December and 1883 December, when the phase of the annual inequality should be the same, but that of Mr. Chandler's inequality should be a maximum in one case and a minimum in the other. Accordingly search was made for observations of the same stars, in months of the same name, at two epochs when such observations should be in one case on the crest, in the other in the trough of Mr. Chandler's wave, and the mean differences formed, weights being assigned according to the formula where m and n are the numbers of observations at the two epochs. The results, with corresponding weights, are given in Table IV.

2mn

• m+n'

The material available must now be restricted to 1880 and following years, for there are systematic corrections to be applied to the individual results in 1877-1879, though these corrections were applied in the Ten-Year Catalogue.

The sign implies that the differences of the observed N.P.D.s are too large, or, according to Chandler, that at the period of maximum the mean colatitude is too large, and at the period of minimum the mean colatitude is too small.

II. So far the theoretical difference between the epochs of maximum and minimum is very fairly confirmed, though it is obvious that there are large accidental errors. As a further and independent test, the epochs were selected where the wave crosses the zero line; but there are of course ascending nodes and descending nodes, and comparisons were made between nodes of opposite senses, and not those of the same sense.

The

results, as will be seen from Table V., were not so favourable to the theory, which may be due to one of three causes

(1) The large accidental errors, in which are to be included systematic errors of a different kind from the one here considered. It will be shown, for instance, in the sequel that there is evidence of a 5-yearly inequality.

(2) A slight error in the epoch of the inequality, owing to which the wave is displaced in the direction of its length. The results of Table IV. would in this case not be appreciably affected; but there would be the maximum disturbance of Table V. It would appear from the sequel that the error, if any, is small.

(3) A want of symmetry in the wave, owing to the rotation of the pole being elliptic and not circular. It is evident from the sequel that this is very doubtful, but combined with (1) might be the explanation of the persistence of sign in Table V.

III. The most satisfactory material for such a discussion as the present is obviously the series of observations of the close circumpolars, and especially Polaris, which extend over many months, and are otherwise numerous, and were accordingly excluded from discussions I. and II., to be treated independently. A treatment of these precisely similar to that in discussions I. and II. need not be further considered here, as it is virtually included in the final discussion; wherein monthly means corrected for RD were formed from the observations of these stars in the years 1880-1891, and compared with an adopted mean place for 1885 0, obtained by combining the places in the Ten-Year Catalogue 1880'0 and the Five-Year Catalogue 1890'0 (shortly to be published), and the differences plotted on paper as

un

in Plate I, on which is also shown Mr. Chandler's theorctical curve. It will be seen that there is a fair agreement

almost throughout, although there are three marked divergences between the two which do not seem to be wholly explicable by

accidental error. These occur about 1881, 1886, and 1891, and the equality of the intervals separating them suggests a fiveyearly disturbance, as mentioned above, though the evidence is, of course, very incomplete.

But when corresponding phases of the wave are taken together and the mean result formed for the whole of the nine periods, there seems to be indisputable evidence of the Chandler wave. There is a little doubt occasionally as to the assignment of particular months which fall between two of the fourteen or fifteen phases of the wave; but, adopting the two different suppositions of a fourteen- or fifteen-month period as roughly agreeing with the varying periods of rotation at the beginning of 1880 and end of 1890 respectively, the observations give the results shown in Table VI.

The results of columns 2 and 4 have been corrected for the general mean correction +o".12.

The 3rd and 5th columns have been corrected as follows:It has been assumed that the R-D correction varies, annually