1906-7.] Normal Take-up in Twisted Threads — Part III. 
107 
XIV. — The Relation between Normal Take-up or Contraction and 
Degree of Twist in Twisted Threads when the Singles 
are of Unequal Sizes. By Thomas Oliver, B.Sc. (London 
and Edinburgh), Carnegie Research Fellow. Communicated by 
Dr C. G. Knott. 
(MS. received December 17, 1906. Kead same date.) 
In a former paper (vol. xxvi., pt. iii., p. 186) the author proved that 
the contraction due to forming a single thread, 
7 r 2 d 2 o 
= > 
and the contraction due to the two-ply twisting, 
where D = diameter of circle in which the axes of singles revolve. 
d = diameter of single. 
I = length of untwisted single. 
L = length of twisted single. 
n = turns of twist in single. 
x = turns of twist in 2-ply. 
Also, if the fibres have perfect freedom of rotation the total contraction 
will be equal to that due to n + x turns in the single state plus that due 
to x turns in the double, i.e. the total contraction from the single to the 
two-ply condition 
t r 2 d 2 f (n + x) 2 n 2 \ tt 2 D 2 „ 
=^ri 4 -j\ + n7- x 
L may be substituted for l, as they differ very little from each other ; 
2 
and writing k for , the total contraction 
= kd 2 x('2n + x) + 4 kT) 2 x 2 
Let the two unequal singles A and B have diameters and d v and degrees 
of twist n x and n 2 turns respectively. Let them receive a further torsion of 
x turns in forming into a two-ply thread. Then the contraction on A will be 
y l = kd^x(2n Y + x) + ikD 2 x 2 
