We can obtain horizontal deformation vectors of the crust of the earth from comparing new geodetic observation with old one in the same area under an appropriate assumption. The map of vectors varies according to the assumption. The map of vectors, that satisfies ΣV = 0, is generally believed to be very plausible. Fig. 2 is drawn under the assumption that the positions of two stations 202 and 205 are common in new and old surveys. The triangular mark in the same figure is the new volcano appeared suddenly in 194445. Fig. 2 changes to Fig. 4 under ΣV = 0. But it is obvious in this case that Fig. 2 is preferable to Fig. 4, because very little deformation is expected in the environs of the district as be seen in Fig. 3. Let p_i be such a value as be inversely proportional to average variation in angle at i-station. If we draw the new map of vectors with V_i-V₀again, where V_o = (Σp_iV_i)/Σp_o, it seems to be very plausible.

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According to the satellite observations of geoidal undulation, gravity anomaly is positive around Scandinavia, if it is calculated basing upon the standard gravity formula including only (n = 2 and 4, m = 0) term in the spherical harmonics expansion. So according to the principle of isostasy, the Scandinavian region must be subsiding as against what is oc curring there. It may be, however, that the mass anomaly responsible to the geoidal undulation exists deep (probably in the C layer, mesosphere) within the earth, and that the process responsible to the Scandinavian uplift occurrs in the shallower depths (probably in the B layer, asthenosphere). If such is the case, in order to get the gravity anomaly closely related to the Scandinavian uplift, we must take as the standard gravity not only (n = 2 and 4, m = 0) term but also higher order terms in spherical harmonics expansion derived from the corresponding expansion of the geoidal undulation. In short it is our proposition here that we must choose the standard gravity so as to get gravity anomaly which is agreeable with other geophysical and geological observations.

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The Chandler wobble may be considered to be a succession of free damped nutations of the earth excited randomly in time. Making use of a method to detect reflection impulses in seismic explorations, we calculate the random time series ξ(t) and η(t) corre-sponding to the χ(t) and y (t) terms in polar motions. The results obtained are shown in the attached figures. The arrows in the figures show the time of occurrence of earthquakes of magnitude larger than 8.6. The correlation between the occurrence of large earthquakes and (ξ(t), η(t)) is not so good. This may be due to our use of χ(t) and y (t) data of rather long time interval, about 35 days.

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New Approximation formulas having necessary accuracy were derived by using Chebyshev's approximation by polynomials in calculation of geodetic functions. The computation becomes much speedy than former formulas, and used area of memory of a computer was saved. In this report, two examples were given as follows;1) a formula of meridian arc from equator when the latitude is given, 2) a formula of latitude when the meridian arc from equator is given.

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In 1956 the author computed the dynamic and orthometric corrections to the results of precise levelling for the purpose to compare the dynamic and orthometric height along the same level-circuit that in 1938 Dr. K. Muto studied the gravity correction to the levelling. Recently it has been widely recognized that the normal height proposed in 1945 by Prof. M. S. Molodensky is quite suitable for the both purpose of theoretical study and practical work. The author tried to calculate the normal height from the already computed orthometric height using the relation that the difference between orthometric and normal height at a point is proportional to the product of Bouguer anomaly and levelled height at that point. After this, comarison is made among these three height systems. Also the relations of the distribution of topography and Bouguer anomalies along the discussed levelcircuit to these height systems are considered. The main results are: (1) Dynamic correction is the largest correction among the three and strongly affected by topography. (2) Normal correction is also affected by topography. (3) Orthometric height is affected by both topography and Bouguer anomalies. In flat and middle mountain region orthometric correction is approximately equal to normal correction. At B. M. 576, the highest bench mark in the discussed level-circuit, the following results are obtained:levelled height = 1453.78 mBouguer anomaly = -53.6 mgaldynamic correction = +278.2 mmorthometric correction = + 166.7 mmnormal correction = +87.3 mm The normal height can be used in practical geodetic work instead of orthometric height, but, in Japan, the following point seems to be emphasized, that is, if we assume that the crust density is 2.67 g/cm³ and the normal vertical gradient of gravity is 0.3086 mgal/m, Helmert orthometric height can be much easily calculated as compared to normal height, because we have already finished to get the observed gravity values on all bench marks in Japan. The difference between normal and orthometric height is very important in the modern physical geodesy. This can give the height difference between quasi-geoid and geoid. Estimation of this value at the highest triangulation point of Mt. Fujisan is 1.1 m. Therefore we can neglect the difference between quasi-geoid and geoid when we discuss the problem of physical geodesy in the order of ± lm in geoidal height.

