94 
Proceedings of the Royal Society of Edinburgh. [Sess. 
Let 
w = weight of unit length of thread. 
c= “ inverse ” yarn number. 
k = weight of unit length of No. l’s, a constant. 
A = average area of cross-section of thread. 
d = average diameter of thread. 
p = volume density of a thread in its commercial state ; 
then C = — but A = - d 2 
w 4 
and w = Ap 
d 2 = 
4 k 1 
7T p C 
Substituting in the expression for the take-up, y : 
y=^x(5x+2n)± . . . .'(2) 
n and C are the only variables on the right-hand side of the equation, if 
we confine the comparison to threads of similar nature in which p may be 
assumed constant. 
If n is a constant throughout the range of threads of different sizes in 
the same quality, then equation (2) may be written 
y = cA. . . . . . . (3) 
where y = take-up, 
c = + 2 n), a constant. 
2p 
A = 1/C, the yarn number reciprocal. 
For experimental purposes, n, the twist in the singles may be kept 
constant or varied at pleasure. In commercial yarns, n is never constant 
throughout a range of different yarn numbers. The degree of twist must 
increase as the thread becomes smaller. 
The precise relation which should hold between the degree of twist 
and size of thread has long been a debatable question. The late 
Mr T. R. Ashenhurst of Bradford Technical College, founding his 
argument on geometrical considerations, enunciated the theory that the 
twist in yarn should be directly proportional to the square root of the 
yarn number, or inversely proportional to the diameter of the thread. 
This theory has been styled by later writers, “ the theory of relative twist.” 
The late Mr M. M. Buckley of Halifax, from the results of experiments, 
contended that twist in yarn should be simply proportional to the yarn 
number. 
