210 FIGURE OF THE EARTH. [11 



4 



Now the volume of th spheroid = - ira'c ; call this M. Hence the 



o 



potential of the whole spheroid at any point upon its axis is 



M 8 c 5 



V= -- ,- 7TC . 



r 15 r 3 



7. Attraction and potential of the same at any point in the equa- 

 torial plane. 



We can find this without further integration. Let the equation of 

 the spheroid be 



where the axis of revolution is that of z, and the point considered lies 

 in the axis of x. Put as before 



where e is supposed small ; then 



e 



r = a -- z; 

 a 



so that the thickness at any point, measured from the circumscribed sphere, 

 is - z 2 . Let V be the potential of this shell at a point on the axis of 



x ; then it is clear that V is also the value of the potential at the 

 same point of the shell 



a* 



which is the same shell turned through a right angle about Ox. Hence 

 the potential of these two shells together is 2 V. 



But together these shells make up a shell of thickness -(y* + z 3 ), that 



ot 



is to say a shell having yz for its equatorial plane ; this is the case we 

 have already discussed, the polar radius being a, and the equatorial radius 

 a (I e). Hence 



_ T , 8 a 3 8 a 5 4 a 3 4 a 5 



2K=-7T -- TT 776 ^' OT ' / =o 7re -- T^rTTe-r, 



8 r 1 5 ' r* Srlor 3 



/4 a 



which, subtracted from the potential of the circumscribed sphere ( - TT 



\3 T 



gives for the potential of the spheroid 



4 ,. . 1 4 a 5 

 -#(l-.)- + I5W -. 



