A general theory of thermal fluctuations, or Brownian motions, in a system in thermal equilibrium is developed on the basis of Gibbs’s classical statistical mechanics. Taking advantage of the deterministic character of classical systems and applying fluctuation formula for a canonical ensemble, a formula for the cross-correlation function for two quantities u and X is given as
‾u(t+τ)X(t)=kT(Ψ(∞)−Ψ(τ)),
where Ψ(τ) is the after-effect function for u responding to a unitfunction change of x, while X is the force associated with the generalized coordinate x. Thus the formula indicates the identity of the statistical after-effect to the dynamical after-effect. It may be shown to comprise virtually all the known results pertaining to thermal fluctuation derivable from classical statistical mechanics. A generalization of Onsager’s reciprocity relation to a system with after-effect is also obtained.

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The hydrodynamical problem of many particles interacting in a laminar flow is discussed on the basis of Stokes approximation. Two methods of approximation are introduced in order to derive the concentration dependence of the viscosity of solutions of rigid spheres. In any case an improper integral is involved which can be evaluated to be of different values according to different ways of integration. This physically absurd result comes from Stokes approximation. A new fundamental equation is proposed for the problems in a laminar flow from an analogous reasoning given by Oseen for the case of uniform flow.

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The numerical calculations for the overlap integrals between 3s, 3p hybrid A.-O. (atomic orbitals) and 1s, 2s, 3s 3p hybrid A.-O. are tabulated in this paper by the use of Slater’s A.-O.

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The physical meanings of concepts such as group velocity and wave path, are first carefuly examined and mathematically founded, and it is then investigated whether the significant concepts of group velocity and wave path do exist or not, when the Maxwell field energy is lost in a medium. It is concluded that, in a rigorous sense, they generally do not exist in absorbing non-uniform medium, and that, when the energy loss in a medium is sufficiently small that only the first order changes in group velocity and wave path due to the energy loss are required to be taken into account, they approximately exist and are concordant with the ones obtained when the field energy is not lost in that medium, so that it is anyhow meaningless to discuss their changes due to the energy loss in a medium.

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The fine structure of the line He II λ4686 was measured, and it was found that the shift of the 3S term relative to the 3²P₁⁄₂ term is 0.135±0.005 cm⁻¹. This is to be compared with the calculated value 0.138±0.001 cm⁻¹. That of the 4S term was found to be 0.056±0.009 cm⁻¹, the calculated value being 0.0584±0.0005 cm⁻¹ .

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Interference color of thin film was calculated taking multiple reflection within the layer and/or dispersion of the layer into account, both having not been taken into account in our previous papers. The result shows that the effect of the former changes the purity of color without altering its hue, whereas the latter changes its hue and purity.

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The method to find the dispersion-formula for the partial cross-section, due to Kapur and Peierls (Proc. Roy. Soc. A, 166 (1938), 277) may be stated as follows :
To express
S=coskr₀φ(r₀)−k⁻¹sinkr₀(dφ⁄dr)_r0
by finding the solution of
\frach²2m\fracd²φdr²+[E−V(r)]φ=0,0≤r≤r₀,
which satisfies at ends
φ(0)=0,[dφ⁄dr−ikφ(T)]_r0=Ie^−ikr₀,
By suitable variable-transformations, however, the above statement will be simplified as follows:
L[Φ]+λΦ=0,0≤x≤1,
with Φ(0)=0,Φ^′(1)−ikΦ(1)=I₀e^−iκ;
and S=e^−iκΦ(1)−κ⁻¹sinκI₀e^−iκ.
Here L[ ] means the differential operator: [d²⁄dx²−q(x)].
By making use of Green’s function, we call express Φ(x) and S in the forms:
Φ(x)=I₀e^−iκΓ(x,1;λ)
and S=I₀{e^−i2κΓ(1,1;λ)−κ⁻¹sinκ·e^−iκ}.
These results involve the same meaning as the expansion (21) in Kapur-Peierls’ paper, and furthermore have the advantage of more simple expressions.

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In the first half of this paper, the behavior of the reflected wave of a weak incident one at an interface of two supersonic flows is studied. The weakness of the incident wave makes it possible to use the isentropic characteristic theory with good approximation and, as a result, complication in the analysis is greatly reduced. It is found in this study that under a certain condition the reflected wave of an incident compression (or expansion) one is an expansion (or compression). The condition is given as a function of the Mach numbers of the flows and the ratios of the specific heats. In the latter half similar treatment is applied to the analysis of wave patterns in a supersonic compound jet. The treatment of the problem in flow deflection and pressure plane simplifies the analysis greatly. It becomes clear in this study that there are two different types of wave pattern in the jet. In the one, pressure is oscillatory in the jet and in the other non-oscillatory. The conditions which discriminate the both cases are also obtained.

