We will now discuss how we can formulate the algebraic expressions that go to make up the general equation taking into consideration these requirements.

The value of R can be shown to be 208.35. In the lea-strength test 160 threads together resist the imposed stress. The theoretically possible maximum contribution per thread can be the zero-gauge fibre bundle tenacity, Z g/tex. Therefore the theoretical maximum breaking stress that the lea can sustain is

Replacing tex by NE, and grammes by pounds, we find that the theoretical maximum load that a lea can sustain is

Even when the same cotton is spun on the same set of machinery, the irregularity of the yarn increases with increasing count. There is an important consequence of this. The English count-strength product, CS, which was originally proposed as a measure of yarn tenacity that is independent of the count of yarn, is in reality not so. The CS of yarns spun from the same cotton, and on the same set of machines, decreases with increasing English count. The expression that is in industrial use to quantify this fall in CS with increasing count is

where S₁ and S₂ are the breaking strengths of yarns of count respectively. The commonly used value of K is 18, irrespective of the cotton used to spin the yarn. This generalization yields estimates that are accurate enough for the purpose for which it is used, namely, for correcting the CS for small discrepancies of the actual count from the nominal count.

This equation accounts for the drop in CS with increasing yarn-count as a result of increasing irregularity. In practice, the increase in yarn irregularity with yarn count is less for the finer of two cottons. This implies that the expression for irregularity effect on CS should be written not in terms of C, but in terms of N, the average number of fibres in the cross-section of yarn, that is in terms of both yarn-count and fibre-fineness, and not in terms of only C, the count of the yarn. Furthermore, practical observation tells us that K is the smaller for the longer of two cottons (26). This is because the increase in irregularity with increasing count is less for the longer of two cottons. Examination of experimental data suggests that a plausible expression for F1 :-

Data offered by E. Lord (24) show that the fibre-bundle tenacity at gauge-length g in units of 1/32-inch is given by

where G is a cotton-specific parameter. If, therefore, the length of fibre-elements that break when yarn of twist multiplier M fails in tenacity testing is g in units of 1/32-inch,

In writing the algebra relating U and g to M we are guided by certain premises, that seem quite plausible on basic considerations. The premises are as follows.

With increasing twist, the number of fibres that slip, and do not contribute to yarn tenacity decreases. The longer the fibre, the steeper will be this fall, with twist, of the number of fibres that slip at breaking point with increasing twist. This fall could also be steeper for a yarn that has more fibres in the cross-section than for a yarn that has less fibres in the cross-section. The rate of fall of U with M is, therefore, determined by fibre-length, fibre-fineness and yarn count. Examination of the two broken ends of a high-twisted yarn shows that all the fibre-ends appear broken. We can therefore, take the asymptotic value of U to be equal to zero.

The gauge length of fibre-break in yarn-break is very likely to be determined by the intimacy of intermingling of fibre assemblies inside the yarn (8). The intermingling is quite likely governed by the number of fibres in the cross-section of the yarn. The fall of g with M is, therefore, not likely to be dependent upon fibre-length, but upon the number of fibres in the cross-section, being steeper for a yarn with more fibres in the cross-section than for a yarn with less fibres in the cross-section. The fall of g to its asymptotic minimum is thus dependent upon fibre-fineness and yarn-count. A corollary is that the minimum value of g may not necessarily be zero, but may itself be dependent upon fibre-fineness. We can therefore write,

where lm, the asymptotic minimum value as M tends to infinity, is dependent upon fibre-fineness, and the increment over lm at any finite twist, is dependent upon fibre-fineness, as well as upon yarn count and twist.

To construct the algebraic expressions for FL, U, lm, and LM, in terms of fibre parameters, we need data on fibre and yarn tests of a number of disparate cottons. Brown and co-workers (7) have provided such data. In our first attempt to formulate expressions for the general equation we made use of these data. We were able to construct a general equation that gives yarn CS estimates of acceptable accuracy over a range of yarn counts and twist. The procedure that we followed for arriving at these expressions is also explained in Part II.

We can assess the predictive accuracy of the equation by examining the proximity of the experimental points to the smooth curves that are generated by the equations. The results of this exercise are available in Figure I - 3 - i. In most of the resulting 18 cases, the experimental points are seen to be close to the smooth curve.

The error in the estimate of CSP in each of the 133 yarns is given in Table I – 3 – ii. Only in 14 cases out the total number of 133 does the error exceed 6%; out of these 14 cases, in only one does the error exceed 10%. With his equations, Neelakantan (6) found, “Only 17 out of 133 cases showed errors of prediction beyond 6% and only in two cases errors were beyond 10%.” Thus, in terms of the accuracy of estimates, The General Equation is not at all inferior to Neelakantan’s equations. At the same time The General Equation has been constructed with a logical framework that renders it capable of explaining the roles of fibre-length and fibre-fineness in governing the contributions of yarn count and twist to the realization of fibre-tenacity as yarn tenacity. Neelakantan’s equations do not help us glean any such insight: they contain nine terms, each a medley of fibre characteristics.

The interacting contributions of fibre length and fibre fineness on the one hand, and yarn count and twist on the other to the translation of fibre tenacity into yarn tenacity is highlighted in Table I – 3 - i.

A closer examination of the graphs in Figure I – 3 - i shows that there are feeble systematic errors, or bias, in some cases. Naturally the question arises as to which of the expressions needs be improved upon. Further scrutiny of the data leads to a surmise: the expression for the irregularity fraction introduces avoidable errors in the estimates. The inference is that the parameter of fibre-length in Brown’s data (6) namely, UHML, is by itself not adequate for quantifying the contribution of fibre-length to yarn irregularity. The algebraic expression for FL, the irregularity fraction, therefore appeared to deserve a re-look. We will do this in the sequel.

Table I – 3 - ii: Brown’s Data : Distribution of % Error In Estimates