136 LAMBERT: CONSTITUTION OF THE EARTH 



SO taken that kW is the potential of the additional forces due to 

 the rearrangement of matter. 



The quantities k and h have no necessary connection with 

 elastic yielding. The yielding may be a plastic one and the 

 nature of h and k may be illustrated by using a result about the 

 figure of the earth that is given in many works on mechanics.'-^ 

 If a spherical body the size of the earth, but with its mass con- 

 centrated at its center, its outer surface being defined by a non- 

 attractive fluid existing for that purpose, were set rotating about 

 its axis in 24 hours, the outer surface would take the form of a 

 spheroid whose flattening is 1/578 or in more general terms 



/ = 



co-a 



2g 



/ being the flattening, g the acceleration of gravity, w the angular 

 velocity of rotation and a the radius, so that oo'-a is the centrifugal 

 force at the equator. The disturbing force is the centrifugal 

 force of rotation, the potential of which is 



IT = — cos'</) 

 2 



4) being the latitude. Thus the swelling of the equilibrium tide 

 at the equator is W/g, as it should be. Our actual earth, however, 

 does not have its mass concentrated at the center. The rotation 

 produces a swelling at the equator, which rearrangement of 

 matter produces a force that draws still more matter to the 

 equator and so on till equilibrium has been attained. When this 

 has happened, our actual earth has a flattening of 1/297 instead 

 of 1/578. The quantity /j represents the ratio of the actual dis- 

 placement to the equilibrium tide, or h = 578/297 = 1.95, 

 for plastic deformation. Since the potential is proportional to 

 the flattening, we have 



, 1/297 ~ 1/578 



2^ E. g., Pratt, Attrcctions and figure of the earth, 4th ed., p. 101; or Thomson 

 and Tait, Natural philosophy (2nd ed.), 2: 370 and 394. 



