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The “Boundary-Perturbation Method” i.e. a general method for estimating the effect caused by a small change in the shape of the boundary (or, of the hypersurface in the many-dimensional space, the time-coordinate being included) upon the solutions or eigenvalues of linear partial differential equations, which was presented for the case of scalar fields earlier (J. Phys. Soc. Japan 15(1960) 2069), is developed to the case of tensor fields and formulated so as to permit their immediate use. Some examples are given to illustrate the method.

A general method is presented for estimating the effect caused by a small change in the boundary shape (or the hypersurface shape in the many-dimensional space, the time-coordinate being included) upon the solutions or eigenvalues of linear partial differential equations (elliptic, parabolic or hyperbolic). When the given boundary is slightly deformed from a shape favorable for obtaining the solutions, this method gives the approximate solutions or eigenvalues based upon the exact or unperturbed ones for the case of the favorable shape, where many solutions can be obtained easier. The following illustrative examples are given, showing the validity of the method presented: (1) The eigenfrequencies of the lateral vibration of a uniform membrane fixed at its boundary close to a circle in shape, e.g. elliptic, oval, cinquefoil-shaped, etc. (2) The eigenvalue problem concerning the heat conduction of a “corned-beef-can” -shaped uniform matter, close to a rectangular parallelepiped in shape and with its tempera...

In order to ascertain the relation between the change in the fibrous fine structure and the dyeability of nylon 6 fiber caused by its heat-setting, the ζ-potential, the surface dye adsorption, and the total dye adsorption of the heat-set nylon 6 fiber in an acidic solution (pH 3.0) of acid dye—Orange II were determined. The crystallinity and the birefringence of the fiber increase in the order of the untreated, the wet-heat-treated, and the dry-heat-treated fiber. The isoelectric points of the untreated, the wet-heat-treated, and the dry-heat-treated fiber are pH 5.4, pH 4.6, and pH 2.8 respectively. With the increase in the dye concentration, the sign of the ζ-potential of the untreated and the wet-heat-treated fiber in a dye solution change from positive to negative because of the electrostatic bonds between the NH3+ of the fiber and the dye anion, RSO3−, while the ζ-potential of the dry-heat-treated fiber increases its negative value because of the van der Waals’ forces between the fiber and the dye. T...

Here are studied some types of differential equations of the fourth order, to which the so-called W.K.B. method can be easily extended, and presented some exemplifying solutions up to the first order of approximation. As a physical example, the equation of lateral bending vibration of a not-completely flexible string, whose cross section of finite size is slowly varying along its length, and whose ends are both clamped, is solved. It is also shown that the terms of the first and the third differentation of a linear differential equation of the fourth order can be eliminated by a suitable transformation.