by the circle. The station where these vertical distances are the smallest will, in general, have obtained the most accurate results. An inspection of the figure shows that the best agreement was obtained at Mizusawa. It is rather significant that at Carleforte, where more than twice as many observations are obtained than at some of the other stations, and the meteorological conditions are exceptionally fine, yet the agreement between the observed and the computed curves is not so close as at some other stations. This is a good illustration of the precept that, in general, little or nothing is to be gained by increasing beyond a certain moderate amount the number of observations made with the same instrument under similar circumstances. In fact it is quite possible that just as good results could be obtained by limiting the number of observations taken at each station to a monthly average of a hundred or thereabouts.
As Tschardjui and Ukiah are separated by nearly 180° of longitude, the curve of the one is almost the counterpart of the other. It may be seen from Fig. 5 that the maximum change in the latitude, during the time represented, is less than 0″.5, which corresponds to about fifty feet on the surface of the earth. The observatory then apparently swings back and forth in the meridian to a distance of twenty-five feet on either side of the mean position.
Having now the actual observed variations in the latitude at six different stations, separated widely in longitude, it is a comparatively simple problem in mathematical analysis to compute what the actual motion of the pole, with respect to its mean position, must be in order to produce the observed changes in the latitudes. If the difference between an instantaneous value of the latitude and the mean value be represented by the rectangular coordinates of the instantaneous pole, with respect to the mean position of the pole, by x and y; and the longitude of the observing station by ; then the following equation, the derivation of which is given in the review mentioned above, may be written,
Early investigations showed that the observations were not represented to the highest degree of accuracy by this equation and Dr. Kimura, the Japanese astronomer, suggested the addition to the equation of a third term, z, independent of the longitude. The observations are satisfied much better by an equation of this form, and z turns out to be a small variable quantity of an annual period. No satisfactory physical explanation of this term has as yet been given. Several have been suggested, one of which is that perhaps there is a small annual shift in the position of the center of gravity of the earth.
In order to solve the problem connected with this term, two additional latitude stations were established in the southern hemisphere in
