An introduction to the mathematical theory of attraction ...

Examples. 157
8. Find the components of the force produced at any point inside an ellipsoid
by a distribution of mass on its surface whose density ao at any point x, y, z is
given by the equation o = p (Lx + ily + Nz), where p is the central perpendicular on the tangent plane at the point, and L, M, N are constants.
Imagine a homogeneous solid ellipsoid E bounded by the given surface to
receive a small translational displacement whose components parallel to the axes
of its initial position are El, E2, E3. The perpendicular distance of the origin,
or initial position of the centre, from the tangent plane at the point whose initial
coordinates are x, y, z, becomes then
px py pz
+ E- + E2 -; + E3
where p is the value of this perpendicular before the displacement. Hence the
normal thickness Sp of the shell comprised between the two positions of the
surface of the ellipsoid is given by the equation
El E2 E3
p =o t x +-2 y + 2
The density o of a surface distribution equivalent to the shell is pb; and by
making p, the volume density of E, sufficiently great and assigning proper
values to ej, e2, s3, we can satisfy the equations
pEJ PE2 pE3
1= Z7, = Yd -, e =
a2 -L) b2 '- c2 N
The components X, Y, Z, of the force exercised at the point P by the surface distribution of attractive mass of density o are now seen to be the changes
in the components of the attraction of E at P, when P receives displacements
- el, - E2, and - e3. Hence X = A-e, Y= Be2, Z= Ce3, where A, B., C are
given by (15) Art. 21.
9. The density c of a distribution of attractive mass is given at any point
x, y, z of the surface of an ellipsoid by the equation o = pf (xyz), wheref denotes
a homogeneous quadratic function, and p the perpendicular from the centre on
the tangent plane at the point x, y, z: find the components of the attraction of
this mass at any point inside the ellipsoid.
Suppose the solid ellipsoid E bounded by the given surface to receive small
angular displacements 80, 8p, 8/, round its axes, and let E' be its new position.
If a, B, y denote the direction cosines of the perpendicular on the tangent
plane to E at the point x, y, z, we have p2 = a2a2 + b2B2 + c272. If p' denote
the perpendicular on the parallel tangent plane to E', the direction cosines of p'
referred to the axes of E' are a + aa, &c., where 5a = 8b3 - 7y8, &c. (' Dynamics' (7), Art. 255), thenp'= p + 8p, where
pAp = a2aSa -t b-l3a83 + c2'y7.
The thickness of the shell comprised between
E and E' is ap;
also, we have a = - &c., and therefore
2 '2
a2 _ b2 b2- eC - a2
a2b2 xya4 + -b2e2 YzO + oa2