238 Wisconsin Academy of Sciences^ Arts and Letters. 



The general expression for the attraction of a homogenous oblate ellip- 

 soid on any outside particle, when^modified by the requisite substitutions, 

 proves that the decrease in attraction from the equator to the pole of the 

 heterogenous ellipsoid, caused by any of the interior component homoge- 

 neous ellipsoid, varies in accordance vs^ith the law for a homogeneous ob- 

 late ellipsoid, vrhen the attracted particle is at any point on the surface. 

 The decrease in attraction then, from the equator to the pole, true for the 

 second power of eccentricity, varies as the square of the sine of the ellip- 

 tic angle, or the angle of geocentric latitude. 



28. To find the decrease in attraction of an oblate ellipsoid on a particle 

 in the surface of any layer from the plane of the equator to a polar ra- 

 dius. 



Under the conditions of this discussion, when a heterogeneous ellipsoid 

 becomes homogeneous, then all the layers become similar and of equal el- 

 lipticity. The layers of the heterogeneous ellipsoid can be made similar 

 by taking away certain crescent pieces. To determine the attraction on 

 any particle in the surface of any interior layer, the attraction of certain 

 outside crescent pieces and the interior mass need only be considered, as 

 all the other exterior mass attracts the particle equally in opposite direc- 

 tions. 



In the last article an expression is developed that can be used to deter- 

 mine the attraction of the interior mass. A method to find the attraction 

 of the crescent pieces become manifest from diagram 4. The attraction on 

 particle P of that portion of the ellipsoid cut out by rotation of the lines 

 Pe and Pg depends upon the length of chord dg or ce, when angle g Pe is 

 small or infinitesimal. The point of tangency a bisects chord d g. 

 The length of chord d g varies as diameter B, B,. The diameter 

 B, B, becomes longer by increasing the elliptici'y. When point P is 

 moved to the interior of the ellipsoid, that portion of the chord d g, as 

 shown in diagram 2, by Pd, is equal to gg,, in case the ellipsoid is com- 

 posed of similar layers, or is of homogeneous density. If the ellipsoid is 

 composed of layers of decreasing ellipticities or with density increasing 

 from surf ace to center, then Pd is less thm gg,. The angle P ax of dia- 

 gram 4 is a right angle by construction, and PaC is less than a right angle. 

 The angle Cax, or angle a, or the angle of the vertical bacomes nothing 

 when the layer, of which a is the point of tangency becomes a circle. 

 When the ellipticity of the ellipse having a in point of tangency is less than 

 that of the ellipse BA BA, then the difiierence between gg, and Pd of dia- 

 gram 3, or the attraction for the external crescent pieces is at ained from 

 the exi.ression already obtained for the sine of angle a or the angle of 

 the vertical. 



sin a = E- sin 5 cos 5 +„ = 2 h sin 3 cos 5 -|-„. 



When point of tangency a, of diagram 4, is in the surface of the layer, 

 and point P is moved in semi axis BC, to the surface of the same layer, 



