To generalize and simplify basic analytical means of studying various processes of stretch- breaking, the authors suggest fundamental formulas which express and correlate several characteristics of changes in fiber quantity during stretch-breaking with two pairs of rollers. They also give and discuss examples of calculation.
The relation between input and output of the probability density of fiber leading ends on a front roller is obtainable if the probability density of the breaking points and the probability density of the delivery points are given. The probability density of fiber length and the staple diagram are obtainable in the same way.
From the output of the probability density of fiber leading ends and the probability density of staple length is obtainable the sliver thickness at each hour. From the output of the probability density of fiber leading ends and the relation between stress and strain of single fibers in the stretch-breaking process, it is possible to obtain the stretch-breaking force at each hour.
For an example of solution to obtain the above-mentioned characteristics practically, the relations presented by simple formulas in Laplace transform are shown if the probability density of the breaking points and the prebability density of the delivery points do not vary with time. And,
1) The staple length, the sliver thickness and the stretch-Breaking force, which vary after tow is fed to the stretch-breaking zone, are shown concretely. In these examples of calculation, the usual correlation between the sliver thickness and the stretch-breaking force is relatively low.
2) Even if there is no distribution of breaking points, the stretch-breaking process which has distribution of delivery points (which has distribution of breaking eongation of single fiber for example”. is “stable.” This is becuase the probability density of breaking points and the probability density of delivery points act equivalently to the realtion between input and output of the probability density of fiber leading ends.
3) Sliver thickness and stretch-breaking force a long time after tow is fed to the stretch-breaking zone can be obtained easilily by using the last value thorem in the Laplace transformation.
4) Some other interesting examples of calculation are given.

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This article expands the theories announced by Livesey-Owen and Grosberg on the bending deformation of woven fabrics. A theoretical equation (11) has been derived. Various factors which determine the bending deformation properties can be arranged as shown in Fig. 4. The bending movement vs curvature curves of various woven fabrics have been measured. The effects of single filament denier, yarn density and various finishing treatments have also been examined. The bending behaviors of woven fabrics of cotton, viscose and other synthetic fibers have been compared. Data obtained from the experiments have been analyzed on the basis of the equation (1).

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