1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 185 
The length of twisted thread producible from length l of 
untwisted thread 
the contraction = 
= VJ 
-\A; 
ion =1- Z^/l - 
7 tH* 
4 
t rW 
4Z 2 
7T 2 d 2 ?l 2 
4Z 2 
Since 7 r 2 d 2 n 2 will, as a rule, not exceed of 4Z 2 , we may adopt 
the usual approximation (that J 1 - a = 1 - Ja, where a is a small 
quantity) without affecting the result for practical purposes. 
*. the contraction = l — l 
TT 2 d 2 
(>-t) 
8/ n ' 
With the same approximation, the length of twisted thread will be 
7 /, 7T 2 d 2 , 
Let us now consider the case of twisting two single threads 
together. If two lengths, each = L , receive x turns of twist, and 
each single be considered to revolve without any motion amongst 
its constituent fibres, then the pitch of the twist 
where D is the diameter of the circle in which the axes of the 
singles revolve. D = d , the diameter of one of the single threads, 
if they are of the same size. The length of 2-ply thread produced 
from lengths L of the singles 
vs 
^ - 7T 2 D 2 
V 1 -'™ 
= L 1 
7r 2 B 2 x‘ 
2L 2 
approximately. 
the contraction 
7T 2 D 2 X 2 
2L 
