186 
Proceedings of Royal Society of Edinburgh. [sess. 
The hypothetical condition that there will not be any relative 
motion amongst the fibres of the single threads will never be 
realised in practice. 
Again, let us consider the twisting of two lengths of untwisted 
single threads, each = Z, and whose diameter = d. Suppose that 
they receive n turns of twist each, and that in forming them inta 
a 2-ply thread, they are subjected to a further torsion of x turns. 
Then D of the general case = d. 
The length of twisted single thread as before 
Therefore the length of a strand in one turn of 2-ply thread 
C 
Fig. 2. — Take-up in 2-ply Threads. 
7 7 r 2 eZ 2 o 
or l- n l approx. 
8 1 
— L of the preceding example. 
x 
.*. the pitch of twist in 2 -ply thread 
the length of single thread 
x ' 
l 
.*. the length of 2-ply thread 
approx. 
the total contraction 
! ) 
approx. 
