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

CHAPTER VIII.

MOVING AXES.
111. We will now discuss more fully the motion of a body
referred to moving axes, that is, to a "frame of reference"
(Art. 81) which is moving in any given manner.
The motion of the body relative to the frame when combined
with the motion of the frame itself will entirely determine
the motion of the body.
112. We will first consider the motion of a point in a
plane referred to two moving axes.
If P be a point x, y referred to two fixed rectangular axes
OX, OY, then the entire motion of P is determined if we know
x, y, and consequently
dx dy and d2x d2y
dt' dd t andt2' dt2'
or in the fluxional notation,
x, y and, yj.
Now suppose that, 0 remaining fixed, the axes turn about 0
but still remain rectangular (Fig. 65). The motion of the point
P is now entirely determined when we know x, y (and therefore
a, y and a, y) relative to the moving frame of reference, or
moving axes, and the motion of the axes themselves. This
latter motion is of course entirely rotational since 0 remains
fixed; and although OX, OY are revolving it must be remembered that at any instant they coincide with two definite,
fixed, directions in space. We shall proceed to determine
expressions for the total velocities and accelerations of the
point P in these two definite directions fixed in space.
113. Linear velocities in two dimensions. Method I. Let
OX and OY represent the positions of the axes at the time t,
the instant in question, and let OX' 0 Y' represent their positions
at the time t + St.
Let x( = ON) and y( = PN) mark the position relative to OX,
OY of the moving point P at the time t, and x +dx(=ON'),