11] FIGURE OF THE EARTH. 217 



Hence g = G \l + (f^-6.) X 2 } , 



\* / } 



or the coefficient of \ 2 in this expression + the compression of the earth 



5 

 = - <. This is Clairaut's Theorem. 



u 



12. In this discussion we have not made any supposition as to the 

 internal fluidity or otherwise of the spheroid. Let us now suppose the 

 interior fluid, and let us consider what forms the surfaces of equal density 

 would assume. 



In the position of relative equilibrium, when the whole rotates as if 

 solid, the surfaces of equal density will coincide with the surfaces of equal 

 pressure. 



Let us consider the following problem : 



Suppose a spheroid ivhich is rotating as if solid to be made up of a 

 number of different fluids that do not mix, to find the ellipticity of the 

 bounding surfaces, given the volume and density of each different mass of 

 fluid. 



Thus we are given the quantities /c,, p l , /c, p.,, ... for each stratum, 

 and it is required to determine e 1; c 2 , 



The advantage of employing K rather than c will be obvious in this case. 

 Since the common surfaces are surfaces of equal pressure 



must be constant over each such surface. 



Let us first solve the case of three fluids, and afterwards proceed to 

 the general case of any number. 



It is to be observed that the portions of V which are due to the 

 different layers of fluid assume different forms according as the point lies 

 within or without that layer. 



Thus if F 1; F,, ... be the parts contributed by the first, second, ... 

 layer (beginning at the inmost layer), the first surface of separation is 

 external to the first fluid and internal to all the rest ; hence at the first 

 surface of division F is the sum of F 19 F 2 , F 3 , where 



A. II. 



28 



