34 Proceedings of Royal Society of Edinburgh. [ sess . 
Hamilton and Cayley showed long ago, to a particular quaternion 
operator. Because of its peculiar form this operator, viz., 
q ( )y~\ involves the same four scalars which enter into the 
analytical expression for the quaternion'^. These four scalars have 
long been known to he remarkably simple functions of the half 
angle of rotation and of the position of the axis of rotation symbol- 
ised by the operator q ( ) q~ x . The modification introduced 
by Klein and Sommerfeld in their passage from the simple Drehung 
to the Drehstreckung is completely symbolised by the quaternion 
form q ( ) Kq , a form already used by Tait (. Proceedings , R.S.E., 
Yol. XIX., p. 196, 1892), while the equivalent form uq ( ) q -1 , 
where u is a scalar multiplier (in fact Klein and SommerfehTs 
tensor of the Drehstreckung), was used by Tait in his earlier paper 
on Orthogonal Isothermal Surfaces ( Transactions , R.S.E., 1873-4 ; 
Scientific Papers , Yol. I., p. 180). 
Thus, in their attempt to base the quaternion calculus on the 
conception of the Drehstreckung, the one novelty to be placed to 
Klein and Sommerfeld’s credit is the identification of a quaternion 
with a very special kind of quaternion operator. Given the 
Hamiltonian quaternion q , it is a comparatively simple matter to 
pass to the required rotational operator q ( ) q~ 1 . But to 
pass originally from the rotation to the quaternion with which it is 
noio known to he so intimately associated would almost certainly 
have proved a feat beyond the powers of any mathematical mind. 
For what is there in the simple conception of a rotation to suggest 
the presence of a quantity or operator and its reciprocal ? 
