96 
Proceedings of the Koyal Society of Edinburgh. [Sess. 
In Part I. (p. 205, vol. xxvi., pt. iii.) it was set forth that, owing to 
the friction amongst the individual fibres in the thread, the analytical 
expression obtained for the “ take-up ” did not give the same result as 
obtained by experiment. It was also shown that for the direct or open- 
band second twisting, 
y = \x(px + 2n) . . . (5) 
and for the inverse or crossband second twisting, 
y = ^kx{p x -Zn) . . . . ( 6 ) 
gave results which did not differ materially from those obtained by 
experiment. In the above equations k = 7r ^ when x and n are expressed 
as turns per inch. 
In equation (6), for convenience x is to be taken positive although 
measured in the negative direction. 
Table II. shows the “ take-up ” per cent, on the hypothesis that equations 
(5) and (6) are true. 
TABLE II. 
Percentage Contraction. 
Turns per 
“ Direct ” < 
or “openband” twist. 
“ Inverse 
” or “ crossband ” twist. 
inch. 
X 
2/12’s. 
2/16’s. 
2/24’s. 
2/12’s. 
2/16’s. 
2/24’s. 
1 
•33 
•29 
•23 
-T6 
-T6 
— T5 
2 
•88 
•73 
•56 
-•18 
-•22 
-•23 
3 
1-63 
1-32 
1-00 
-•06 
-T8 
-•24 
4 
2-59 
2-07 
1-53 
+ •19 
-•04 
-•19 
5 
3*77 
2-96 
2T7 
•59 
+ •20 
-•07 
6 
5T4 
3-96 
2-91 
1T3 
•54 
+ •12 
7 
6-72 
5T5 
3*75 
1*80 
•99 
•37 
8 
8*52 
6-48 
4*69 
2-62 
1-53 
•70 
9 
10-54 
7*97 
5-73 
357 
2*18 
108 
10 
4-66 
2-93 
1-54 
11 
5-89 
3-78 
2-06 
12 
753 
4-73 
2-65 
Table III. shows the “ take-up ” per cent, if 
y = x(5x + 2 n) + 4C for the direct second twisting ; . (7) 
and y — x(5x - 3n) + 6C for the inverse second twisting. . . (8) 
C = single yarn number, as before. 
