1901-2.] Dr Peddie on Quaternions in Theory of Screws. 317 
M?q = - (1 + p^VrfJVr-VVr - ( 1 + pJVYrfJVr- UVr , 
M > 2 = - (1 + _p 2 ) S Vr 2 UYr •UYr - (1 + j> 2 )VVr s UVr. UVr . 
When the condition VV^UVr-f- VVr 2 UVr=0 holds, the motion 
is free. When the condition does not hold naturally, we may 
suppose that it holds in consequence of constraint which prevents 
rotation except around an axis parallel to Y r. Let 0 be the angle 
between Y r and Yr lt and put TY?q = ^cos^, TVr 2 = ? 7 sin 0 . The 
condition for complete freedom gives £= 77 . When constraint has 
to be applied to make V?’ the resultant axis, the coordinate motors 
are M'^ = Mr x + VV^UVr-UVr and MV 2 = Mr 2 + VVr 2 UVr-UVr „ 
and the pitch of the resultant is 
„_ &i cos2 fl + W in 20 
** $cos 2 0 + rjsm 2 0 
If we write y = xtsm0 , and 2 = TS, where 
^ ( p ^ -p 2 y)cos6smd 
fcos 2 0 + yjsin 2 0 
we find that the resultant screws for different values of 0 are the 
generators of the ruled surface 
+ yy 2 ) = {jpf -p 2 y)xy . 
When ^-y, this becomes the well-known equation of the 
cylindroid. 
To find the nature of the more general surface, consider an 
ellipse whose semi-axes are 1/^/1 and l/Jy . Take the former of 
these as the a>axis, and take the origin at one extremity of the 
diameter. Intersect the elliptic cylinder whose normal section is 
the above ellipse by a plane which passes through the se-axis and 
makes an angle <f> with the plane of the ellipse. Take as the 2 -axis 
the generator of the cylinder which passes through the origin, and 
draw perpendiculars to it through the points of intersection of the 
cylinder by the inclined plane. Consider further a circular cylin- 
der of radius 1 /J£ surrounding the elliptic cylinder. We have 
2 = yt&ncf) = \/ — y' tariff) 
1 
