected by attraction. 
Draw BM and BM' perpendicular to Ax 
and Ax': let AM=a, BM=^, ZzAz'=v, 
ZBAM=o 5. Then AM is the true hori- 
zontal distance between A and B, and AM' the calculated horizontal distance. Hence 
the calculation makes this portion of the arc too short by 
AM— AM'=Am[i — ^^^^^^^)=«(tan a.sim'-fl — cosv) = A.v-l-^«.v% 
neglecting the cube and higher powers of v. 
Hence the whole arc is made too short by 
+ + + •••• T*'™)) 
hjiju...hn being the heights of the various stations of observation above the trne 
horizontal line. When the Station B is below A then h is negative. These heights 
are all extremely small compared with a, as the arc lies through a comparatively flat 
country. Hence the expression for the error in the length of the arc is made up, as I 
said, of small terms of no higher order than the second ; whereas the error in the 
difference of latitude has terms of the first order. 
6. That this expression for the shortening of the arc is a minute quantity utterly 
inappreciable, may easily be shown by taking an extreme case. The quantities 
are some of them positive and some of them negative, in such a manner 
that their algebraical sum equals the difference of height of Kalianpur and Kaliana 
above the level of the sea. From Colonel Everest’s work on the Indian Arc (pub- 
lished in 1847) I gather, that between Kalianpur and Kaliana there are forty- seven 
principal stations, or, including the two terminal ones, forty-nine : and the Survey 
