194 
Proceedings of Royal Society of Edinburgh. [sess. 
Since normal folded threads with more strands than three are 
of little practical importance in cloth manufacture, it is unnecessary 
to discuss these in detail. 
If the origin of the graphs be shifted to the point of maximum 
elongation whose co-ordinates are 
n lc n 2 
4p 2 + 1 ’ 4 4p 2 -f 1 
, where h = 
2L 
If X , Y he the co-ordinates of any point on the graph measured 
from the new origin, 
v n 
v+i 
y = Y- 
4 if + 1 
Substituting in the expression for the take-up, 
+ j)*’} 
Y " l .TO ' M l( x -4^,) + (" = + i)( x -cti)’ ! 
Keducing Y = Jc(p 2 + i jX 2 . 
Substituting the values of p in particular cases gives 
for 2-ply threads, Y = - /cX 2 or 1 ’25/cX 2 
„ 3-ply 
„ 4-ply 
„ 5-ply 
Y =— *X 2 or 1-58M 2 
12 
Y = -kX 2 or 2" 25/cX 2 
4 
Y = 37 X~ v /5 / cX 2 or 3-14/tX 2 . 
20 - 4 J5 
Since working out this theory my attention has been drawn by 
Professor Barker of Bradford Technical College to a long article 
in the September 1903 issue of the L’Industrie Textile , Paris, 
entitled “ Baccourcissement d’un hi par l’effet de la torsion et 
consequences pratiques que Ton peut en deduire ” (The con- 
traction of the thread by the effect of twist, and practical results 
which we can deduce from it), written by Professor Bartolome 
Amat, Tarrasa, Spain. He elaborates a theory which is quite at 
