488 Proceedings of Royal Society of Edinburgh. [sess. 
The mean projection width = the mean value of the mid- 
ordinates of the nine strips into which the diagram is conveniently 
divided. 
= |(l-996 + 1-966 + 1-906 + 1-819 + 1-707 + 1-574 + 1-423 
+ 1-259 + 1-087) 
= 1-637(7. 
Now, if the twist had been taken as equivalent to a single thread 
of twice the sectional area of one of the component singles, the 
conclusion would have been arrived at that the projection width 
= jZd or T414 d. Thus an error of about 14 per cent, would 
have been made, following the usual assumption. 
Fig. 6. 
In practice, however, it will be found that the discrepancy is not 
so great as shown above, because, for the sake of simplicity in intro- 
ducing the subject, a hypothetical case has been considered which 
would never arise in practice, i.e. an unstretched thread. When 
yarn is formed into a warp it is necessary that it should be sub- 
jected to a relatively large longitudinal stress in order to secure 
uniformity in weaving. The result of this is that the spirals in 
the twist tend to become straight, and consequently each single 
thread exerts a transverse pressure on the other along the spiral 
line of contact : in practice, contact takes place along, not a line, 
but a surface, the extent of which depends upon the compressi- 
bility of the material of which the thread is composed. A thread 
also presents this deformation to a lesser degree, even when not 
subjected to longitudinal stress. Because, in the process of form- 
