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Because of the twofold ambiguity in the angle 0 mentioned above, there are 24 possible sequences of rotations, counting rotations through different angles as different rotations and ignoring rotations by multiples of 360 deg. The axis sequences divide naturally into two classes, depending on whether the third axis index is the same as or different from the first. Equation (12-20) is an example of the first class, and Eq. (12-22) is an example of the second. It is straightforward, using the techniques of this section, to write down the transformation equations for a given rotation sequence; these equations are collected in Appendix E. In the small-angle approximation, the 123,132,213,231,312, and 321 rotation sequences all have direction cosine matrices given by Eq. (12-24a) with the proviso that <f>, 0, and xp are the rotation angles about the 3, 1, 2 axes, respectively. Comparison with Eq. (12-13) shows that in the small-angle approximation, the Euler symmetric parameters are related to the Euler angles by

0 0