314 
Proceedings of Royal Society of Edinburgh. [sess. 
On the Use of Quaternions in the Theory of Screws, 
By Dr W. Peddle. 
(§§ 1-2 communicated on February 6, 1902 ; §§ 3-7 on July 21, 1902. Other 
illustrative sections communicated on the earlier date, and since found to 
have been given in substance by Joly, are omitted.) 
1 . A quaternion r = Sr + Y r denotes the sum of a scalar and a 
vector, the former being an essentially undirected quantity. In 
many cases, however, and specially in the theory of screws, we 
have to deal with two co-directed quantities. In the usual nota- 
tion, the components of a translation, A, which are parallel to,, 
and perpendicular to, a rotation, /x, are represented respectively 
by the first and second terms in the identity A = (SAyx _1 + YA/x _1 )/x.. 
The axis of the corresponding screw of pitch SA /x _1 has the direc- 
tion of /x, and YA/x _1 is the perpendicular upon it from the origin, 
A certain advantage in point of unity would arise from taking Sr 
and TYr to represent respectively the magnitudes of the transla- 
tion and the rotation in a screw whose axis passes through the 
origin, and has the direction of Y r. When such a use is made of 
a quaternion, it is necessary to attach a special symbol. Thus, Mr 
may be taken to denote the motor S?**UYr + TYr-UYr whose axis- 
passes through the origin and, whose pitch is Sr /TYr. 
2. To determine the form of the general expression for a motor,, 
not passing through the origin, we may consider three motors pass- 
ing through the origin 
Mr, = Sr,.UYr, + TWyUYr, , 
M> 2 = Sr 2 -UYr 2 + TV?yU Vr 2 , 
Mr 8 = Sr 3 -TJYr 3 + TYr 3 -UYr 3 ; 
and we may assume these to be rectangular. 
The sum of these motors is 
Mr, + Mr 2 + Mr 3 = (Sr,.UYr, + Sr 2 U Vr 2 + S»y UVr 3 ) 
+ (T Yr r UYr, + T Vr 2 -UYr 2 + TVr 3 .UVr 3 > 
= (Srj-UYr, + Sr 2 -UYr 2 + S%ITYr a ) + p 
