1905-6.] Mr T. Oliver on Take-up in Twisted Threads. 189 
as the singles are always measured in the twisted condition. The 
“sliver,” as the untwisted woollen thread is technically called 
after it leaves the condenser or last machine in the carding process, 
is always drawn out about 50 per cent, in length on the mule 
before any twist is put in. But, in order to understand what is 
going on in the second twisting, it is necessary to introduce the 
hypothetical length l in the discussion. To compare the theory 
with the experimental evidence, the contraction should be 
referred to the length of the twisted single thread, and only 
that part of the contraction due to the second twisting need be 
considered. 
The contraction due to the twisting of the singles 
t r 2 d 2 
21 
4 
7 r 2 d 2 n 2 „ 
2L ‘ 4 p ractlcall y- 
\ the contraction due to the second twisting; 
t r 2 d 2 J (n + x) 2 o n 2 \ 
~ZL \ “X" 4 j 
8L 
x(hx + 2 n ) . 
. •. the contraction = 0 when x = 0 or when x = — ■§ n , 
,, is negative ,, x lies between the limits 0 and - 1 n , 
,, is positive „ x lies beyond these limits. 
By differentiating the above expression, we get the minimum 
value for the contraction or the maximum value for the length of 
the 2-ply thread, 
V = 
dy 
dx 
——x(5x + 2 n) 
8L 
^{lOx + 2 n) 
8L 
d 2 
if we take — as a constant, which is not far wrong. 
JLj 
~ will be zero, and y will have a minimum value when 
(XtC 
\0x + 2n = 0, i.e. when x = — 
n 
5 
