Pressure coefficient of bulk modulus of rocks was estimated from data of high pressure experiments. These values for both granite and dunite are estimated to be nearly equal to 7 and 4 respectively. These obtained from high pressure experiments of dunite are well in acccord with those deduced from seismic observations.

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Some polymorphic transitions of lattices were treated with relation to the Earth's mantle. Some consideration about the radioactivity of the mantle were also made. It was concluded that dunite is the most probable constituent rock of the mantle.

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This is a note describing algebraic and numerical details which were omitted in paper (1) referred to below. In §2 of the present paper, explanations are given on an approximate way to solve the eigen-value problems (3.6), (3.8) and (3.11) and (3.12) in the above paper.
In §3 and §4, are given the details for getting the results (3.14) and (3.15) in the same paper.

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The first shock is recorded to have occurred about 8a.m. 14 October, 1709. The map on page 152 shows best the whole character of this earthquake. Every place marked by _??_ or _??_ is based on reliable documents respectively. The circles denoting isoseismals in Japanese intensity scale are those corresponding to the magnitude Mk=6.5 or M=8.1 (h=ca 80k.m.). The longest epicentral distance is 864k.m., to Pu-hsien, Shantung Province.
The most heavily damaged place was Chungwei-hsien, where the city walls, buildings, including the neibouring Great Wall, were all collapsed, and moreover the land-slide happened, the river flooded, the water gushed from the earth as deep as above the knees, the water in wells sprang up some feet high, and the dead from pressure numbered more than two thousands. The damages were also seen everywhere in Ninghsia Province and in some places in Kansu. In Chungwei-hsien, the aftershocks were continuously felt over fifty days and dwindled more than one year, and their record, from ten days to three months, are found in many places within the whole disturbed area.

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1. In this paper, we would like to deal with the vibration of buildings standing on the surface of a semi-infinite homogeneous elastic solid. As the following investigation shows, a remarkable fact is involved in the present problem, namely, the fact that the motion of buildings is affected by the existence of elastic base and the elastic base by that of buildings.
To the analysis of this phenomenon we applied the theory developed by I. Toriumi. Consequently the assumption he used is also adopted in the present paper. The authors' hearty thanks are due to him to which we owe very much.
2. Vertical motion. When the incident wave is a purely vertical harmonic oscillation traveling in the direction perpendicular to the free surface, the problem may be solved by the process explained in Fig. 1. The amplitude and phase-lag of the building oscillation are given by the formula (3.8) and Fig. 2 and Fig. 3, where S₀=(ρ₀/ρ)·(l₀/r₀) is used as a parameter. Since the ratio of densities of the building and the ground does not differ much from 1/4, S₀ may be approximately determined by the geometrical configuration of a building only. We find it is nearly equal to l₀/4r₀=1/4β. The graph of the amplitude is similar to that of the magnification of a seismometer. Fig. 4 shows that the value of the maximum amplitude is considerably smaller than in the case of rocking or horizontal motions.
3. Rocking motion. When a wave of purely horizontal oscillation is incident upon the surface where a rigid cylindrical building is standing, a rocking motion is induced. Fig. 5 and Fig. 6 explain the process by which the analysis is perfomed. The results are shown in Fig. 7-Fig. 15. Fig, 7 and Fig. 8 give the amplitude and the phase-lag of the horizontal displacement of the center of gravity. If the displacement of the building is same with that of the ground, X becomes unity. In Fig. 9 and Fig. 10, the resonance amplitude and the corresponding frequency are given. The amplitude and the phase-lag of the center of gravity for the angular motion can be seen in Fig. 11 and Fig. 12, and its resonance amplitude is shown in Fig. 13. If the center of gravity remains stil, Y′=-1. Parameter in the figures are the same with those adopted in the former case.
Since the waves are incident upon the surface, it is natural for the base of the building to move accordingly. However, the motion of the ground is also affected by the existence of the building. The results of the calculation of this effect are given in Fig. 14 and Fig. 15, which show the amplitude and the phase-lag of the additional displacement of the center of the base. We must notice the fact that seismometers installed at the basement do not record the same motion with those installed where the building is absent.
4. Horizontal motion. Next, we shall consider the horizontal vibration of a building assuming a model of one-mass system. (See Fig. 16.) The process of the calculation is illustrated in Fig. 17, and the analytical expression of the motion of the center of gravity is expressed by Eq. (5.13). The results are given in Fig. 18 and Fig. 19. Fig. 20 gives the horizontal displacement of the center of the base. In this figure, the interesting fact is that the curves all meet at a point on the a₀-axis. This point (a₀=1) gives th condition of resonance, which is shown in Eq. (5.19). Fig. 21 shows the maximum amplitude. In this figure we used S₀ as a parameter, the values of which are assumed to be 0.3, 0.4 and 0.5, which represent fairly well buildings with different heights. (We assumed that the building has no damping and v=1.)
Fig. 22 gives the apparent damping coefficient when the building itself is without damping. On the analogy of the magnification of a seismograph we u

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If we consider the ground motion determined by the data observed at stations when a severe earthquake occurs, each time reaching a certain phase (namely, here, the first maximum part of the initial ground motion) appears to show us a regular geographical distribution. Now, let us call this time the initial period.
This distribution may be fairly well accounted for, by assuming that a great crack runs in a given direction along one of the nodal lines, and the wave generates accompanying it.
If the wave source moves in an elastic medium, we are to recognize the evidence of the Doppler effect.
In conformity with this idia, we shall be able to explain the above time distribution in a severe earthquake.
In the first case, we carry on the data of the earthquake of June 28, 1948 at Fukui. Each initial period calculated on this idea at observatories agrees considerably well with each observed value.
Still more, the Abuyama seismogram of Fukui-earthquake provides us with the concept that the fracture mechanism of its occurrence is composed of three cracks which have different amplitudes and cracking times. But, examining the length and duration time of each crack, we may regard a train of three cracks as one crack.
That is to say, in the case of Fukui-earthquake, assuming that a great crack runs from north to south along the nodal line showing the direction of N 20° W at the epicentre, and considering the Doppler effect of the wave accompanied by the crack, an even better agreement with the observations satisfies the explanation of the geographical distribution of each initial time.
Hence, propagation speed v, and length l of the crack in Fukui-earthquake are described as follows:
v=2.1±0.3km./sec. l=27±4km.
Further, if we apply the above mentioned idea to Totori- (Sept. 10, 1943), Tonankaido- (Dec. 21, 1944), and Nankaido (Dec. 21, 1946) earthquake respectively, we shall be able to obtain the similar results for each geographical distribution as regards the initial period, assuming that the crack runs a'ong either of the given nodal lines.

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