240 Wisconsin Acadeinij of Sciences, Arts and Letters. 



An oblate ellipsoid having semi-minor axis B, and ellipticity H. and den- 

 sity increasing from surface to center can be made up of a homogeneous 

 oblate ellipsoid, having semi minor axis b and ellipticity h, with density 

 the same as layer having semi minor axis b and ellipticity h, and of other 

 homogeneous oblate ellipsoids having semi minor axis less than b and el- 

 lipticities less than h, and also an outside part, composed of layers having 

 semi minor axes varying fx'om B to b, with ellipticities varying from H 

 to h. 



Let 1 be the density of the layer having semi minor axis b. For the out- 

 side crescent masses there can be substituted one crescent mass of the same 

 volume as the sam of the crescent masses, with a density, so as to give an 

 equivolent attractive effect. Let this density be c times the surface den- 

 sity of tbe heterogeneous ellipsoid. Let M bs the mass of the whole he- 

 terogeneous ellipsoid, Ji" the mass of the homogenejus ellips )id, having 

 density 1, and m, m,, rug. ete., the masses of the homogeneous ellipsoids 

 having semi minor axis less than b. 



It is now evident frcm investigations already made, that for the surface 

 of any layer, the atti'action at the pole of the layer, less that at the equa- 

 tor, can be expressed as follows: 

 « 



Diff. of Att.= — fi h + ^^ (H— h)l-|-^(m + m, + m., + etc.) 

 M L^" 5 1 J M ' I - 



9 



— (m h + m, n" h, + m., n,'- h., + etc.) 



5M - - - 



The result from the general expression for the attraction of a homogen- 

 eous cblate ellipsoid on any outside particle, when the requisite substitu- 

 tions for the case under consideration are made, proves that the decrease in 

 attraction on the surface of any layer from its pole to its equator varies in 

 accordance Avith the law for the homogeneous oblate ellipsoid at its sur- 

 face. 



29. To find the figure of equilibrium of a fluid rotating mass, increas- 

 ing in density from surface to center. 



The attraction at the surface of any layer at its equator being less than 

 at its poles, the thickness of any layer at its equator^is greater than at its 

 poles. As the decrease in attraction on the surface of any layer varies per 

 law for the surface of a homogeneous o^ilate ellip>oid, the increase in the 

 ' thickness of the layer as caused by attraction, must follow the law for the 

 increase in the radii of an oblate ellipsoid. The effect of the centrifugal 

 force to give its addi clonal increase of ellipticity follows for the layer the 

 law for the surface of the homogeneous ellipsoid. The combined effects, 

 then, of attraction and centrifugal force, or gravity, cause each layer to be 

 an oblate ellipsoidal layer, or the figure of equilibrium of all layers to be, 

 also that of an ob'ate ellipsoid. 



30. To find the density of the earth from surface to center, the mean 

 density being |5 times greater than that of a surface layer 127 miles deep. 



