The theory of the fibrogram has been derived based on the probability of randomly catching and holding a fiber. The amount axis has been shown to be proportional to the relative mass when related to the original specimen. Tangential equations are used to show that theoretically, for a length l or longer, the proportional mass, the mean length by number, the relative number, and the mean length by mass of fibers in the sample can be obtained from the fibrogram, as well as the number and mass arrays. Finally, the theoretical percentage of fiber by number or by mass extending a length l or longer from a clamped random sliver of the sample is obtainable from the fibrogram.The theory of the fibrogram was first published by . Hertel [2]. He considered the abscissa as the length axis and stated that the ordinate for I = 0 is NL where N is the total number of fibers in the sample and L is their mean length by number. This multiple is equivalent to the total length of all the fibers. Since Hertel presented this theory, this axis has been considered as the weight axis [6,7], the number or relative number axis [1, 2, 3, 4], or the probability axis [6,8]. In this work, we show that the ordinate axis can be considered as a proportional mass or mass probability axis, and we examine the fibrogram for additional information pertaining to the length of the sample. ' . Theory . Statistical texts generally write P[x) for the probability that the stochastic variable is less than or equal to the real number x. P[x] is called the cumulative distribution function of the variable. The derivative is called the distribution function of x.The general function that leads to the fibrogram must be defined differently. Consider the probability that the stochastic variable is greater than or equal to the real number l; q[l is then the complement of the cumulative distribution function. Since the fibrogram relates to fiber lengths, constraints must be applied to q[/]. Fibers shorter than 0 and longer than the longest fiber within the sample do not exist. The constraints applied to the probability q[I are that where 1m is the length of the longest fiber(s) in the sample. q[l is then the probability of the existence of a fiber in the sample having a length I or longer. The derivative is the distribution function of l.Since the total number of fibers in the sample is N, where N(l ) is the number of fibers in the sample having a length equal to I or longer. This gives Differentiating Q(1) gives the frequency distribution or

Determining cotton fiber tenacity with high volume instrumentation (HVI) requires an accurate measure of specimen linear density. Indirect measurements of linear density such as light attenuation, by which the Zellweger Uster Spinlab HVI system accom plishes this, require additional knowledge about fiber numbers or fiber fineness, usually by introducing a Micronaire correction factor. The effectiveness of this correction is less applicable than fiber fineness determinations obtained by direct weighing methods. Furthermore, fiber fineness distribution is not accounted for, and this factor may be important to enhanced HVI technology. For purposes of discussion, length-fineness distributions by number count and direct weighing methods have been determined for twenty cotton samples using Suter-Webb sorting techniques. Ramifications of fine ness distributions across length groups on cumulative fiber fineness distribution within a randomly prepared test specimen are presented in the context of breaking load normalization by a fiber fineness indicator.

Five cotton samples were tested for 3.2 mm-gauge tenacity and elongation five times daily as relative humidities (RH) were cycled within the limits of 70 and 48%. Tenacity and elongation responded almost instantaneously to changes in RH. In a controlled laboratory maintained at a standard atmosphere of 70 ± 2°F and 65 ± 2% RH, up to 4% range in tenacity and in elongation may be expected due to the oscillating RH alone.

Theoretical fibrograms were generated from fiber length distributions, and new fiber length distributions were then generated from these theoretical fibrograms for two, three, four, five, and six significant digits using differential and algorithm methods. A minimum of four significant digits of fibrogram data appears to be generally required to generate fiber length distributions for short, medium, and long varieties of measured fibers to closely simulate actual fiber length distributions using either the differential or algorithm method. The algorithm method takes fewer steps to generate a fiber length distribution than the differential method.

The Digital Fibrograph length uniformity index is usually misinterpreted. The theoretical maximum for this index is 51.28. The theoretical maximum index requires that: 1) all cotton (Gossypium sp.) fibers within the beard being scanned by the Fibrograph have the same length, 2) that the fibers are randomly caught and held, and 3) that the optical scan of the beard begin at the point of holding. The latter condition does not exist, because the scan of the beard begins at 3.81 mm from the point of holding. The result is that maximum possible uniformity index is increased to 56.0 for 40 mm and to 60.7 for 20 mm fibers. Direct comparisons of the uniformity index are valid only in a narrow range of fiber lengths.