184 
Proceedings of Royal Society of Edinburgh. [sess. 
The length of fibre which forms one turn will be - . A fibre 
n 
on the surface of the thread will trace out a helix whose 
diameter will be d , while a fibre in the axis of the thread 
would remain straight. It would be clearly possible to find 
fibres tracing out all sizes of helix between these limits. 
It must not be inferred from these statements, however, 
that some fibres which are central will gain ground on others 
which are peripheral in their position. The position of a fibre 
is not fixed in any size of helix throughout its length ; it 
may be gradually changing from axis to circumference, and 
vice versa. 
Thus the average diameter of helix can be taken as-^-. 
Let ADB represent one turn of the average spiral or helix in 
the completed thread, 
Its length = — , 
n 
AB = the pitch of the twist, or the distance between two 
consecutive turns, 
Since the fibre ADB when uncoiled would form the hypothenuse 
AC of a right-angled triangle ACB, whose base is the circum- 
ference of the helix, i.e. — , 
5 2 
c 
Fig. 1 . — Take-up in Single Threads. 
