212 FIGURE OF THE EARTH. [11 



corresponding to equal elements of the spherical surface by two other 

 elements the thickness of each of which is equal to the mean of the 

 thickness of the two former. 



If <f> be increased by TT all the terms are the same as before except 

 the term involving cos ( ; therefore the potential at M and the attraction 

 in OM will not be altered if we suppose the thickness of the shell at 

 any point P to be the mean of the two, or 



ec {X 2 ( 1 - /A 2 ) - (1 - X 2 ) ( 1 - /r) cos 2 <}. 



But if </> be increased by - , the thickness of the shell at each of 

 these points will now be represented by 



we may suppose the thickness at the four points referred to, to be equal 

 to the mean of these two quantities, that is to 



We have thus brought down the shell to one of uniform thickness, 

 and the same law applies as in the case of a shell included between a 

 spheroid of revolution and a sphere touching it at the poles, the quantity 

 e being replaced by 



In the term multiplied by e it matters not whether we take c or 

 OL. Thus we have 



T/ M 8 c'/3 1 

 -T5 ff M2 X -2 



where M is the volume either of the original spheroid or of its substitute. 

 The attraction in the direction MO is 



dV M 8 



: I - X* - 



(If i* 2 ^ " *~ ^ \ 9 O / ' 



tt / / o I \t LJ ] 



and perpendicular to MO it is dVjds, where it is easily seen 



d\ = - sin CMdsjr, 



dV 8 c 5 



so that it is r = - ire- X(l X 2 )* 



as 5 r 4 v 



or it varies as the sine of twice the latitude and is positive towards the 

 equator. 



