1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 191 
elongation which would result from opening out all the twist from 
one of the singles separately. 
(4) The 2-ply thread is longer than the singles until two- 
fifths of the number of turns in the singles are put on : when this 
number is exceeded the 2-ply thread is shorter than the singles. 
The latter is the important case in nearly all yarns used in cloth 
manufacture. 
(5) When the direction of twist is the same as in the singles, 
the contraction rapidly increases from the beginning. 
The assumption that is a constant is further from the truth 
at the point of maximum length than at any other point, because 
the diameter decreases throughout with increase of twist, while L 
is actually longer than at first. It will be useless, however, to 
attempt to find out by mathematical analysis how much this 
assumption will affect the results ; for there are other disturbing 
factors in operation of much greater importance. Prominent 
amongst these are — 
(1) When two threads are twisted they lose their cylindrical 
forms. 
As shown in my former paper on “ The Diameters of Twisted 
Threads,” communicated to this Society last year,* the circular 
eross sections are deformed into ellipses. The effect would make 
D, the diameter of the circle of axial revolution of the singles, 
= 2 b, where b is the semi-minor axis of the elliptical cross 
section. 
(2) In the above analysis the fibres which constitute the thread 
are considered as arranged parallel to the axis of the thread before 
twisting. This is only approximately true for worsted yarns, and 
very far from the truth for woollen yarns. This condition, though 
probably modifying the amount of the contraction, does not alter 
the general form of its expression. 
(3) In the initial stages of the second twisting, the fibres are 
free to rotate about the axes of the singles, and the experimental 
results are practically the same as those deduced from the above 
analysis. But the movements of the fibres become more restricted 
as the torsion proceeds, and long before the operation is completed 
* Proc. Roy. Soc. Edin ., 1905, vol. xxv., part vii. 
