An introduction to the mathematical theory of attraction ...

40 Resultant Force.
The attraction at the extremity of the longest axis can be found from those
at the extremities of the other two by means ofthe relation given in Ex. 1. The
construction in this example is due to Mac Cullagh.
14. Find the mutual attraction between the portions into which a homogeneous ellipsoid is divided by a central plane perpendicular to a principal
axis.
Let E1 and E2 be the semi-ellipsoids into which the ellipsoid E is divided
by the diametral plane. The attraction of E on E2 is compounded of the
attractions of E1 on E2, and of E2 on itself, but the latter is zero, being the
resultant of pairs of equal and opposite forces, and therefore the attraction R of
Ei on E2 is equal to that of the whole ellipsoid on E2.
If the diametral plane be perpendicular to the shortest axis of the ellipsoid,
then, for the corresponding attraction R3, we have -R3 = Cp fff z dx dy dz taken
through the volume of the semi-ellipsoid, C being given by equations (15).
If we change the variables by assuming
a k' b k' c k'
we obtain R3 = Cp k4 d d d
taken through the volume of the hemisphere whose radius is k; whence.R pabc2 3MJ
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where M denotes the mass of the ellipsoid, and C is given by (15) or (27).
In like manner, for diametral planes perpendicular to the other two axes of
the ellipsoid, we have
R1= Aa, R2= Bb.
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15. Show that the mutual attraction of the two semi-ellipsoids situated on
opposite sides of a principal plane is equal to the attraction of the entire ellipsoid
on a particle of half its own mass situated at the centre of inertia of one of the
semi-ellipsoids.
16. Show how to determine the attraction of a homogeneous solid of revolution at any point O on its axis.
If we take the point O for origin, the axis of revolution for axis of x, and a
perpendicular to it for that of y, and if X denote the required attraction, by (3),
Art. 14, we have
x = 2rp 1 - (~ —2 + —dx.
17. Determine the form of the homogeneous solid of revolution, of given
density and mass, whose attraction at a point O on the axis of revolution is the
greatest possible.