NO. 8.| REMARKS ON THE EARTH’S CRUST. 69 
The equation for the flux of force now gives 
hy 
oR do! — wae dw =tale R? dw dh, 
0 
where dw under the integral sign, is to be considered as a function of h. 
If we call the normal acceleration upon the earth’s surface g,, the mass 
of the earth M, and its average density oy, then 
putting this in the above equation, we obtain 
hy 
g do’ — 9) Eh do = 3.5 [FF do Fr VI. 
0 
As the thickness of the earth’s crust must always be considered a small 
quantity in proportion to the earth’s radius, R,, it will be seen that the 
integral on the right will have to be regarded as a quantity of the third 
order, if the quantities on the left are of the second. If therefore the 
density, @, does not undergo finite changes, an alteration in the course of the 
lines of force in the interior of the earth’s crust will have no perceptible 
influence upon the magnitude of the acceleration, g, as the change undergone 
by the integral will have to be regarded as a quantity of the fourth order. 
If, however, the surface-element, Rj dw’, cut off by the tube of force, differs 
from the normal, the acceleration will prove to be changed — augmented, as 
equation VI shows, where the element cut off is less than normal, and dimi- 
nished where the element is greater than normal. It will generally be difficult 
to express an opinion as to the manner in which the transverse section of a 
tube of force will change in its course through the earth’s crust to the external 
surface; but I think it will be clear from the above that we cannot straight- 
way conclude that there is an accumulation of mass deep down below a 
place, because the acceleration there appears to be somewhat too great, or 
a deficiency of mass if it is rather too small. If the tube of force meets the 
surface of the earth near the coast-line of a continent, we may infer, as we 
shall also see subsequently, that its transverse section at the free surface 
is smaller than normal if the tube of force intersects that surface in the conti- 
