In our first attempt to derive algebraic expressions for F1, U, L_M,l_m in terms of readily measurable cotton fibre-characteristics we made use of data reported by Brown [7]. Brown and coworkers spun each of six cottons to three counts, NE 18, NE 36, and NE 72; and each count to a number of twist multipliers ranging from 2.75 to 5.75. The fibre test data on the six cottons are reproduced in Table II– A – 1 - i and the yarn CS in Table II – A – 1 - ii

For any cotton, at any one count, the plot of CS against M, the twist multiplier, yields the typical ‘inverted parabola’ -- the smooth curves in Figure II – a – 1 - i for example, with the curve of a finer count being below that of a coarser count. We have seen that the general equation for these curves is

We can rewrite this equation as

As M increases, U and (l_m + L_M tend to their respective asymptotic minimum values, namely, zero and l_m. Therefore, for any one count, as M increases,

For convenience we write

If, therefore, for yarn of any one count, we plot the quantity (CS x D) against M, we will get a curve that reaches this asymptotic maximum - - Table II – A – 1 - iii, and Figure II – A – 1 - ii. This is what would happen if there were no obliquity effect. With increasing twist, the fraction of fibres that slip, and the length of fibre-element that breaks when yarn fails in tenacity testing, will both reach their respective minimum; therefore the CS will reach its maximum, and with further increase in twist will continue to remain the same.

For any cotton, from plots of {(CS x D)} against M, Figure II – A – 1 - ii, we can read off the numerical value of X for each of the three counts spun from that cotton. We can do this for each of the six cottons. These values are also given in Table II – A – 1 - iii.

At any intermediate value of M,

Substituting for F2 and F3,

The important point to note is that, for any yarn, we can calculate the numerical value of the RHS of this equation from its CS and M. Therefore, we can prepare a table giving the experimental values of  for each of the 133 yarns in Brown’s spinnings - -  six cottons, each spun to three counts, and each count to a number of twists. These values are in Table II – A - 1 - iv. We use these reference data to formulate algebraic expressions for U,Lm,lm  We do this by trial and error until the numerical value of the LHS calculated from the algebraic expressions is as close to the corresponding experimental value in the RHS, for each of 133 yarns.

In this context, we recall the earlier enunciated premises regarding U,l_m,and L_M.  The fall of U with increasing M is governed by fibre-length, fibre-fineness and yarn count. Based on the observation that all fibres appear broken when the two ends of a broken high-twisted yarn is examined, we assume that the asymptotic minimum value of U=0  However, the asymptotic minimum value of the gauge-length of fibre-break in yarn-break as M tends to infinity, is not zero, but a finite value l_m. This value is dependent upon fibre-fineness. Furthermore, L_M, the increment over l_m at any other discrete value of M is dependent upon fibre-fineness and yarn-count, but not on fibre-length. Therefore, fibre-length does not influence the fall of (L_M+ l_m) to l_m with increasing twist.

The final set of equations that we constructed for U,l_m,and L_M is follows:-

Let us denote the fibre effective length in mm by L₀, the fibre fineness in millitex by H, the Stelometer zero-gauge tenacity in g/t by Z, and the gauge-length effect on fibre tenacity by G.

Figure II – A – 1 - iii is a plot of the observed value of  against M, for Akala cotton; over the points, is the smooth curve generated by the algebraic expression for estimating , F2 x f3 namely   . The smooth curve for each one of the three counts passes quite close to the points representing the observed values. A point of interest is that at twists of 2.75 and 3.25 the estimated value of F2 x f3 decreases from NE 18 to NE36, and again from Ne 36 to NE 72. This trend is discernible in the observed values when one scans the data over all the six cottons - - Table II A – 1 – iv. The estimated values of F2 x f3, namely, are in Table II – A – 1 – v. In each of the available 133 cases, the estimated value  is in good agreement with the observed value  namely, . The scatter diagram of estimated against observed, Figure II – A – 1 – iv, has a straight line, Y = 0.997 x X. Plotted serially, the observed and estimated values almost merge into each other. The functions

and the algebraic expressions for U,l_m,and L_M are, therefore, valid to quantify the increase of CS with increasing M. We already have,  to quantify the fall in CS with M as a result of fibre obliquity.

We have, thus, algebraic expressions for F2,F3 and F4 in terms of fibre-characteristics and yarn- count and twist to quantify the three-fold contribution of twist to yarn tenacity.

It now remains to derive the expression for F1, the irregularity fraction. For this we recall that

Therefore, for yarns of the three counts spun from any one cotton, a plot of  The numerical values of the intercept and slope of the line can be related to cotton characteristics, to give us F1. The actual plots did yield straight lines for all the six cottons. The algebraic expression for the irregularity fraction turned out to be 

a = -0.209, b = 0, and L is effective length in mm.

Therefore the final equation is

Also

Table II – A –1 - i  Brown’s Data :
Fibre-Characteristics of Cottons  Used In The Spinnings

Notes :-
 Source ; Tex. Res. J., April, 1957, Page 332
 E.L., comb sorter effective length, mm. = UHML, inches x 1.154 x 25.4
             Fineness, millitex = Suter-Webb micro-g/inch x 39.4 x 1.10
             0-gauge tenacity g/t, Stelo level = Pressley Index x 5.36 x /1.152
             2.5-mm tenacity g/t, Stelo level = Pressley 2.5-mm g/t / 1.26

TABLE –  II – A - 1 – ii Brown’s Data: Experimentally Observed CS

TABLE II – A - 1 – iii Observed Values of (CS x D)

TABLE II – A – 1 – iv Experimentally observed values of F2 x f3

Figure II – A – 1 - i Inverted parabola shape curves of CS – M plot

Figure II – A – 1 - ii Plot of (CS X D) against M

Figure II – A – 1 - iii Plot of F2 x f3 against M
with smooth curves from expression for F2 x f3 drawn over the points

Figure II – A – 1 – iv Showing good agreement between observed and estimated
values of F2 x f3