HIMALAYAS ON THE PLUftIB-LlNE IN INDIA. 
65 
Hence formula (4). becomes, — Attraction of mass standing on any one compartment 
^sin^jS-— 
yw U 2' r ^ 
Let h be expressed in parts of a mile ; § being the density of the superficial crust 
of the earth, we shall take =275, which is the density assigned to the mountain 
Schehallien ; reducing to numbers, we have 
Attraction of mass standing on any one compartment 
= 0-000005523 X A sin i/3.g (5.) 
This gives the attraction of each mass in terms of gravity. 
22. We may from this easily deduce the deflection of the plumb-line caused by the 
attraction of the mass ; for the tangent of deflection evidently equals the expression 
(5.) divided by g, by the simple law of the resolution of forces. Hence 
Tangent of deflection =0-000005523 X ^ sin ^ j3 
Jt 
= tan (l"-1392) X A sin ^/3 ; 
Deflection of the plumb-line caused by the mass standing on any one compartment 
= 1"-1392 h sin -j3. 
( 6 .) 
23. It remains to calculate the dimensions of the successive compartments as 
indicated by the Law of Dissection which I have adopted. The equation which ex- 
presses the law cannot be solved directly; we must therefore resort to approximation 
or trial. All pairs of values we thus find for a and (p must satisfy the equation ex- 
pressing the law. That equation becomes, on our replacing the arc p by the angle p. 
?"= 
A HP 
21 
or 
^=log * 
11-0379639 
d-log sinQa-1-^^) 
— 2 log cos {loi+\p) 
( 7 .) 
This is the test which all corresponding values of a, and p must satisfy. 
24. The solution of equation (3.) expressing the law may be facilitated, for values 
of a not exceeding 38°, by expansion and approximation. Expand in powers of a and 
p, and it becomes 
+(2+4) } 
-^(“+l){^+6(2+4) 
5 /« 
