1906-7.] Normal Take-up in Twisted Threads — Part III. 
y Y = M 1 %(2w 1 + x) + ik(2z) 2 x 2 
= kf 2 d*x(2n 1 + x) + 4 : Jc( 
|/ 2 (2w 1 + x) + 4( 
109 
//+ 1 
= kd 2 2 x 
= &d 2 2 x 
/ 2 + l 
/+_! 
./ 2 + 
/6 + 2/4 + 5/2 + 8/+ 4 
2 ^ 2 2 ^ 2 
iM 
X + 
2/X | 
Similarly, 
(p+\f 
y 2 — kd^x(2n 2 + x) + 4&(<£ 1 + “ 2z) 2 a? 2 
= M 2 2 * | (2re 2 + x) + 4(/+ 1 -//y* j. 
= kd 2 h: | (2 n 2 + x) + 4 / IP - | 
(3) 
= kd^x 
4/ 6 + 8/ 5 + 5/ 4 + 2/ 2 + 1 <, 2 l 
. c Ff + / 
By substituting suitable numbers in the formulae, it will be found that 
y x is in general greater than y 2 taking the first assumption as correct, viz., 
that corresponding points in the axes revolve in the same circle. But if 
we assume that the corresponding axial points revolve about the centroid 
of section, then numerical substitution shows that y 2 is vastly greater 
than y v Obviously, between these limits there can be some centre of 
revolution about which the torsion will give equal values for the con- 
tractions in the singles. 
Let s = the distance of this point from the axis of the thick thread A ; 
then y 1 = kd l 2 x(’2n 1 + x) + kk(2s) 2 x 2 
y 2 = kd e) 2 x(2n 2 + x) + 4 k(d 1 + d 2 - 2s) 2 x 2 
Let y 1 = y 2 and /= d x jd 2 as before ; 
kd-^x{2n 1 + x) = kd 2 2 x(2n 2 + x) + 4 kx 2 {(d Y + d 2 ) 2 - 4 s(d 1 + d 2 )} 
/ 2 (2ra l + a;)=2»2 + « + 4« j (/+ l) 2 - 4(/+ 1)1 j. 
then 
or 
2 
S = 
3/ +5 n 2 -f 2 n , 
lT + 8(/+ij J: 
3 f +5 d + n 2-P n l d 
2 + 8(fTW 2 
From this result, it is evident that in general there is no single point 
which can serve as a centre of revolution throughout the second twisting 
so as to give equal contractions on both singles. Because s will be 
dependent on x except in the special case when n 2 =f 2 n v 
or 
2 — f-2 = ( ^iY 2 
! 1 W 
Considering threads as cylinders of uniform density, this is equivalent 
to the case : the degree of twist proportional to the “ yarn number ” or 
