100 
Proceedings of the Royal Society of Edinburgh. [Sess. 
These are equations of the first degree in A, and therefore represent 
straight line graphs, which in the first instance cut the ^/-axis above the 
origin, and in the second instance cut the ^/-axis below the origin. The 
crossband line should cut the axis as much below the origin as the open- 
band line for the same value of x cuts the axis above the origin. Any 
divergence from this condition noticeable in fig. 1 is due to the fact that 
in the actual yams experimented with, the degrees of twist in the singles 
were not exactly linear functions of the yarn number. This fact may be 
seen from fig. 3, in which the ordinates represent degrees of twist in turns 
per inch, n, and the abscissae represent yarn numbers, C. The straight line 
does not pass exactly through the points. 
To test the theory further, the same yarns were taken, but the twists 
in the singles were altered to seven turns per inch in each case. 
Since the “ direct ” twist gives a two-ply thread, which is too hard for 
any useful purpose, and the results obtained are therefore of no practical 
value, the remainder of the investigation was confined to the “inverse” 
twist. 
Table V. shows the experimental results for the case when all the single 
threads had seven turns per inch each. 
Fig. 4 shows the results of Table V. As before, ordinates represent 
percentage contractions ; and abscissae, yarn number reciprocals. 
The point of convergence of the lines is practically the origin. This 
is quite in accordance with the theory. Since n is constant in all the 
threads, 
_ x(&x - 3 n) 
V 6C 
may be written 
1 
= JcA where A = — , the yarn number reciprocal, and & is a constant. 
