THE PRESSURE WITHIN THE EARTH 67 



form requires, of course, a change in area of surface, which change 

 is noted in lines 5 and 6 of this table. The change in shape of 

 the spheroid would likewise change the values of the attraction 

 of gravitation at all points of the surface. The value of the 

 attraction at the poles would be greater than the mean attrac- 

 tion on the surface of the present earth, while the attraction at 

 the equator would be less. These values are placed in lines 9 

 and 10 of Table I. The determination of the attraction has 

 been made in terms of the eccentricity from accurate formulas. 1 

 The values could have been computed in terms of the ellipticity 2 

 from the following approximate formulas, in which the square 

 of the ellipticity has been neglected : 



attraction at pole =(1-)-^ e ) ^ 



attraction at equator = (1 — T ^ e) g . 



Here g is the attraction at the surface of the same mass in 

 spherical form. 



It should be noted in this connection that the ellipticities of 

 the spheroids under consideration are so large as to render the 

 omission of their squares unsafe, although, as is the case in the 

 present paper, no great importance is to be attached to the actual 

 figures of the results. For a like reason, Clairaut's theorem may 

 not be used with much accuracy in checking results. 



Besides the reduction in the attraction at the equator due to 

 the change in the shape of the earth, there was formerly a still 

 further loss due to the high centripetal acceleration accompany- 

 ing the short rotation period. In the case of ^ = .5, this 

 amounted to 107 dynes, and in the case of ^ = .4 to 66 dynes; 

 these values subtracted from the values of the attraction pre- 

 viously determined, give the value of equatorial gravity placed in 

 line 13 of Table I. 



The values of gravity and pressure at any point on the polar 

 or equatorial axis of the spheroid may now be determined. If 



1 See Pratt's Figure of the Earth, 4th ed., p. 98. 



2 The ellipticity is the difference between major and minor axes divided by the 

 major axis. I have represented it by the Greek e, and have represented the eccen- 

 tricity by e. 



