64 0. E. SCHI0TZ. [NORW. POL. EXP. 



In the following pages, we will consider the earth to be spherical, and 

 pay no regard to its rotation. Thus on a spherical surface concentric with 

 the earth's surface, at a sufficient depth, the density must be regarded as constant; 

 and as the density, starting from this surface, may be considered on the 

 whole to change in one and the same manner inwards towards the centre, the 

 acceleration at the depth in question must have the same value over the whole 

 earth, at any rate beneath that portion of the earth's surface, which we are 

 here considering. Let the radius of this spherical surface be R^ , and that of 

 the earth's surface R^. 



According to the potential theory, the flux of force from a closed surface 

 is equal to kit times the sum of the active masses in the same. This can 

 be immediately employed upon the gravity, only taking the inward flux instead 

 of the outward flux through the surface, as the force between two ponderable 

 masses is in the opposite direction to the force between two similar electric 

 or magnetic masses. We will consider a portion of the shell outside the 

 spherical surface with radius R lt which is cut off by a conical surface with 

 its vertex at the centre of the earth, and subtending a solid angle dw. We 

 may consider the force of gravity along the sides of the cone, at any rate at 

 a sufficient distance from the boundary between land and sea, as running 

 parallel to those sides. The flux of force through the boundary of this part 

 of the cone will thus be limited to the two end surfaces, one of which is in 

 the open surface of the earth, the other on the spherical surface farther in. 

 If the acceleration at the earth's surface equals g a , and below at the spherical 

 surface with radius R lt g v , the flux of force, K, will be 



->&; <*-%;&, 



where / is the gravitation constant. Thus this expression gives the entire 

 mass that is found in the part considered, multiplied by 4?r. It follows from 

 this, since g according to the above, has the same value over the oceans 

 as in the lowlands of the continents, that over each surface unit of the inner 

 spherical surface with radius R lt beneath such portions of the earth's surface, 

 there must be the same quantity of mass, whether it is beneath land or sea. 

 If this be so, we can, in the following pages, when we leave out of conside- 

 ration those parts of the earth's crust that contain the boundary between land 

 and sea, regard the depths of the ocean as constant, equal to fe 1( and assume 



