11] FIGURE OF THE EARTH. 



thus the elementary surface is 



Now the thickness of the shell at P is PL. 



Thus if a be the equatorial axis of the spheroid and 

 L + e), or PP' = e.NP; so that 



PN 

 PL = --. 



c 



= c(l+e), then 



but 



Hence the volume of the elementary zone of the shell is 

 TTC ,, 



To find the potential, divide by p, and integrate from p = r c to 

 ) = r + c. This gives 



and the attraction, 



V=^-n^(---\, 

 15 \r 1*) 



dV _ 8 ,/5 3c 2 \ 



; ic TCC ~i ~~ -- ) , 



ar 15 \r r / 



consisting of two parts, one varying inversely as the square of the distance, 

 and the other inversely as the fourth power. 



The method of 1 may also be adapted to find this quantity. 



To find the potential of the whole spheroid add for the inscribed 



sphere the quantity 



4 .1 



3 



-7TC 3 -. 



3 r 



A. II. 



27 



