Thus e 1 e 2 = - e2ei, and if q, q are any two quaternions, qq is generally different from q'q.
The values of x and y are different, unless V (qq o) = o.
If Q and Q' are commutative, that is, if QQ' = Q'Q, then Q and Q' have the same centre and the same radius.
Draw Pp and Qq touching both catenaries, Pp and Qq will intersect at T, a point in the directrix; for since any catenary with its directrix is a similar figure to any other catenary with its directrix, if the directrix of the one coincides with that of the other the centre of similitude must lie on the common directrix.
Similarly Qq must pass through the centre of similitude.