248 
Proceedings of the Royal Society of Edinburgh. [Sess. 
is 765 grams, the average load on each fibre is 8*5 grams. At this point 
the five weakest fibres break ; there are now only 85 fibres to carry the 
load of 765 grams, which gives an average of 9 grams per fibre. But 
this is the load at which the second weakest group of fibres break. 
Therefore these break immediately after the first group, before the total 
load has increased. Then 75 fibres are required to carry 765 grams, 
i.e. an average of 102 grams ; but this is a higher load than 40 
of these fibres can carry. The result is that the thread tears across 
almost simultaneously with the rupture of the weakest fibres. If the 
average strength of the fibres were deduced from this result, it would 
be set down as 8*5 grams, but we have seen that this is merely the 
strength of the weakest group of fibres. 
In general, if there are n fibres in a thread, the strength of the weakest 
fibre is a, and the strength of the second weakest is a + d. 
Then, the condition that the total load will increase between the times 
of rupture of the weakest and second weakest fibres is that 
7 t*/ 
or d> . 
n — 1 
(3) Another discrepancy is introduced, because the fibres are not all 
continuous throughout even 1 millimetre of thread under test. This 
would not, however, be an objection against the hypothetical testing length 
of infinitesimal magnitude. If we take a carded yarn made from a Botany 
clothing wool, 1 inch is a common length of fibre. With 100 fibres in 
a cross section, there are 100 fibre ends in 1 inch on the average, and 
therefore four ends on 1 millimetre; i.e. there are only 96 fibres con- 
tinuous throughout the length of 1 millimetre under test. 
Therefore the common practice of setting down the breaking strength 
of 1 millimetre, and sometimes even of 1 centimetre, of thread as the 
sum of the strengths of its constituent fibres is misleading. 
The further consideration of Professor Hartig’s formula raises several 
very interesting questions, but these do not come properly into the scope 
of this paper, and will be reserved for discussion in another communication 
which deals with the influence of the length of the test sample on the breaking 
strength observed. The weakest point in the formula is that /a varies 
between such wide limits. Although noting that the variation of /x is due 
to variation in twist, he apparently made no attempt to ascertain if a 
relation held between /x and the degree of twist. Twist is by far the most 
important factor in determining the strength of a thread of a given weight 
