32 Proceedings of Royal Society of Edinburgh. [sess. 
Drehstreckungen could ever have led the mind to the true con- 
ception of a quaternion, or to the powerful vector analysis which 
clusters round it. 
However this may he, the authors, either in ignoration or in 
ignorance of what Hamilton has done, seem to think it necessary 
to try to “attach a precise meaning to Hamilton’s definition,” 
and they proceed to consider what relation connects Q, v, and V, 
when the Wendestreckungen v and V have their axes perpendicu- 
lar to the axis of the Drehstreckung Q. They find 
( )vT=( )QQV, 
or symbolically, if Q' be written for Q 2 and v for v T, 
Q'=v'V-\ 
This, be it remembered, is a symbolic equation connecting 
operators , and not an equation connecting quantities. It is, of 
course, again an identity in quaternions. The assumed condition 
means that /3 and therefore a are perpendicular to VUg, and hence 
fiq = K q . /3, a q = . a, etc. 
Hence, multiplying K q into both sides of 
q a-Kq = P(Tqy, 
we get 
aKq = Kq.p(Tqy- = f3q(Tq)\ 
and multiplying into q we have finally 
°-=P <? =/¥, say. 
The rotational equation then becomes 
q'( )K q=q 2 { )K q* = P~ l a( )KaK /3" 1 * 
Regarding their form of this equation, Klein and Sommerfeld 
s a y: 
“ The quaternion Q! is represented as the quotient of two vectors 
v and V, whose directions are perpendicidar to the axis of Q! and 
make with one another an angle equal to half the rotation-angle of 
Q', and whose lengths are in the ratio of the tensor of Q! to unity. 
“ This definition of quaternions is obviously somewhat par- 
* In the absence of the tensor T the quaternion form is simpler than the 
Drehstreckung form. 
