OF THE PLUMB-LINE IN INDIA. 
7rjl 
Fig. 2. 
§ 4. Calculation of the Effect of Defciency of Matter, according to the 
assumed hypothesis. 
9. I now proceed to calculate the effect of the attenua- 
tion caused below by the upheaval of the mountains. I 
shall suppose that the masses standing upon the sefveral 
compartments are concentrated into the middle points of 
the meridian-diameters of the compartments. This may 
be done without sensible error, as shown in the note*. 
And next that these masses are distributed into vertical 
bars or prisms, aU to the same depth d, each bar or prism 
being of uniform density in itself, but containing the 
whole amount of matter belonging to the compartment 
from the centre of which it is drawm down. The differ- 
ence betw'een the attractions of the mass thus distributed 
downwards, and of the mass as it lies in the Himmalayas, 
will gi'se the horizontal attraction on the plumb-line at 
the three stations A, B, C, on the hypothesis of the attenuation I have described. 
/. ^^^^^^centrated in the middle of its compart- 
meu IN the bar or prism into which it is distributed. Join AM, AN, AO NO. 
et . . —c, AN—^, A0_7, MN=f7, ZAOM=J It is a property easily proved, 
that the attraction of the prism MN on A in the direction AM= — 
c.k' 
But the attrac- 
tion of M on A in the same direction =^. The ratio of these is |. * In this ratio, 
therefore, musd the attractions of the masses on the compartments, as deduced in my 
ormer I aper (and re-calculated in this), be reduced, to obtain from them the attractions 
of the same masses when distributed into vertical prisms. 
Deflection along the middle line of any lime, caused by the mass on the compart- 
ment when distributed down to a depth d, becomes 
= r'-1392A|sin^/3; 
deflection in the meridian =1''-1892 sin ^ (3 cos Az. 
* Area of compartment =r/3 (cos a-cos t^^ + 4-)), r=rad. of earth. 
Afass on the compartment ^rd^cos a-cos (a + <{,))7A=L’AM sin|sin(^| + |j cos + 
. 1 
. attraction of the mass, con centrated in the centre 2'^ 
ditto (from last note) ~ i — 1 + ^ 
+ ... 
The largest value of this is when and ^ is evanescent, when it is about j-i^th part greater 
than iinitv. 
