NO. 8.] REMARKS ON THE EARTH'S CRUST. 69 



The equation for the flux of force now gives 



f' 

 ^Rl da>'yR\dw = ln \QR*diodh, 



o 



where dm 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 ff , then 



, M 4 , D 

 do I ^T = * n l a o M o ' 

 * 



putting this in the above equation, we obtain 



, 



j. VI. 



As the thickness of the earth's crust must always be considered a small 

 quantity in proportion to the earth's radius, R 6 , 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, p , 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, Rl du', 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- 



