CAMBRIDGE Philosophical Society, October 21.—Dr. Pearson read a paper on a series of lunar distances taken by him during the years 1875-77, mostly at Cambridge and at a place not far distant, the longitude and latitude of each spot being accurately known. He said that they entirely bore out the conclusions at which he had arrived some time back from a much smaller number of observations, and which were communicated by him to the Society m a paper read by him, March 13, 1876 (see Proceedings, ii. pp. 414-418), viz., lhat the errors are such as cannot be called errors of observation of any kind, and may probably arise from the solution of the spherical problem on which the result depends not being, as at present given, strictly accurate. It was mentioned that there is much to justify this conclusion; for example, this method of obtaining longitudes is not much resorted to now in practice (from which it may be argued that it is actually found inaccurate). It is not formally adopted in Germany, though it still is retained in the Nautical Almanac, and in the corresponding publication, the Connaissance des Temps, issued at Paris. Capt. Toynbee, F.R.A.S., in a paper in the Nautical ATagazineiox February, 1850 (of which there is an abstract in the Monthly Notices of the R.A.S.), distinctly states that lunars taken eact of the moon give always a result thirty or forty seconds different from those taken west, though his mean result he says was entirely satisfactory; and until the early part of this century all East Indian longitudes were in error nearly 3m. to the east, a result which very nearly agrees with the errors resulting from these observation?, supposing them to have been deduced from the new moon of five to eight days old, probably the most convenient time at which to take them from the sun. The whole series, it was stated, consists of 250 separate distances, each distance being either a mean of three or two, or else only one observation, there being about an equal number of each class, though there is no reason to think that the last are less trustworthy than the others in any serious degree; the Greenwich-mean time for each being established, with the exception of a very few, within certainly ten seconds. Only 200 of these, the number at present thoroughly verified, were discussed on the present occasion. Classing these in groups of about forty, it was found that the first group gave thirty-two results where the measured distance was in defect of the theoretical distance, and thirteen in which it was in excess. Assuming the rule given in p. 417 of the paper referred to to be correct, this result exactly agrees with what might be expected, it being almost always most convenient, especially for a beginner, to take lunars, at any rate from the sun, under such circumstances as will give this result, while the example of India, founded apparently on observations made at Madras, seems to imply this probable facility, and also that they wrere made on the new moon, these being more easily taken in our hemisphere than those made on the old one. In the four remaining groups the proportions are 26 to 18: 28 to'15: 25 to 17: 17 to 14: giving a total of 128 observations in defect, and 77 in excess. Rejecting three or four certainly questionable results, the"greatest errors occurring are 2' 59' in defect, and 2' 48"in excess. The true mean has not yet been ascertained, but is certainly in each case not far from 1'—i' 20"; which, on an average, will give the observer an error of about half a degree of longitude, or of twenty to thirty-five miles, advancing from our own latitude to the equator. There are probably not a dozen clear exceptions to the rule suggested in the communication of March, 1876, that if the luminaries are both on the same side of the meridian, the observed distance is always in defect of the true if the moon be nearest to it, and in excess if she is farther distant: while the same rule holds good, but with less certainty, when the two luminaries are on different sides. The only exceptions seem to arise wfiere the one more distant from the meridian has a greater altitude than the other, or is of a considerably higher declination, and when the distances are very great, i.e. from 1200 to 1300, in which case the measured distance seems generally to be slightly in excess of the true; but, as might naturally be expected, these last distances cannot often be taken in cur own climate. It was explained that all the reductions had been made by Borda's formula, stated in the Philosophical Transactions for 1797 to have been the first strictly mathematical solution of the problem. But the results vary only by a few seconds of arc from those given by the system adopted in the large folio published in 1772 by Mr. Shepherd, Plumian Professor at Cambridge University under the superintendence of the. Commissioners of Longitude, and while Dr. Maskelyne was Astronomer-Royal; or from other methods which it is believed are allied to this. Two examples were also exhibited of distances reduced according to the el ah orate method suggested by Bessel in the Astronomische Nachrichten of 1832; Bessel's results, however, do not differ to any great extent from those obtained otherwise. It was suggested that the problem is really one of spherical trigonometry, and from the fact that the errors seem to depend on the position of the luminaries towards the meridian, whereas the old methods of solution depend on their altitudes, and also that the different ways suggested for eliminating the error due to the difference between the geocentric and geoirraphical latitude of the place of observation give different results, a hope was expressed that if these two circumstances were thoroughly reconsidered in dealing with the question, means might be found of discovering a farther correction of the observed distance, which would give a really accurate result.