1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 205 
Experimental: (1) y + a = c(x±b) m 
Analytical : 
when the fibres have perfect freedom of 
rotation about the axes of the singles 
during the second twisting. 
(3) y =kx*\ 
when the single threads revolve as elastic solids about 
the centre of the 2-ply thread without any relative 
motion amongst the fibres. 
The average value of m taken from Table III. is 2 - 02, and the 
extreme limits are 1*91 and 2T5. Therefore the analytical value 
deduced for m — i.e. 2 — may he considered as confirmed by experi- 
ment also. In the actual yarn, the fibres are neither absolutely 
free to rotate in the singles, nor absolutely constrained to move as 
a whole, so that some intermediate condition may be expected to 
hold good. On examining the open-band side of fig. 14 we find 
closely at first, but as torsion proceeds it moves away towards the 
curve y = kx 2 , which it never reaches. The fibres, perfectly free 
for the first one or two turns, gradually become more constrained 
in their relative movements, though never becoming absolutely 
so. It will be seen from fig. 14 that the curve of the equation 
ceeds, and never differs much from the latter at any point. 
Considering now the cross-band side of fig. 14, we find that the 
yam curve differs widely from either of the curves corresponding 
to the analytical relations deduced. Much of the discrepancy, 
however, is due to the fact that (a) the turning-point of the 
graph of y = kx 2 is at 0; (b) the turning-point of the graph of 
y = ^x( 5x+ 2 n) is at M ; (c) the turning-point of the graph of the 
yarn equation is at C. The differences will be much less if the 
origins of the hypothetical graphs be transferred to C. Curve A 
positive although measured to the left hand from 0. This curve 
practically coincides with the yarn curve until the twist becomes 
this is the case. The yarn curve follows 
approaches closer to the yam curve as torsion pro- 
2 
in fig. 14 is the graph of y = — kx(5x - 3n), where x is taken 
1 0 
excessive. 
