An elementary treatment of the theory of spinning tops and gyroscopic motion, by Harold Crabtree.

ACCELERATIONS

113

far as rotation about G is concerned, G is equivalent to a fixed
point.
We will also adopt the following convention for the algebraic
signs of rotation about OX, OY, OZ:
Rotation about OX is positive in the direction Y to Z,,, OY, Z to X,
OZ,,,,,, Xto Y,
namely, the positive direction is that of a left-handed screw as
seen from 0 when looking along the axis; or, is determined by
taking the cyclic change of the letters X, Y, Z.
119. Linear velocities in three dimensions. If u, v, w are
the (total) component velocities in the directions of the axes,
we have (Fig. 67)
z= component velocity of P relative to IK+that of K
relative to iV+ that of IN relative to 0,
and...=x-Y6 + z,
and, similarly, by considering relative volocities,
V = - z1 + X03,
w -t = z -02 + y
It should be noticed that the dimensions of the expressions
are correct.
120. Accelerations. Again, if we take OR, RL, LP (Fig. 67)
to represent these velocities u, v, w, the rates of change of
u, v, w, i.e. the total accelerations, along OX, O Y, OZ are
given by -03 + 02
- tV01 +?03,
': - t02 + v01.
121. Angular velocities. Since angular velocity and angular
momentum are vector quantities, the foregoing results are also
applicable to them.
Let w%, W2, w3 be the (total) angular velocities of a rigid body
about the axes OX, 0 Y, OZ; that is, about the lines fixed in
space with which OX, O Y, OZ happen to be coinciding at the
instant under consideration: it follows that the rates of change
of the angular velocities about the moving axes are
1- W203 + w302,
w2 - w301+ 3103,
3- e102 + 201'
122. Angular momentum. If 0 is a fixed point in a body
referred to the three moving axes OX, 0Y, OZ, and h1, h2, h3
are at any instant the components of the angular momentum
H