22 
Proceedings of Royal Society of Edinburgh. [sess. 
Thus, finally, 
r - *(s(« + /3)(ft - a,) + V(/S - a)(ft - a,)) 
where a is, of course, indeterminate. This value may be put in a 
great variety of other forms, in consequence of the necessary 
relations amongst a, /3, cq and P x ; all of which may obviously be 
regarded as unit-vectors. Perhaps the simplest of these is 
r — x(P(p 1 — a 1 ) + (/3j — a 1 )a). 
6 . 
Thus, generally, the expression for a Drehstreckung in terms of 
the necessary data is 
a i) P (Pi a i) a )( 
) 
1 
— aj) + (/?]_ — aj)a 
This is in all respects in marked contrast to the extremely 
simple expression for a Quaternion in terms of its necessary 
data, viz., as above, 
/3/a. 
Treating for a moment /3 and a as unit vectors (for we may at 
once do so by neglecting the tensors, which are mere numbers, 
commutative with everything), a unit Quaternion presents itself as 
P/ a or - /3a , 
and a Rotation as 
+ /3a( )a/3 . 
Their respective effects are : — 
on a, /3, and - /3a/3 = - a - 2/3Sa/3 ; 
on (3, - Pap , and /3a/3a/3 = + /3(4S 2 a/3 - 1) + 2aSa/3 ; 
and on Va p = J(a/3 - /3a), 
they are /3aSa/3 - 1, and -l-Va/3. 
In the case of the rotation the results are, of course, all vectors ; 
but the quaternion necessarily changes Vap into a quaternion, 
because that vector is perpendicular to its plane. 
7. 
With regard to Prof. Klein’s statement that Gauss had explicitly 
given the formula for the multiplication of two quaternions, it is 
