24 
Proceedings of Royal Society of Edinburgh. [sess. 
Professor Klein’s View of Quaternions ; a Criticism. 
By Prof. C. G. Knott. 
(Read December 18, 1899.) 
In the first part of Klein and Sommerf eld’s treatise “Ueber 
die Theorie des Kreisels,” there is a section entitled, Excurs uber 
die Quaterionentheorie. In the preceding paper, Professor Tait 
has discussed the main conclusion contained in this digression; 
and I here propose to sketch the line of argument by which Klein 
and Sommerfeld have arrived at their curious mis-interpretation 
of Hamilton’s Quaternion. 
In Chapter I. ( Die Kinematik des Kreisels) the authors discuss 
the analytical representation of the rotations involved in the 
motions of a top of which one point is fixed. On page 21, they 
introduce four parameters A, B, U, D, satisfying the condition 
that the sum of their squares is unity. These are defined in terms 
of four other quantities, which have already been defined in terms 
of the well-known asymmetric representation by means of Euler’s 
angles 0, <£, i/'. In terms of these angles, A, B , C, D have con- 
sequently the values 
A • 0 c6 — ilr 
A = sin — cos - — m 
2 2 
. 6 • d) — \p 
B = sin — sin r — j 
2 2 
n 0 • cf> 4- if/ 
0 = cos — sin ^ - 
n 0 cf> + \b 
D = cos — cos 
They have also (p. 38) the values 
A = sm — cos a 
a at 
= sin — cos c 
B = sin 
cos b 
D 
where cos a, cos b , cos c, are the direction cosines of the axis of 
rotation, about which the single rotation through angle w is the 
rotation determined by the angles 6 , <p, if/. 
Hence the quantities A, B, C, D correspond to Cayley’s B , C , 
D , A in his Philosophical Magazine paper of 1845, and are 
