141 



QQ'Q 1 , 



represents the quaternion Q' after its axis has been turned 

 through the angle 20 round the axis of Q. 

 " In other words, 



Q ( ) Q 1 



is the symbol of rotation round the axis of Q through an angle 

 equal to the double of its amplitude. 



" I 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 



e , 

 where 



denotes rotation through the angle 9 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 



F (x, y,z) = 



be the equation of a surface, 



e e *F{x i y i z) = Q 3 



will be the equation of the same surface after it has undergone 

 the rotation already described. 



" By means of this symbol, and purely in virtue of the 

 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 

 has furnished; just as my fundamental symbol of rotation is 

 less simple than that which Sir William Hamilton has made 

 known to us." 



