302 The Geophysical Structure of the Sea 



then, as a first approximation, the gravitational acceleration at sea-level is given by 



M 



while at a certain depth h where m is the mass of the Earth shell {R — h) it reduces 

 to 



M — m 



* {R - hf ' 



so that in a first approximation 



^0 + 2gQ - 



m R 



^ ~ 2M h 



If the mean density of the Earth p,„ is 5-5 and p is the density of the shell 

 {M = 3(47T^R^)pm and m = A-rrR-hp], then the expression in brackets becomes 



-^ , P ~ 1-05 and ^ = 3-086 x 10-^. 



^ Pm R 



In that way one obtains for the change in gravity within the sea the relation 



g = g^^ 2-303 X 10-«r, (IX.2) 



where z is the depth of the point under consideration. 



Equations IX. 1 and 2 give the intensity of gravity, and its direction is fixed by the 

 direction of the plumb-line. A surface which is always at right angles to the direction 

 of gravity is termed a level surface (Niveau-Flache). Considering only the gravita- 

 tional force, no work will be expended in the displacement of a body along such a 

 surface. The most important gravitational level surface is the free surface of the sea, 

 the ideal sea-level (see p. 6), wliich forms a part of the geoid. Every other level 

 surface is uniquely fixed by the amount of work that must be expended in moving a 

 particle from the ideal sea-level to any point on the surface under consideration. For 

 a surface at a depth h this work measured along the plumb-line direction is given by 

 the product gh. The level surfaces are thus also surfaces oi equal gravitational potential 

 with the ideal sea-level as principal potential surface with zero potential. In this way 

 the entire oceanic space can be regarded as intersected by a finite number of equi- 

 potential surfaces each of which is separated from the next one by a unit potential 

 layer. The thickness of this layer varies as g alters from point to point but the product 

 gh must always remain constant. The level surfaces must carefully be distinguished 

 from surfaces of equal depth below the sea surface. The two sets of surfaces will inter- 

 sect, and where a surface of equal depth is not at right angles to the vertical there will 

 be a gravitational component in the direction of this surface. If the two surfaces 

 were solid and smooth, a ball on a level surface would remain at rest, but on a sur- 

 face of equal depth it would begin to roll away from the equator towards the pole 

 under the influence of the gravitational component directed towards the poles. 



A point within the sea may be fixed by taking three co-ordinates, either (1) ^ the 

 geographical latitude, A the geographical longitude of the projection of the point 

 under consideration on the surface of the sea along the plumb-line and //, the geo- 

 metric depth of the point itself or, (2) the co-ordinates (j) and A as in (1) and as third 



