1901-2.] Dr Peddie on Quaternions in Theory of Screws. 
319 
where (pf -p 2 y) = k(p 1 - p 2 ) . 
Taking p and zjk as current coordinates, we get a plane repre- 
sentation of the elliptic cylindroid by a circle. If we draw a line, 
parallel to the z-axis, in that plane at a distance (Pi+p^/Z from 
the centre, the distance of a point in the circle from that line is 
the pitch of the corresponding screw ; and k times the distance of 
the point from the ^-axis gives the value of z. 
Otherwise, we might take p and z as current coordinates and get 
the plane representation by an ellipse. 
The representation of other quantities, such as the angle between 
two screws, is not so simple as in Ball’s plane representation of the 
•ordinary cylindroid. 
The whole class of elliptic cylindroids, for which, with different 
values of £ and y, the condition p l £-p 2 y=Pi ~P% holds, have the 
same representative circle as the ordinary cylindroid has, so far as 
p and z are concerned. 
From the relations x' = £cos0, f = ysm6, where x, y are the 
coordinates of the extremity of TVr', we see that the locus of the 
extremity of TV/ is the ellipse 
This ellipse is therefore the amplitude conic for TV/, i.e. for the 
axes of the unconstrained screws, the name being more suitable in 
the present case since the radius is the rotation vector. 
The locus of the extremity of TVr is the sextic 
A simple geometrical construction for both loci is got by drawing 
through a fixed point a line of constant length which carries at 
its end a circle of constant size whose centre is in the prolonga- 
tion of the line. Through the junction a line, whose direction is 
fixed, is drawn, and the whole system revolves round the fixed 
point. The intersection of the circle by the line whose direction 
is fixed traces the free locus; and the foot of the perpendicular, 
