HIMALAYAS ON THE PLUMB-LINE IN INDIA. 
91 
ordinate P/>, so that the three masses on QRr^, Q'R’r’q', and Q''R"r''q" are of the same 
length and coincide precisely with each other, except in a very trifling degree at the 
two extremities. AB'C' are the three attracted points for the three curves : ahc are 
the three points where the attracting mass may be conceived to be concentrated, 
corresponding respectively to AB'C'. ABC is the meridian through the three stations 
Kaliana, Kalianpur, and Damargida. It is evident that the attractions of the mass 
on A, B, and C, in the direction parallel to Aa, will not differ from the attractions on 
A, B', and C' by any appreciable quantities. 
Let Aa=a, B'b=b, C'c=c. By art. 49, we have shown that a=1688 miles, 
6=2692, c=3544. I will now show the consistency of these results, as flowing from 
the property I have enunciated. 
The Trigonometrical Survey shows that AB=37l miles, and BC=430. AB makes 
22°^ with AB', and cos 22° 30'=0'92388. Hence AB'=343 miles, and AC'=740. 
Taking A as the' origin of coordinates, and the axes as before, the equations to the 
three curves give {Am—x) 
— P^'= (a^ + 343j2 — (-^+343)^ 
and 
If we put a=1688 and 6= 2692 in the first and second of these and equate them, 
the equation is satisfied by x=222 miles. And when this value of x is put in the 
equation formed by equating the first and third values of Pm^, we obtain 
962^1^^-222^+962^} 
=3559 miles. 
The very close agreement of this result with that obtained from the calculations of 
this paper^ and shown in art. 49. to be 3544 miles, shows how exactly the law here 
deduced represents the facts of the case. The value of x, viz. 222 miles, places the 
line Pp on the part we have called the Plateau, running W.N.W. and E.S.R. about 
thirty miles north of Gertope. 
53. The law thus developed by aid of the curve enables us to interpolate the 
amount of the deflection of the plumb-line at any station of the arc between Kaliana 
and Damargida. Thus suppose X is the distance of the station from the fixed line 
Pp, and A the distance of the centre of concentration for that station. Then 
X^=Pm^, and this = 21 ^ — 222^ ; 
A^_ f (X.^ -222^)222^] 
'■ «■* “222^1^ + I* 
Since X is never greater than 962, and «= 1688, the second term within the bracket 
