1899-1900.] Dr Knott on Klein's View of Quaternions. 25 
identical with the quantities x, y , z, w used by Tait in his expres- 
sion (Tait’s Quaternions , § 375) for the quaternion 
q=xi + yj + zk + w 
in terms of which the rotation is symbolised by Hamilton’s 
remarkable form 
<z( )r l 
Klein and Sommerfeld call the quantities A, B, C, D the 
Quaternionengrossen (p. 21), and speak of them as supplying the 
transition to Hamilton’s Theory of Quaternions. This seems to 
be, at first reading, correct enough; for undoubtedly the quantity 
Ai + Bj + Gk + D 
is a Hamiltonian Quaternion when i,j, k are used in the Hamil- 
tonian sense. 
But now let us pass to § 7, pp. 55-68, and consider carefully 
the authors’ Excurs iiber die Quaternionentheorie. 
In the first place the “ Drehstreckung ” is introduced, being 
defirfed as “an operation which is compounded of a rotation about 
the origin 0, and an isotropic expansion with reference to 0.” 
If the length of every line is changed in the ratio T : 1, then the 
Drehstreckung can he symbolised by the four magnitudes A, B, 
G, D , which, however, instead of having the sum of their squares 
equal to unity, satisfy the equation 
A 2 + B 2 +G 2 + D 2 = T 
Two Drehstreckungen acting in succession produce a resultant 
Drehstreckung, and the equations connecting the twelve quantities 
of the type A, B , G, D , are obviously the same as those that hold 
when the Drehstreckungen are simple rotations (T=l). These 
.are given, and then the authors say : “ The primitive ( ursjpriing - 
liche) definition of the word quaternion we base on our conception 
of the Drehstreckung : A quaternion signifies nothing else than the 
operation of the Drehstreckung. It is completely determined by 
the magnitude of the Streckung (T), by the axis of the rotation 
(a, b, c) and the magnitude of the half-angle of rotation 
The Drehstreckung Q, determined by the four magnitudes 
A, B, G, D , is then written in the form 
Q — iA +jB + kG + D 
