The equation: which appears freqpuently in the problems of linear heat conduction was integrated, when A(x) is a function of x, in the interval First we give the case The corresponding solution when Next a general method of integration was given when A(x) is any given function of x and for the interval, of course being able to be extended to the case By a method similar to Stokes', the solution for is give by where Xs, s=1, 2, 3, .... are the solutions of the equation subject to the conditions that is, Xs(x) are orthogonal functions and the constants λs are also determined at the same time. Several examples were also given.
The time-distance curves of distant earthquakes obtained hitherto by various authors are considered to be insufficitent with regard to the observation made at a small epicentral distance. But, as there occur many large earthquakes and a fine net-work of seismological stations is framed in our country, the study of near large earthquakes can be easily made. We have obtained thus time-distance curves of three recent destructive earthquakes; they are the great Kwantô E. on 1st. Sept. 1923, the great North-Tango E. on 7th Mar. 1927 and the Abukuma E. on 6th Aug. 1927. And moreover, that of the large deep-focus earthquake having the depth of about 400km. occurred on 29th Mar. 1927 is also obtained. The observational materials were taken from the seismological reports appeared in the “Kisyô Yôran”, “International Seismological Summary” and etc. The curves are drawn for the epicentral distance as far as about 100°. It is affirmed that the focal depths of destructive earthquakes are generally very shallow and can be assumed to be zero _??_o far as the propagation of seismic waves is concerned. The three time-distance curves of destructive earthquakes mentioned above are all alike with each other, and it is interesting that the reduced time-distance curve of the deep-focus earthquake become to be very alike to them. These curves, however, do not always agree with those obtained by many authors of the foreiga countries from the observations of distant earthquakes, especially with those obtained from the mean values of the observations of many earthquakes. This discrepancy is mainly due to the indistinctness of the first phase of seismic waves as some authors were already noticed. The travel time of P-wave obtained up to this time seems to be more or less too small, since the time of occurrence at the seismic origin cannot be accurately determined on account of the scantiness of observations made near the epicenter. The influence upon the travel time due to the irregular construction of the upper earth's crust seems to be not so large and it is not investigated in the present report. The mean time-distance curve obtained by the destructive earthquakes of our country, which is called as “the Japanese mean time-distance curve”, is considered to be the most accurate one for the large earthquakes whose foci can be assumed to lie at the earth's surface; at the least for those occurred near our country. As for deep-focus earthquakes, some corrections must be applied to the values of the time-distance curve mentioned above, which are given in the preceding paper (this Journal p. 39). For the time-distance curves of other phase, we shall write in the next report.
The velocity ratio of P and S waves is obtained by many authors to be nearly constant everywhere in the earth's crust. In the present investigatio_??_, is treated this problem by a method of tp-Tp-s diagram which gives the relation between the arrival time of P and duration of (P-S). Examining in many cases of large earthquakes of both shallow and deep origins occurred in our country, the relation is obtained to be approximately linear as is expected, especially in the cases of deep-seated earthquakes. From these investigations, the velocity ratio of P and S waves is obtained as about 1.73 in the upper part, it may be probably the so-called Mohorovicic layer, but its value seems to differ in different localities; while, in the deeper part nearly a constant value of 1.79 is obtained. Thus, using the result we are able to obtain the depth of seismic focus by a simple method, especially in case of deep-seated earthquakes.