484 Proceedings of Royal Society of Edinburgh. [sess. 
due to the memory of Mr Ashenhurst in connection with his 
pioneer labour in this field of research, to recognise that it was 
only a forward step in the evolution of a difficult subject in no 
way detracts from that honour. Textile students would do well 
to consider the foundation on which Ashenhurst’s assumptions 
rest, and to investigate the limitations to which they are subject, 
as set forth in his own words in the second section of his Textile 
Calculations ; so that by the aid of experiment and reasoning 
the next twenty years may be more fruitful in results than the 
same period which has just passed. 
As the author’s experiments on the absolute diameters of threads 
do not admit of generalisation at the present stage, we shall pass 
on to consider what is the main subject of this paper, viz., the 
diameter of a twisted thread compared with the diameter of its 
component singles. The subject is admittedly a difficult one 
both on the analytical and experimental sides, which may 
doubtless have deterred textile writers from discussing it. But 
it is, nevertheless, a logical consequence of Ashenhurst’s teaching. 
Single threads for purposes of calculation may be assumed to 
be flexible cylinders if not subjected to lateral stress, since to this 
form single threads approximate according as they approach per- 
fection in structure. Writers on textile calculation have always 
tacitly reckoned twisted threads to have the same form also, in 
order to avoid the mathematical difficulties which more complex 
forms must introduce. If the thread is twofold, i.e. consists of 
two threads twisted together, then its diameter is considered to 
be the same as the diameter of a single thread of twice the weight 
and volume per unit length, or twice the sectional area. A little 
consideration, however, will show that this is an erroneous idea, 
and sufficient in many cases to vitiate the results arrived at. It 
is very evident from fig. 1 that a twofold twist consists of two 
spirals interlocking each other, a form differing very markedly 
from that of the cylindrical single thread. 
The dimension of a thread which is of practical importance in 
the theory of cloth-setting is its horizontal projection, since in all 
ordinary cases cloth is constructed by the interlacing of two series 
of threads which cross each other at right angles. The series 
which is stretched lengthways in the loom is called the “warp,” 
