The Variation in Attraction Due to the Attracting Bodies. 235 



The expression foi- any radius of an oblate ellipsoid in terms of ellipticity 

 and sine of elliptic angle (O) in which angle aagle (O) is measured from an 

 equatorial radius is: 



A, = A r 1-h sin- 0-\-\ h- sin- 0-* h- sin' 0±lr' (,.)±,, 1 



For any point (a) on the surface or within the ellipsoid: 



'[ 



1-h sin- 04-i h- sin- 0-i h- sin^ O+lv' (,,)- 



The above expre-sions for gravity and radial distance prove that gravity 

 for all points in the same layer of the ellipsoid varies inversely as distance 

 from point to center of body. To comprehend how this result satisfies the 

 second test for equilibrium in full it must be understood in the expedient 

 of the infinitesimal layers and sones, that the same system of cones is used 

 only to divide up one or any one layer. The point now required to be 

 proven is that when any cone is so moved that its axis is changed to the 

 normal or direction of gravity, without changing the point of base on the 

 layer, the cone cuts the same mass from the layer as when in first position. 

 The expedient used to divide up the layer requires the ellipse to be de- 

 scribed with a variable radius with the center fixed at the center of the 

 ellipse. Per law of ultimate ratio the radius is constant ia describing an 

 infiuitesimal arc. The same ellipse can be conceived described with the 

 same variable radius, and with that radius kept on the normal or indirect- 

 ion of gravify, providing the center so vary in locations that the describ- 

 ing end of the radius be kept on the ellipse to be dsscribed. In this case 

 the ellipse is known, because it is the one describ d by the fir.4 method. 

 For any point of the curve, then, the center in the normal, or in direction 

 of gravity is known. Per law of ultimfite ratio by this method an infiui- 

 tisimal arc of the ellipse is described with the same constant ra"dius as in 

 first instance, and with a fixed center. The same infinitesimal arc of the 

 ellipse, then, is described by either expedient. Therefore the cone in either 

 position cuts from the same layer the same volume or mass. 



A fluid mass takes on a spherical figure from the mutual attraction of 

 component particles. If such spherical body receives an initial impulse of 

 rotation sufficient to cause one complete rotation during infinity of time, 

 then the ultimate ratios initiated for change of figure are those, and those 

 only, that are due to change from a sphere to an oblate ellipsoid. This 

 should be taken as demonstrat on that the oblate ellipsoid is the figure of 

 equilibrium and alone that figure, unless there is real ground for positive 

 proof otherwise. 



