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Let us take two symmetry operations: the first one corresponding to a rotation
of an angle around one axis, and the second one to a rotation of an
angle around another axis. Let us call the angle between the
two axes. Then, the product of the two rotation matrices is also a rotation
matrix. The rotation axis of the product matrix is, in general, oriented in a
different way than the other two. We can obtain the matrices corresponding to
symmetry operations in the following manner: for a given orthonormal vector
basis *A _{1}A_{2}A_{3}* (Fig. 1), the symmetry operation corresponding to a
counter-clockwise rotation of an angle around the

If, on the other hand, the rotation takes place around the axis,
which lies on the plane determined by *A _{1}* and

where:

represents a counter-clockwise rotation of an angle around

In explicit form we have:

The counter-clockwise rotation of an angle around the *A _{3}* axis is
given by the matrix:

The combination of two rotations (one of an angle around the *A _{3}* axis
and the other one of an angle around the axis which
forms an angle with

The trace of the *R* matrix given by the sum of the elements of the principal
diagonal, is:

i.e.

(12) |

This rotation *R* must be compatible with the lattice as well. Therefore, the
value of the trace, invariant with respect to a base system transformation, must
be an integer. The possible values of the trace are: +3, +2, +1, 0, -1.
These numbers give the order of the resulting rotation axis.

When we assign to and in the expression (12) all the possible
values, depending upon the order of the rotation axis, we obtain the second
degree equations in listed in Table 1, where *m* is an integer
representing the trace of the *R* matrix.

In Table 1 those solutions for which is greater than 1 are obviously not shown, as well as those that do not give as a result both and 180 - . This last condition is evidently necessary if two axes intersect.

On the basis of the results listed in the table, we can obtain the axis combinations shown in Fig. 2, i.e. the point groups 222, 32, 422, 622, 23, 432.

**Copyright © 1980, 1998 International Union of
Crystallography**