1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 187 
We have directly, since the total contraction must be equal to 
the sum of the contractions due to the consecutive twisting 
operations of forming the single and double threads, 
But as these values will not differ by more than 1 or 2 per 
cent, in all cases which would arise in practice, we may safely take 
the first expression. 
This analysis takes no account of relative motion of the fibres in 
each thread. As the second twisting proceeds, the degree of twist 
Fig. 3 — Two-ply Thread with Open-band Twist both in Singles and 
in Double. 
in the singles will also vary. The result will be differently affected 
according as the second twisting is in the same direction as that 
which the singles possess (shown in fig. 3), or is in the opposite 
direction (shown in fig. 4). 
The former is scarcely ever used in practice, because of the 
rapid increase in hardness of the thread as the twist increases. 
Further, the friction between the surfaces of the fibres is not 
sufficient to prevent the elasticity of the thread from asserting 
itself in opening out the twist. The excessive degree of twist 
caused by both operations being in the same direction produces a 
thread of little use for cloth manufacture. 
Fig. 4 — Two-ply Thread with Open-band Twist in the Singles and 
Cross-band Twist in the Double. 
If the fibres have perfect freedom of motion, the total con- 
traction will be equal to that due to n + x turns in the single state 
plus that due to x turns in the double, 
if x be reckoned positive when in the same direction as n, and 
negative when in the opposite direction. In the latter case the 
opening out of the twist from the singles at first completely masks 
^. 2^72 -- 2T)2 
Total contraction = n 2 + — — x 2 . 
8 1 2L 
i.e. the total contraction 
