11] FIGURE OF THE EARTH. 215 



In the former case, let K' = K + 8/c ; then for a point within the shell 



for a point outside 



a 17 



8 V= 



-- TTO 

 rf Mi * 



15"' dK : -|*2 



If *, e refer to the stratum passing through the attracted particle we 

 may express r thus 



and then to obtain the potential of a solid heterogeneous spheroid at any 

 internal point we must take the sum of the latter expression for 8V inte- 

 grated from /c = to K K O , and of the former expression integrated from 

 K = K O to its value at the bounding surface ; but for an external point the 

 former expression must be used exclusively. Thus at any point of the 

 bounding surface or external to it 



47T f 8 TT /3 . 1\ f d( K s e) , 



V - DK-dK rF -3 1 S A, o \p -j ' dK. 

 r J r 15 r 3 \2 2/ J r dK 



But the whole mass of the spheroid is 



M = 4?r I pK*dK ; 



and we may write E = - TT \p -\ - dK, a constant depending only upon the 



structure of the spheroid ; thus 



v M E / l\ 

 = 7 ~ ? \ 3/ ' 



11. Let us apply this result to demonstrate Clairaut's theorem on 

 the variation of gravity at the surface of the earth. 



Consider the equilibrium of the fluid portions at the surface of the 

 earth. The equilibrium of this requires that at the surface 



T/ M EL, 1 

 where = ~ X 3 



if we suppose the analysis of 10 to be applicable to the earth ; that is to 

 say, if we suppose the surfaces of equal density within the earth to be con- 

 centric coaxial spheroids of small ellipticity. 



