The fixed frame and the body frame share the same origin. Hence, we can transform from one to the other by means of an appropriate rotation of our coordinate axes. In general, if we restrict ourselves to rotations about one of the Cartesian axes, three successive rotations are required to transform the fixed frame into the body frame. There are, in fact, many different ways to combined three successive rotations in order to achieve this goal. In the following, we shall describe the most widely used method, which is due to Euler.
We start in the fixed frame, which has coordinates , , , and unit vectors , , . Our first rotation is counterclockwise (if we look down the axis) through an angle about the axis. The new frame has coordinates , , , and unit vectors , , . According to Section A.6, the transformation of coordinates can be represented as follows:
The angular velocity vector associated with has the magnitude , and is directed along (i.e., along the axis of rotation). Hence, we can write Clearly, is the precession rate about the axis, as seen in the fixed frame.The second rotation is counterclockwise (if we look down the axis) through an angle about the axis. The new frame has coordinates , , , and unit vectors , , . By analogy with Equation (8.45), the transformation of coordinates can be represented as follows:
(8.47) 
The third rotation is counterclockwise (if we look down the axis) through an angle about the axis. The new frame is the body frame, which has coordinates , , , and unit vectors , , . The transformation of coordinates can be represented as follows:
(8.49) 
The full transformation between the fixed frame and the body frame is rather complicated. However, the following results can easily be verified:
It follows from Equation (8.51) that . In other words, is the angle of inclination between the  and axes. Finally, because the total angular velocity can be written Equations (8.46), (8.48), and (8.50)–(8.52) yieldThe angles , , and are termed Euler angles. Each has a clear physical interpretation: is the angle of precession about the axis in the fixed frame, is minus the angle of precession about the axis in the body frame, and is the angle of inclination between the  and  axes. Moreover, we can express the components of the angular velocity vector in the body frame entirely in terms of the Eulerian angles, and their time derivatives. [See Equations (8.54)–(8.56).]
Consider a freely rotating body that is rotationally symmetric about one axis (the axis). In the absence of an external torque, the angular momentum vector is a constant of the motion. [See Equation (8.3).] Let point along the axis. In the previous section, we saw that the angular momentum vector subtends a constant angle with the axis of symmetry; that is, with the axis. Hence, the time derivative of the Eulerian angle is zero. We also saw that the angular momentum vector, the axis of symmetry, and the angular velocity vector are coplanar. Consider an instant in time at which all of these vectors lie in the  plane. This implies that . According to the previous section, the angular velocity vector subtends a constant angle with the symmetry axis. It follows that and . Equation (8.54) gives . Hence, Equation (8.55) yields
This can be combined with Equation (8.44) to give Finally, Equation (8.56), together with Equations (8.44) and (8.57), yields(8.59) 
(8.60) 
