116 Proceedings of the Royal Society of Edinburgh. [Sess. 
of the former cut the corresponding graphs of the latter at different points 
(Eh would cut I. outside the limits of the diagram). 
The condition represented by graphs Eh and B 2 is evidently extreme, 
but it must be borne in mind that the analytical results have all been 
deduced on the hypothesis that the fibres have perfect freedom of rotation. 
This is not true ; indeed, it is very far from the truth except in the initial 
stages of the second twisting. In the actual thread, the individual fibres 
are neither absolutely free to rotate in the singles nor absolutely con- 
strained to move as a whole. The fibres, practically free to rotate for the 
first one or two turns, gradually become more constrained in their relative 
movements, though never becoming absolutely so. On account of this 
constraint, it is therefore reasonable to expect that we should find, as we do, 
that the hypothetical curves shoot out above the experimental curves in the 
contraction parts of the diagram, and also that the former dip down 
beneath the latter in the elongation part of the diagram. The friction 
of the fibres rubbing on each other will oppose motion in either 
direction. Another fact which must be taken into account in explaining 
any divergence which occurs, is that the fibres projecting from the surface 
of the singles seriously interfere with the free motion of the latter as a 
whole about the centre of revolution in the second twisting. The ratio 
of the contractions would be very much greater but for this interference. 
We should naturally expect to find that the ratios for the woollen threads 
would be smaller than for the worsted threads, because the typical woollen 
thread possesses a large number of surface fibres projecting from the body 
of the thread. The aim in making a worsted thread, on the other hand, 
is to develop parallelism so far as possible amongst the individual fibres. 
That this inference is just, is borne out by a consideration of fig. 3 and 
fig. 4. The individual members in the pairs of graphs in fig. 4, representing 
the results of twisting woollen threads, are closer together than those in 
fig. 3, which represent the results of twisting worsted threads. This is 
especially noticeable on the positive sides of the diagrams, which represent 
the “ harder ” twist. 
En conclusion, the author has pleasure in acknowledging his indebtedness 
to the Carnegie Trust for the Universities of Scotland for the financial 
assistance which has enabled him to prosecute this research. 
{Issued separately May 30 , 1907 .) 
