By means of elliptical functions the rectangular line with inner central conductor of circular cylindrical shape is transformed into a coaxial line with circular outer conductor and nearly circular inner conductor, then by employing a finite number of terms of circular cylindrical harnomics we can fit the boundary conditions at the outer conductor and at a finite number of points at the inner conductor, thus the problem of the rectangular line with inner central circular conductor is solved. Similarly, by means of trigonometrical functions, the trough line with central inner conductor can be handled, i.e., it is transformed into a wire of nearly circular cross-section parallel to and in front of a grounded plane, then by means of the bipolar coordinate transformation, the problem of this trough line can be solved by using a finite number of rectangular harnomics. When the distance between the axis of the inner conductor and the bottom of the trough tends to infinity, the results obtained in this paper go into that of the well-known slab line.

The theory of heat conduction in the presence of radiation inside a transparent body is developed, and a general equation of heat transfer has been obtained. The general equation of heat transfer in the stationary state is solved for four special forms of bodies, namely, plane, circular cylinder, sphere, and prolate spheroid. In the plane problem, numerical calculation is carried out, with results. differing frpm those obtained previously by Kellet. It is pointed out in the present paper that Kellet has neglected the property of radiation emitting in all directions, but only considered the radiation along the direction of temperature change, so that the equation of heat transfer obtained by Kellet is different.

In this paper, the author endeavors to study the diffraction of electromagnetic wave by a long slot with conducting environment on opposite sides (i.e., a slot-coupled system of arbitrary shapes) from the viewpoint of coupled waves. A novel theory, which is justified by clear physical reasoning, is presented for a practical solution of the relevant boundary-value problem. In the light of this theory, problems of coupled waveguides, like problems of a single waveguide structure, are attacked by the same mathematical approach, namely, the theories of development in the orthogonal functions. Thus, two types of problems heretofore treated diferently are brought under a unified and coherent point of view. To illustrate the applicability of the present theory, the author solves the problem of a long slot directional coupler, whose usefulness in the development of long-distance waveguide transmission is well-known, and derives a set of original formulas for computing the coupler parameters. As concluding remarks, some further appliations of the theory are freely discussed.

In two previous papers the author presented a generalized theory of local normal modes by introducing the method of slowly varying coefficients-a mathematical method to solve systems of first order linear differential equations with slowly varying coefficients, which are associated with problems of multi-coupled modes in irregular waveguides. However, the theory presented therein was not in a form to adapt itself to problems involving abrupt Changes of the waveguide parameters, which are inevitable in any practical transmission system. The purpose of the present paper is to consider such abrupt changes in the light of the above theory, so that the concept of local normal modes may be used to describe a broader class of problems.