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The non-parallelism of the geopotential surfaces at different heights may cause some amounts of the closing discrepancy of a levelling circuit having no connection with the observation errors. In order to evaluate the closing error, the well-known formula after Helmert; ΔΣΔg₀-g_i/g₀ Δh_i has been widely used, where gi, Δhi and g₀ are the observed gravity, levelling increment and a constant respectivelly. Starting from the equation of the astronomical levelling after Molodenski, a new formula for computing the closing error is given as a line integral along the circuit, as follows; ∫dhw=-∫Δθdl_h
=-1/gΔΣΔ(Δg_i′Δg_i′+1/2Δhⁱ⁺¹_i+γ_i′+γ_i′+1/2Delta;hⁱ⁺¹_i)
where Δθ denotes the difference θ'-θ between the components of the vertical deviation to the direction of the route θ at a bench mark and θ' at the point which is located above the bench mark and on one external geopotential surface named Exogeoid, Δgi' and γi' being the station free air anomaly and the normal gravity, both reduced to the values on the Exogeoid above i-th bench mark respectivelly. For the purpose of calculating Δg'(P') from the distribution of observed gravity on an undulated physical surface of the earth, an iterative approximation for computing the anomaly of the vertical gradient of gravity was proposed, where the first approximation for Δg'(P') on the Exogeoid is the station free air anomaly Δg(P) itself on the ground, and its first correction Δg₁(P) is obtained from the value of ∂g/∂z anomaly computed from the distribution of the first approximated value Δg (χ, y, h) insted of Δg'(χ, y) on the Exogeoid surface, An example of the closing error was computed on the levelling circuit; Tokyo-Takasaki-Suwa-Yamanashi-Tokyo, the closing discrepancy amounting to +21.45 mm to the direction of clock wise, using the result of levelling with the normal orthometric correction.

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Since 1957, the year of the launching of Sputnik 1, efforts have been done in Japan for the tracking of the artificial satellite in various ways and by many persons. Visual groups, so-called moon-watch teams, have been distributed all over Japan using specially designed telescopes. Some of them are still active. Photographic observations were introduced in 1957 with the use of our meteor camera. In April, 1958, optical tracking with the use of the Baker-Nunn camera began as the cooperating work between the Tokyo Astronomical Obeseratory (T. A. 0.) and Smithsonian Astrophysical Observatory (S. A. O.). The geodetic tracking of the artificial satellites has been one of our important study items since 1957. For such purpose, special instruments had been designed and constructed by Geographical Survey Institute (G. S. I.) and Hydrographic Office (H. O.), respectively, and test observations and mutual comparisons between these three types of instruments, Baker-Nunn of T. A. 0. and other two Japanese tracking cameras of G. S. I. and H. O., were done in cooperation manner. Now-a-days, the project of the isolated islands of H. O. and that of the national survey of G. S. I. are going to start. The method adopted to our geodetic projects is so-called the trailing method (Hirose 1963). The G. S. I. instrument is an equatorial camera equipped with a photo-electric knife-edge device as the timing unit (Tsubokawa 1965). The H. O. instrument is also an equatorial camera, but, equipped with a traveling slit device as the timing unit (Ono 1966). Both of these two timing devices can evaluate the time of observation with the accuracy of several tenths of 1 ms and these instruments may also be used for the simultaneous observation, if necessary. For our geodetic purpose, we have already set up several base stations as given in the following Table :

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