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The uniform supersonic flow Past a body of revolution with a finite radius of curvature at its nose is studied. Various hydrodynamical quantities are expanded in power series of the coordinates in the neighbourhood of the stagnation point of the body. The coefficients of the series are determined by the boundary conditions at the surface of the body as well as by the shock wave conditions at the shock wave. Numerical results are compared with the existing experiments as well as with the theoretical results obtained previously by the writer.

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Velocity distribution in the axisymmetric laminar inlet-flow through a circular tube is obtained, under the assumption of “almost similarity” of velocity profiles. Theoretical velocity distribution curves obtained agree fairly well with the experimental one in a rather limited region near the entrance where the approximation is valid.

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As a continuation of Part I, the stability of the inlet-flow is now discussed, obtaining the result that there exists a stability limit whose minimum critical Reynolds number is 9700 at the point 17 times of the radius of tube downstream from its entrance.

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Many influences of physical, chemical and geometrical properties of the diffused substances on the Lagrangian correlation coefficient for the turbulent diffusion in a wind tunnel flow are considered theoretically. Influences of the decaying phenomena of a wind tunnel turbulence are also taken into consideration. Influences of these factors on the Lagrangian smallest eddy λ_η introduced by Sir Geoffrey Taylor are discussed in detail making use of the concept of the smallest turbulon Λ_∞∼ν^3⁄4ε^−1⁄4 due to the similarity theory of turbulence, where ν and ε mean the kinematic viscosity of a fluid material and the dissipation rate of turbulon energy respectively.

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Making use of informations on the fine structure of turbulence due to the modern similarity theory, influences of hot-wire length or instrumental sensitivity on measurement of G.I. Taylor’s smallest eddy in a turbulent fluid are dealt with theoretically in this paper. It is assumed that the hot-wire anemometer of length L cuts off the contribution of turbulons (turbulence elements) smaller than L, and the result is obtained that the practically measured diameter of the effective smallest eddy is in proportion to L^2⁄3, provided L is slightly greater than the smallest turbulon length given by ν^3⁄4 ε^−1⁄4, where ν and ε denote the kinematic viscosity of the fluid and the mass rate of turbulent dissipation respectively.

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On the basis of his previous works, the author derives the equations of the decay of isotropic turbulence in magneto-turbulence in compressible media. All the random variables fulfill the requirements of isotropy and homogeneity and are dependent. The author assumes that the reader is acquainted with previous papers on the subject, and consequently, only a few items from those papers will be cited.

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This paper deals theoretically with one of the three-dimensional isostatical problems in the theory of plasticity.
Fundamental equations for plastic states of stress in a solid of revolution are established under the assumptions that the circumferential principal stress is equal to one of the other two lying in the meridian section and that materials obey the theory of constant elastic strain energy of distortion.
The author discusses the method of solving these equations for the case when the equations for slip-lines are known and two plastic states of stress having slip-lines of logarithmic spirals and straight lines. are considered in details as practical examples of applying his method of solution.

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The author discusses an analogy between the statically determinate state of stress in the perfectly plastic material and the steady irrotational flow of a hypothetical non-viscous fluid introduced by him, the pressure in which is proportional to the logarithm of its density and which has the same local Mach number \sqrt2 throughout the whole field of flow.
The hodograph method for this fluid flow is developed in detail, and besides an application of the W.K.B. method is considered.
Several exact solutions of the fundamental differential equations illustrate the present new method of solving the isostatical problem in the theory of plasticity.

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Noises generated in a selenium rectifier were measured at 600 kc for various reverse voltages and temperatures. They were analyzed by the method of F.E. Haworth and R.M. Bozorth, and attributed to electron multiplication in the barrier layer owing to very high fields in the layer. Because the barrier layer is composed of selenide of about 10⁻⁵ cm thickness, the internal field will be as high as 10⁶ volt/cm at reverse voltages of the order of ten volts, thus creating impact ionization. The observed negative temperature coefficient of the avalanche noises will account for the abnormal temperature dependency of hard flows at high reverse voltages, since the latter may also be attributed to the electron multiplication.
At higher temperatures and lower voltages, however, the noises seemed to be of the flicker type.

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According to the “avalanche theory” of dielectric breakdown, the breakdown occurs by the energy dissipation of electron avalanche composed of sufficient number of secondary electrons. Therefore, the breakdown probability is equal to the average frequency with which the avalanches of such a sufficient sizes occur. On the other hand, the breakdown probability can be obtained by the measurement of the statistical time lag of the breakdown. In this paper, we measured the statistical time lag of breakdown of mica and the pulse size distribution of avalanche noise in the pre-breakdown region. Correlating the both results from the view point of avalanche mechanism, we obtained 10⁸ as the avalanche size which is just sufficient to produce disruption for our specimens which were 3∼5×10⁻⁴ cm in thickness.

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