30 
Proceedings of Royal Society of Edinburgh. [sess. 
For, writing g ( ) K^ as one Hamiltonian form and unam- 
biguous symbolic equivalent of Klein and Sommerfeld’s Dreh- 
streckung Q, (T q) 2 being equal to their T , we find, putting a and 
P instead of the vectors v and V, that the equation vQ=VT 2 
takes the form 
qaKq = P(Tqy. 
Also the symbol ( )QFQ _1 is, in quaternion symbolism, 
q-'fiqi ^qKpKq- 1 = q~ l qaK.qq{ ^qqKaKqKq-^Tq)' 8 
= a( )Ka(T q)~\ 
the required result. 
It is well to note here that, although v and a are the same 
vectors, a( )Ka and Klein and Sommerfeld’s Wendestreckung 
v are not quite the same operators. Their tensors differ, the 
Wendestreckung v being equivalent to (Tg)~ 2 a( )Ka. 
Immediately following the demonstration of the equation just 
discussed there is given on pp. 64-65 an analytical investigation 
essentially the same as that given long ago by Cayley, from which 
the direction cosines of the new positions of a set of rectangular 
axes with reference to the original positions are expressed in terms 
of the quantities A, B , C, D. This investigation is of course quite 
correct, because, for the moment , the authors use the quantities 
Q, v, V really in their true quaternion significations and not as 
Drehstreckungen. 
Thus, in the analytical part of their work, Klein and Sommer- 
feld simply reproduce long known results and follow accurately 
Hamilton and Tait. But they leave true quaternion lines when 
they regard Q ( = iA-\-jB + kC+ D) as a complete symbol for the 
operation which they call a Drehstreckung. In the symbolic 
equation 
( )vT- l = ( )QVQr 1 
Q, V, v are rotations or very particular types of strain. They are 
neither true quaternions, nor true vectors. Yet, for reasons which 
are plain to the quaternionist, these Drehstreckungen depend in a 
most intimate manner upon the quaternion (iA +jB + kC + D) and 
the vectors (ix +jy + Jcz) and ( iX +j Y + TcZ). In all this there is 
nothing new. Nevertheless the authors proceed to claim that 
their “ geometrical definition of the Drehstreckung leads to a 
