When the velocity of a body moving in air reaches a certain critical amount less than the velocity of sound, condensation and rarefaction waves are produced respectively on the head and tail surfaces of the body. The position and direction of these waves relative to the body depend on its velocity relative to that of sound in air. In an airscrew, tip condensation and rarefaction waves are also produced at the critical moment at the head and tail edges respectively of the blade tip.
In the present paper, these critical waves and the waves which are produced when the velocity of the body is greater than that of sound are dealt with. Mathematical equations useful for calculating the motion of these waves are also given, and it is made clear that the energy consumed by these waves depends on the 4th power of the velocity of the body.

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Let us assume a lap-joint as shown in Fig. 1, and assume that, from the condition of thin plates, the theorem of two dimentional elasticity can be well applied. If we further assume the state of connection as equation( ), …i.e., the force of connection is proportional to the relative displacement between the two plates, …the equation of equilibrium can be expressed only by the amounts of slip between the two plates, and be reduced to the equation (6) or (8), according to the situation of the point of consideration inside or outside the range of connection.
In sections 5 and 6, we reduced the fundamental equations for both long and round joints respectively, for these two might be the typical cases from the engineer's view point. In section 7, solutions for some individual problems are developed. Generally, the connecting force is much concentrated to the edge or surrounding part of the connected zone, except when the connecting rigidity is very small compared with that of the plates.
This feature, in other way, causes the earlier yielding of connection at the surrounding part, and, to some extent, contributes to the equalization of the connecting stresses. But, when the load is released from this condition, the system never recovers its first state; one set of residual stress is induced at its no-load condition, which would have very important meanings for the case of repeated loadings.

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In this paper, two forms of failure of long thin circular tubes under combined bending and uniform internal pressure are discussed, namely, the flattening and the local buckling. It is shown that during bending of thin circular tubes with internal pressure, a flattening of the cross-sections takes place and especially in the case of negative internal pressure, or positive external pressure, the cross-sections becomes more and more oval unitl a critical point is reached at which the resistance to bending starts to decrease. If the bending moment is increased above this value, the thin tubes must collapse.
Thus in the case of bending of tubes with negative internal pressure, the failure can be considered to be mostly due to the flattening of the cross-sections. The critical bending moment Mcr and the critical pressure pcr for this case are given by the following formula: M_cr=(2√2/9)(Eπrtt²/√1-μμ²)[1+{4(1-μμ²)r³/t³}(pcr/E)]1/2, or, McrM_cr²/(M_cr)(M_cr)2p=0+p_cr/(p_cr)M=0=1, where r=radius of the tube, t=thickness of the wall, E=Yonng's modulus, and μ=Poisson's ratio. For the special case when the bending moment alone is acting (or pcr=0), the above result is in good agreement with the result obtained by Brazier, namely, (Mcr)p=0=(2√2/9)(Eπrtt² /√1-μμ²). For the case when the external pressure alone is acting (or Mcr=0), the above result is also in agreement with the well-known result obtained by many investigaters, namely, (p_cr)M=0=-Et³/4(1-μμ²)r³,
In the case of bending of thin tubes with positive internal pressure, the author's ext periments about celluloid tubes shows that local buckling occurs before the bending momenreaches the critical value due to flattening. The calculation for this case is developped on the assumption that the local buckling may occur when the stresss of the compression side of the tubes reaches a certain value which is determined by the ratio of the thickness to the radius of the lateral curvature at the flattened cross-sections and independent of the internal pressure. However the experiments about celluloid tubes shows values for the bending moment much lower than the calculated values. Mean experimental results for the case of postive internal pressure is shown by the equations (25) & (28) in this paper. It should be noted that the increase of the bending strength of thin tubes due to positive internal pressure is comparatively small, while negative internal pressure decreases the bending strength considerably. The results may be of use for the design of the pressure cabin of monocoque construction for high altitude flying.

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The study of fatigue strength of material is practical importance, but a testing time is very long and a cost of ordinary testing machine is very high Then the many arrangements of testing machines are practically difficult. In order to exclude these difficulties, the very simple apparatus is designed here, for instance, such as the elastic part (the test piece) and the inertia block with electrical magnets are joined together and the nodes of inertia block in the natural vibration are suspended by two fine wires. This apparatus vibrates in the natural frequency of the resonant state making use of the electrical contacts. Then the consumption of electrical energy is very few. Also the study of dynamical properties of the elasticity and the internal friction of the test piece can carry out.

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