412 Archdeacon Pratt on the Fluid Theory 



6. I proceed now to find the effect of the change in distribu- 

 tion upon the formula for gravity. Let V, Vj, and V 2 be the 

 potentials for the whole mass of the earth and for the spheroidal 

 portions of it lying within the outer and inner surfaces of the 

 shell. Then* 



v=? + (.-f)?a-4 



v.-M-.-?)¥G-4 



fx= sine of latitude, also?w = ratio of centrifugal force to gravity 



1 



at the equator = ^r ; 



m x E tfj 3 W2 2 E # 2 3 

 m h { a d m L 2 a d 



Now Vj — V 2 is the potential of the shell : this must be replaced 

 by the potential of the same mass as it is differently distributed, 

 viz. by (E,— E 2 )-rr, Let U be the potential of the whole earth 

 under the new distribution ; then 



U=V-(V 1 -Y 2 ) + (E 1 -E 2 )-r 



E 



r 





E Ea 2 f m E, a, 2 ma, 5 E 2 «/ »ia 2 5 \ /l A 

 E Ea 2 f m ~WI A 



= 7 + ^ U~2 _G J V3 -/i 7 suppose- 



Let r/ and (J be the value of gravity before and after the change 

 in distribution, 



dV . f dV E 



.-. #= - — - cent, force = - -^ — »i -zj (1 -/* 2 ), 



9 = --j: -m^ 1 -^)'' 



dr dr a" 



?U E 



' l + 8(e-5)g^)-m(l-^ ' 



* Sec ' Figure of the Earth/ p. 93. 



.'/ 



