141 
RQ? . : 
represents the quaternion Q’ after its axis has been turned 
through the angle 26 round the axis of Q. 
~ In other words, 
EN ORES 
is the symbol of rotation round the axis of Q through an angle 
equal to the double of its amplitude. 
‘«¢ T am not aware that any symbol, expressed in the terms 
of our previously existing calculus, has been assigned to the 
same operation. It is not difficult to prove by geometrical 
considerations, that the symbol 
where 
eiga sp d_id Hae eS cme 
~=cosa ag ez +608 P| to — Zz Tee ee A 
_ denotes rotation through the angle @ round the axis which 
passes through the origin of rectangular co-ordinates, and 
makes with the axes the angles a, 3, y 
* So that if 
de (2. yy, ay = 0 
be the equation of a surface, 
. eY F (2, y, z) = 0, 
_ will be the equation of the same surface after it has undergone 
__ the rotation already described. 
v *< By means of this symbol, and purely in virtue of the 
q 4 laws of the received analysis, I have succeeded in demon- 
_ strating the known theorems concerning finite rotations and 
their composition. I freely admit, however, that my proofs 
are less simple than those which the calculus of quaternions 
i has furnished ; just as my fundamental symbol of rotation is 
less simple fan that which Sir William Hamilton has made 
known to us.’ 
