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LXXIV. On the Limits of the Oblateness of a Rotating 

 Planet and the Physical Deductions from them. By Prof. 

 Percival Lowell*. 



1. TT\Q Clairaut we owe the first important work on tbe 

 JL relation of the oblateness, tbe rotation spin, and the 

 matter-distribution of a planetary body. By taking into 

 account only terms of the first order in the ellipticity he 

 showed that, whatever the law of density from surface to 

 centre, the oblateness must be comprised between the values 

 -|- and J of the ratio, <£, between the centrifugal force so-called 

 and gravity at the equator when the body is homogeneous. 

 In the present paper it will be shown that if we extend the 

 investigation to higher orders we can prove not only that 

 the upper limit is very approximately J $, but that the lower 

 limit is not dependent on a series but may be expressed 

 exactly in terms of the mean density and the rotation. 



2. To find the conditions for equilibrium in a fluid mass, 

 consider the little parallelopiped dx, dy, dz. The increase of 

 pressure between its two faces in the direction of x is 



- d £.d X .dy.dz, 



the minus sign being used because the pressure decreases x. 

 The resolved part of the force acting on it in the direction 

 of x is 



pX . dx . dy . dz. 



Similarly for the other coordinates. 



.For equilibrium these must balance. Whence 



or dp = p(Kdx + Ydy -f- Zdz) . 



This is the differential equation for fluid equilibrium. 



3. To find the integral equation for the pressure at a point, 

 suppose the fluid in equilibrium and canals drawn from any 

 point within to a point on the surface. The pressures pro- 

 duced by the fluid in the several canals must all be equal at 

 the point since otherwise there be a flow to or from it. The 

 pressure due to any canal must therefore be independent of 

 the form of the canal. 



* Communicated by the Author. 



