The Variation in Attradion Due to the Attracting Bodies. 239 



then the radius for the angle a for any direction of chord dg cutting the 

 layer, varies in It^ngth from zero to semi axis b. The general expression 

 for any of the radii is, b sin 5. 



Let E be the eccentricity and H the ellipticity of the outside layer. The 

 expression for the sine of the angle of the vertical for the ellipse having 

 semi minor axis b and eccentricity E is, sin a: = b E- sin- 3 cos 5. 



The expression for the sine of the corresponding angle of the vertical for 

 a layer having semi minor axis b and eccentricity e is, sin tr, = b e'' sin^' 

 5 cos 5, 



gg, — Pd = 2 sin a — 2 sin a.. 



The expression for any chord of a layer drawn from a pole of the layer 

 is, 2 b cos 5. 



^^' ~ ^^ = (W — e") sin'2 5 = 2 (H - h) sin= 5. 

 2 b cos 5 



The attraction of the mass cut from a homogeneous oblate ellipsoid hav- 

 ing semi minor axisb and eccentricity e, by any two chords drawn from 

 a pole, the one chord making an infinitesinal angle with the other is: 



Att.= -^ (i - # e- sin'^ 5 +„). 

 b- 



The attraction then of the mass cut from the whole ellipsoid having 

 semi minor axis B and eccentricity E by the two chords extended, on a 

 particle at the pol-e of the iaterior layer, having semi minor axis b and ec- 

 centricity e, providing the density of the whole mass is homogeneous, is 



Att.= -^ (1 - 4 e- sin- 3 + E" sin- 3 +„) 

 b^ 



To make summation for whole ellipsoid we must put, as hertofore: 



sin- 3 = f . 



sin'' S =-^ — , etc;, for higher powers. 



For whole ellipsoid as above described, then, on particle at pole of layer: 



Att.= -^ (1 - e-^ + i W +, ) =— (1—0 + 1 E-^ + J. 

 b- ■ a- 



Likewise obtained for particle in equator of same layer: 



Att.= .-™ (1 + A e-' — ! E-^ +J. 

 it- 

 Attraction at pole, then, less that of the equator, is: 



Dif. of Att. = — (0 E-' - A e'^ + ) = — [\ h + HH - h) + , 1. 



a- ' ' ■ a" L' -J 



