GRAVITATIONAL METHODS 261 



postulates that the amount of material standing upon a unit area will be the same 

 regardless of whether it is under highlands or lowlands, continents or ocean depths. 

 The unit area, of course, cannot be taken indefinitely small ; also, the state of isostasy 

 is not perfect. In general, a circle 100 miles in radius is large enough to serve as a 

 unit area, and frequently a much smaller circle may be used. 



Two theories have been advanced as an explanation of the phenomenon of isostasy. 

 One of these, published by J. H. Pratt in 1856, holds that there is a definite depth of 

 compensation. The material constituting the outermost layers of the earth's crust is 

 assumed to be less dense than the material below these layers. As a result, the total 

 mass standing on any unit area is substantially the same. This hypothesis lends itself 

 more readily to computation than the subsequently described theory. Topographical 

 computations give 60 miles as an average depth of compensation. 



Another theory published by Sir George Biddell Airy in 1855 is somewhat more 

 in accord with geological theories. According to the Airy theory, blocks of the earth's 

 crust are floating in a relatively dense plastic material, usually called magma or sima. 

 The excess of mass corresponding to high blocks is compensated for by a displace- 

 ment of the denser plastic magma or sima. Thus, blocks of crust containing excess 

 mass produce a greater displacement of the magma than blocks containing a "normal" 

 amount of mass. Calculations on this theory give an average depth of about 30 miles 

 for the lighter crusts. This depth is, of course, less under the oceans and more under 

 the continents and highlands. 



Isostatic Method of Determining the Figure of the Earth. — In one method of 

 determining the figure of the earth, the deflections of the plumb line are used to deter- 

 mine the ellipsoid which best fits a relatively small region. If it is desired to make the 

 region representative of the earth as a whole, it is necessary to correct the deflections 

 for the visible surface topography and its isostatic compensation. This isostatic method 

 was applied by Hayford to observations extending over the United States, and the 

 figure of the earth deduced by him was adopted in 1924 by the International Geodetic 

 and Geophysical Union as the best available figure of the earth as a whole. The 

 ellipsoid thus determined is known as the International Ellipsoid of Reference. The 

 semi-major axis (or equatorial radius) equals 6,378,388 meters and the ellipticity is 

 equal to 1/297. The mass of the ellipsoid, assuming a mean density of 5.527, is 

 5.988 • 10^^ metric tons. Later measurements of the deflection of the plumb line in 

 Europe gave isostatic results which are substantially in agreement with Hayford's. 

 Observations on the value of gravity discussed in the next section in general support 

 the conclusions regarding isostasy. 



"Normal" Variation of Acceleration Due to Gravity. — If it is as- 

 sumed that the earth is an elHpsoid of revolution revolving about an axis of 

 symmetry with a constant angular velocity and that the surface of the 

 ellipsoid is a gravitational equipotential surface, it can be shown that the 

 value of gravity at any point on the surface is given by the expression : 



g^ = g^{l+asm^c}.-bsm^2cf>) (20) 



where g^ is the value of gravity at sea level in geographic latitude <f), gE 

 is the value of gravity at the equator, and a and b are constants which 

 depend upon the gravity at the equator, the angular velocity of rotation, 

 and the departure of the shape of the earth from a true ellipsoid. The 

 constant b is small and may be taken as 0.000007. This leaves two coeffi- 

 cients gE and a. A commonly accepted value of g^ is 978.039 cm./sec.^ 



