[375
axial
right
some
A 2 ? +1 , (2q + 1) X 2 ; A 2 ?, qL 2 , qL' 2 ; A 2 ?, 2qL\
12. The meaning of the rotation symbol is as follows : viz. if in general we have
a rotation 6 about an axis inclined at the angles X, Y, Z to any three rectangular
axes, and if II be the rotation symbol,
n = cos 16 + sin 6 (i cos X +j cos Y + k cos Z),
then if x, y, z are the original coordinates of any point of the body, and x', y', z' the
coordinates of the same point after the rotation ; the values of x', y', z' are given in
terms of x, y, z by the formula
ix' +jy' + kz' = II (ix +jy + kz) II -1 .
This is in fact the form under which, in the paper “ On certain results relating to
Quaternions,” Phil. Mag., voi. xxvi. (1845), p. 141, [20], I exhibited the rotation formulas
of Euler and Rodrigues. See also my paper “ On the application of Quaternions to
the Theory of Rotation,” Phil. Mag., voi. xxxm. (1848), p. 196, [68].
We have, it is clear,
IP = cos s9 + sin s9 (i cos X +j cos Y+k cos Z)
which shows that II s is the symbol for the rotation II repeated s times : (more
generally performing on the body, first the rotation II and then the rotation P about
any axis, the same or different, the symbol of the resultant rotation is = PII). If II
2^
be the symbol for a rotation through the angle —, then the rotation which corre
sponds to the symbol II? is a rotation through 360°, that is the body returns to its
original position ; it might at first sight appear that we ought to have II? = 1, and
that the symbols 1, II, II 2 , ... II? -1 would form a group; this however is not so, for
we have not II? = 1, but II? = — 1 ; in fact, it is to be observed that to pass from
ix +jy + kz to ix +jy' + kz', we have to multiply by II ( ) II -1 , so that the symbol of
the rotation is indifferently ± II, and that the rotation symbol — 1 is thus equivalent
to the rotation symbol +1. But as regards the formation of the group, the only
difference is that it is not 1, II, II 2 , ... II? -1 which form a group of q symbols, but
+ 1, ± II, + II 2 , ... i n? _1 which form a group of 2q symbols. And so in the axial
system of any polyhedron, if II be the rotation symbol of any ^-axis, then taking for
each axis of the polyhedron the set of symbols ± II, + II 2 , ... + II?- 1 , and besides the
two symbols ± 1, the whole series of symbols form together a group.
14. Thus in the before-mentioned case B(q = 2) we have the eight symbols
± 1, ± % ±j, ± k