188 Proceedings of Royal Society of Edinburgh. [sess. 
the contraction due to the second twisting. As a result, an 
elongation is really observed as twisting proceeds. The point of 
maximum length is, however, soon reached when the rate of con- 
traction due to the second twisting is equal to the rate of 
elongation due to the opening out of the twist from the singles. 
After this point the rate of contraction increases. 
A is the curve of contraction for the singles in the first twisting. 
A' ,, ,, ,, second direct twisting. 
A" ,, ,, ,, ,, inverse twisting 
after all the single twist is opened out. 
B is the curve of contraction for the 2-ply if no fibre rotation (inverse). 
C „ „ „ „ (direct). 
D ,, ,, ,, free ,, (inverse). 
The results are represented graphically in fig. 5. The ordinates 
represent contraction, and the abscissae turns of twist. 
7T 2 d 2 
In the diagram is written as k. The ordinates of any 
point on curve D is obtained by measuring the vertical distance 
between curves A and B at the same abscissa, i.e. — + kx 2 . 
4 
The practical problem in connection with this subject does not 
involve the consideration of the length of the untwisted thread I, 
