192 
Proceedings of Poyal Society of Edinburgh. [sess. 
each single thread revolves as a whole about the axis of the 2-ply 
thread without any relative motion amongst its fibres. 
For a twist composed of three or more singles, the analysis is on 
similar lines. 
When three singles of equal size are formed into a folded thread,, 
the diameter of the circle of axial revolution, D, would be 
= CD = BC sec 30 c 
2 
JSd 
= 1*055 d. 
The total contraction for folded threads in general has been shown 
to be 
rfdf 
8/ 
(n + x) 2 + 
r 2 D 2 
2L 5 
.*. the contraction in the formation of the 3-ply thread is (if l is 
taken = L) 
t r 2 d 2 / / 
y “-8L l ( 
n + x) 2 — h 
•} 
_ 7r 2 d 2 /’2nx + x 2 4x 2 
2L V 
TT 2 d 2 
"2ir a 
/1 9 , ?z\ 
\Ef + §)* 
+ 
2LVV3, 
d 2 x 2 
This expression is positive when x is positive, and when x is 
greater numerically than -~n in the negative direction. 
It is equal to zero when x = 0, and when x = - 
It is negative between the limits x = 0 and x = — 
