An introduction to the mathematical theory of attraction ...

Examples. 147
Let the direction angles of the initial axis of rotation be a,, y; then
W2 W3 = W2 cos 3 cos 7, so that if
3m y z x dlw dw2 doW
2 -, cos 3-,cosy=, cosa=-, wehave - —, -
Cc) r3' r 3^ r dt dt dt
initially zero. Also x, y, and z are initially zero. Hence, the successive differential coefficients of cl, o2, w3, with respect to the time, are all zero initially, and
therefore wi, w2, w3 are constant.
If If be the moment of the impulsive couple required, its components round
the axes are, therefore,
3în. /3m y /3
r5.. 5. CZ'
and the invariable axis of rotation is the line joining the centre of inertia of K
to the centre of the sphere.
If K be free, let G denote its centre of inertia, and O the centre of the sphere;
then, if GO be made to coincide with a principal axis of K, and if K be projected with a velocity - along another principal axis, and be given an angular
velocity.- round the third, one principal axis of K will always be directed
towards 0, and K will continue to rotate uniformly round an axis perpendicular
to its plane of motion, whilst G describes in this plane a circle round O as centre
with a constant velocity.
6. Find the potential of the mass distributed over a plane area at a distant
point Pin its plane, the force due to an element of mass varying inversely as
the distance.
Let a be the moment of inertia of 1 round the principal axis perpendicular
to its plane at G its centre of inertia, r the distance of P from G, and Ithe
moment of inertia of 1M round GP; then V, the potential at P, is given by the
equation
1 ('- 21
V= Mlog + C -
-M - 2r2
7. If the potential energy due to the mutual action of two invariable mass
systems, A and B, be zero for all positions of B outside A, and if the total mass
of B be not zero, show that at all points outside the system A its potential is
zero.
If V denote the potential of A at a point Pin B w-hose coordinates are x, y, zy
at any other point Q, whose coordinates relative to P are $, 7, (, the potential
of A is
dV dV dV
V+ + -~Ç- + C + &c.,
dx dy dz
and if p denote the density of B at Q, and W the energy due to the mutual
action of the systems A and B, we have
VL2 d d pd/d
L2