1899 - 1900 .] Prof. Tait on a Claim made for Gauss. 
21 
case as a quaternion operator, not as a quaternion. And it is 
specially to be noted that the angle of the quaternion r is only 
the half of that of the Drehstreckung . 
5. 
The utter difference in kind between the two concepts conies 
out even more clearly when we consider the vector data necessary 
to specify them respectively. 
To determine, fully, a Quaternion, requires but two vectors. This 
would ordinarily involve six scalar conditions ; but two of these are 
not required, because the aspect and angle and the ratio of the 
legs of the biradial are the sole essentials : — the orientation of the 
biradial in its own plane, and its scale of size, being immaterial. 
To determine a Rotation we must have two pairs of vectors, 
but there are other specifications, or necessary limitations, as to 
their lengths, etc., which reduce the number of really necessary and 
independent scalar data to three. These will be obvious from the 
results of the subjoined analysis. [What is essentially requisite 
amounts to two pairs of points on the unit sphere, those of each pair 
having the same arcual distance. This is at once apparent when 
we consider the nature of the possible displacements of a cap which 
fits a sphere, and which has, therefore, three degrees of freedom only. 
Of course the factor for Dilatation makes up the Tetrad required 
for the Drehstreckung .] 
Let cfia = /3, cji a 1 = /3 1 , or as above 
ra = /3r, rcq = p x r , 
so that we must have Ta = T/3 , Tcq = T/3 1 . 
[Hence, by the way, ra a 1 = fir . cq = (3 . ra Y = P/3^ ; which shows 
that the data are at least sufficient ; and that JSacq = s/%.] 
We have S(/3 — a)r = 0 , S(j3 1 — a 1 )r = 0 , so that 
Vr = xV(P- aXfr-a,). 
But /3(Sr + Vr) = (Sr + Vr)a. 
Substitute the above value of Yr, and we have 
(0 - a)Sr = X(V 08 - a)(& - a,) . a - £V(j8 - a)(ft - a,)) 
= - a)(S(ft - a x )a + S/3^ - cq)) 
= »(/3-a)S(a + 0)(0 1 -a 1 ) 
