106 Proceedings of the Royal Society of Edinburgh. [Sess. 
straight lines which nearly coincide with the curves between the limits 
A =*025 and A = '04 intersect at 0 2 . 
In summarising this subject, we find the following main conclusions 
have been arrived at : — 
(1) When the twist in the single thread is a constant throughout a 
range of crossbred worsted numbers, the contraction is simply proportional 
to the yarn number reciprocal for any fixed degree of two-ply twist. 
(2) When the twists in the single threads can be expressed as a linear 
function of the yarn number throughout a range of crossbred worsted 
numbers, the contraction is a linear function of the yarn number reciprocal 
for any fixed degree of two-ply twist throughout the same range. 
In conclusion, the author has pleasure in acknowledging his indebted- 
ness 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 .) 
