HIMALAYAS ON THE PLUMB-LINE IN INDIA. 
89 
In the same manner it may be shown that 
distance from B of the point of concentration =2692 miles; 
distance from C of the point of concentration =3544 miles. 
The differences of these two last from the first are far greater than the distances 
of B and C from A, viz. 5° 23' 37" and 11° 27' 33". From this it is easily inferred, 
what indeed did not need this proof, that the mass in no sense whatever, even an 
approximate one, attracts as if concentrated in a fixed 
point. 
50. It is extremely difficult to obtain a simple law 
of attraction, even an approximate one, of such a mass 
as that under consideration. I have, however, arrived 
at one which appears to represent the facts with a con- 
siderable degree of exactness, and by which we can 
interpolate for the amount of deflection at any station 
of the arc intervening between Kaliana and Damargida. 
It depends upon the properties of the curve of which 
the equation is — Let Ay be the axes of 
X and y, and Q.aq be the curve ; the axis of y is an 
asymptote, and the curve cuts the axis of x at right 
angles at a distance A«=a. The property of this 
curve which I am about to use is as follows. The 
attraction upon A of any slender prism of matter Qq, 
parallel to the axis of y and terminated at both ends 
by the curve, is the same as if the mass of the prism 
were concentrated in the point a. This property is 
easily demonstrated. For by the process pursued in 
the proof of the Lemma (art. 13.), it can be shown that 
the attraction on A of the half prism QM in the direc- 
tion AM 
mass of dM . 
~ AM.AQ ’ 
.*. attraction of (^q on A in AM= ^^j°^q^ 
mass of Qiq mass of Qiq 
X V X^-\- if’ 
by the equation to the curve. This demonstrates the property. The same, then, is 
true of any line parallel to y : and it follows easily that a mass of the form QRrg^ and 
of uniform thickness will attract the point A as if concentrated in a. But more than 
this ; the property is true also for any prisms parallel to y and lying above the plane 
ocy : so that the mass lying on QRr^- need not be of uniform thickness, but may vary in 
Fig. 6. 
