Figure of the Earth. 485 



The potential of the earth, regarded as a heterogeneous 

 spheroid such as I have supposed, its polar radius being R, is 



V 3 = |KttJ% ± {l'(l + 2e)P- ^(3 cos* 6-1) ~}dfl 



o \r v y or 3 J * 



where A f E d ,+ ^^^-.^ 



By putting r=R and = in the expression for Y 3 we find 

 that the potential at the end of the polar radius 



_4 rA 2B ) 



And by putting r = R(l + e) [e being the value of *? at the 



7T 



earth's surface] and Q ■=.—-, the value ofithe potential at the 

 earth's surface on the equator is found to be 



4 f A B ~\ 



3 \R(l + e) + 5R 3 (l + e)* J 



4^ fA A B "1 



= 3 K7r iR- e R + 5Il3|. 



Since B already contains the first power of e we do not need 

 to retain eB &c. 



The potential of centrifugal force at the equator, due to the 

 earth's rotation, is 



iR 2 (l + e)V or JRV, 



(&) being the earth's angular velocity of rotation) provided 

 we assume, as in the case of attractions, that the space 

 variation of potential gives the force on unit mass, and not 

 the negative of this force as is usual. Since the earth's sur- 

 face is a level surface the whole potential at the equator is 

 equal to the whole potential at the pole. Hence 



4^ f A 3Bl l p . 



3 

 Now 4 



= E, the earth's mass. 



