20 
Proceedings of Royal Society of Edinburgh. [sess. 
4. 
In its initial conception the quaternion had no direct connection 
whatever with rotation. But, of course, as an organ of expression 
capable of dealing with all space-problems, it can be employed to 
describe the effect of rotation. 
Thus, if we are to represent the effect of turning a vector p 
(conically) round an axis e (a unit-vector) through an angle A, it 
is obvious that p must be resolved into components parallel and 
perpendicular to e. Of these the first is unaltered, the second is 
made to rotate round e through the angle A. Hence, if <f) be the 
operator (not, it is to be carefully observed, a multiplier) which 
produces the rotation, we have, since 
p — — eSep — eV ep , 
<£p = — eS ep - (cos A + e sin A)eV ep 
= p cos A - eSep(l - cos A) + Yep sin A . 
If we multiply this by e (the conjoined dilatation) the right hand 
side represents the effect of a Drehstreckung on any vector p. I 
say effect , because a Drehstreckung is not a space-reality like a 
quaternion, it requires a subject before it can obtain embodiment. 
Introducing, instead of A, a scalar w, such that 
• a 2m? . m? 2 - l , ■, A 
sin A = — — - , cos A = — — - : or w = cot 4 A : 
w l + 1 w* - 1-1 
and remembering that 
€ 2 = - 1 
in this case, we have 
<£p = — ( (m? 2 + e 2 )p — 2eS ep -1- 2 mY ep\ 
id 1 — e 2 \ / 
= s^?{ (ro+e)p(ro_€) } 
If we write r for the quaternion w + c, this becomes 
cf>p = rpr~ x 
a remarkably simple expression given by Hamilton ( Proc . R.I.A., 
Nov. 1844), and shortly afterwards by Cayley (Phil. Mag., Feb. 
1845). This shows that Gauss’s Drehstreckung , like everything 
else in space, can be represented by means of quaternions, but in its 
