260 Proceedings of the Royal Society of Edinburgh. [Sess. 
on the assumption that the strength of the thread is directly propor- 
tional to the twist. If their inference had been correct, it would have 
placed a premium on bad spinning. The careless spinner and unskilful 
blender might equally cease from troubling, and rest in the assurance 
that the twist would effectually cover up the flaws in their work. 
Now, every spinner is well aware of the fact that, the more uniformly he 
can distribute the fibres along the axis of the thread, the better value will 
he give to his employer from every pound of wool or other material 
committed to his care. The uniform thread, although containing exactly 
the same number of fibres as the uneven thread, can stand a higher stress, 
and fewer breakages will result in weaving. 
That the analytical problem of this paper is not the easy deduction 
performed by Professor Hartig will be readily admitted by anyone who 
has followed at all closely the various points discussed. The author has 
attempted a mathematical synthesis from the fundamental constants of the 
fibres constituting the thread and the degree of twist impressed in its 
formation, but he does not venture to present the results of the attempt 
with this communication. As each fibre is arranged in a spiral, the helices 
of which vary in diameter, increased torsion introduces both a bending 
moment and a twisting moment. The measurement of these involves the 
tension modulus and the torsional rigidity of the fibre. The strengthening 
influence of increased twist on the thread is due mainly to the resultant 
increase in the normal pressure of the fibres on each other, and therefore 
the resistance to slipping must be greater. Lesser causes are the increase 
of the surface of contact of the fibres and the interference of the twisted 
fibres with each other. It will not be a case of pure slipping, because the 
convolutions of a twisted fibre may prevent the motion of its neighbour by 
simply blocking the way. Here again, the extent to which the fibre can 
offer such resistance will depend upon its flexural and torsional rigidities. 
Taking these and other difficulties into account, it is probable that no 
satisfactory mathematical synthesis of the results obtained by experiment 
can be effected. 
While the aforesaid conclusion may be true, an investigation of the 
graphical pictures of the relations between strength and twist in the 
various threads experimented on leads to very interesting results. The 
graphs are evidently of the family of curves to which mathematicians 
apply the name of “witch.” 
The general equation of a “ witch ” curve, as shown in fig. 6, is 
7 (6 - af 
y = b ' „ , 
lex 2 + (b - a ) 2 
