190 
Proceedings of Royal Society of Edinburgh. [sess. 
Substituting this value of x in the expression for the contraction, 
the minimum value of y may be found, 
7t 2 c£ 2 /5w 2 
^ min ' = ~8L \25 
7 rW 2 rc 2 7 rV 2 n 2 
8L * 5 ° r 2L * 20 
Summary of Results arising from the Theory of 
Free Fibre Rotation. 
(1) The 2-ply thread has the same length as the original 
length of each of its constituent singles when the turns of twist in 
Fig. 6. — Curves of Contraction during Second Twisting. 
A is the curve for the single twist. 
B ,, ,, 2-ply (inverse) if no fibre rotation. 
C „ „ „ (direct) „ „ 
D ,, ,, ,, (inverse) if free fibre rotation. 
E „ ,, ,, (direct) ,, 
the second twisting amount to two-fifths of the number which the 
singles possess, but in the inverse direction. 
(2) The 2-ply thread has its maximum length when the turns 
of twist amount to one-fifth the number of turns in the singles in 
the inverse direction. 
(3) The maximum elongation is equal to one-fifth part of the 
