1901 - 2 .] Dr Peddie on Quaternions in Theory of Screws. 315 
where p is the resultant rotation. The first three terms may be 
written as 
S?’ 1 *UY/ , 1 + (1 + m)Sr 2 -UVr 2 + (1 + w)Sr 3 -TJYr 3 - raSr 2 -UVr 2 - wSr 3 *TJYr 3 
in which we may choose 
.. Sr, TYr 2 S?^ TV?' 3 
S n> LVrj Sr 3 1 V r^ 
If we put Sr l =p l TWr 1 , S? 2 = Vr 2 , Sr 3 =^ 3 TVr 3 these give 
7\ -Po Pi ~ V% 
m = , n = — — — : 
P 2 Pz 
and the vector 
S^U-Y^ + (1 + m)S?yUYr 2 + (l + w)Sr 3 .UVr 3 = <r 
is parallel to p. AYe thus get 
Mr x -l- Mr 2 + Mr 3 = (Tp + ??zSr 2 ■ UYp 2 Up + ?iSr 3 • UVr 3 lJp)IJp + <r 
= (1 + + faSr 3 ■ SUVr 2 TJp' 
+ nSr g ■ SUVr 3 Up)Up + cr 
, TYr.TVr,(p, -jJUW, +TVr a TVr 1 (p, -p s )UVr 2 + T\>/n>, Q., - p t ) UVrJ 
1 + T 2 p ' 
2? x T"V r A +p . 2 T 3 Yr 2 +^ 3 T°Y ? 3 
'Fp ' "" 
The last term represents the component of translation in the direc- 
tion of p. Hence the pitch, tt, of the resultant motor is 
7 r = p 1 COS 2 ^ + £> 2 COS 2 </> 2 + 2^COS 2 0 3 
where the cosines are the direction cosines of the axis with refer- 
ence to the axes of the three component rectangular screws. This- 
is a well-known relation. 
Since the product of the second term within the bracket into p- 
represents the component of translation perpendicular to p, that 
term must itself represent the vector perpendicular upon the re- 
sultant axis, 3 say. Thus we have 
/x = Mr x + Mr 2 + Mr s = (1 + 3 + tt)(V r x + Y r 2 + Vr 3 ) . 
