[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