1899-1900.] Dr Knott on Klein's View of Quaternions. 29 
springs from the attempt to make a quaternion mean a rotation. 
A mathematical Janus has come into being. Had the authors 
realised or distinctly stated that their i, j, 7c are not always 
associative, so that Bi.i is not the same as B.ii , they might have 
saved their readers considerable confusion ; hut then their i, j, 1c 
would no longer have been the same as Hamilton’s, and they 
could not, with any show of propriety, have used the term 
Quaternion at all. 
Cayley showed in 1845 (see Phil. Mag.) that the four scalar 
quantities in the quaternion iA +jB + JcC+ D were the quantities 
symmetrically involved in Rodrigues’ expressions defining the 
rotation 
(iA +jB + JcG + D)( )(iA + jB + kG+D)~ \ 
and some further investigations are given in a later paper (Phil. 
Mag., 1848). The question is also treated in Tait’s paper “On 
the Rotation of a Rigid Body” (Trans. Roy. Soc. Eclin ., 1868; 
Scientific Papers , vol. i. p. 99). Klein and Sommerfeld’s innova- 
tion is to make iA +jB + JcO + D symbolise the rotation, or, more 
generally, the Drehstreckung. 
Passing on now to the analytical part of their discussion, we are 
introduced to the vectors 
v — ix +jy + hz and V = iX +j Y + TcZ 
which are such that the turning part of the Drehstreckung Q 
changes the direction of v into the direction of V, while its tensor 
part (T) changes the length of V into the length of v ; in symbols 
vQ= VT 2 , 
where for simplicity V is understood to have unit length. Here 
v and V are simply directed lines. 
The next step, however, is to consider v and V as WendestrecTc- 
ungen and to combine them in a particular way with the Dreh- 
streckung Q. The result of the investigation, which extends over 
nearly two pages, is the demonstration of the formula 
( )vT~ 1 = ( )QVQr\ 
where the empty bracket represents any system acted upon by the 
operators v and Q. 
But this equation is simply equivalent to a quaternion identity. 
