§ 9. The System of Weights.-The summary cards showed at once that the observatories were not entitled to equal weight. At the same time it was not possible to derive a rigorous system of weights from a discussion of the divergences between one observatory and another, or between the individual results of an observatory. These divergences were necessarily much affected by the fact that five or six separate determinations of a star place made at one observatory would depend on the same set of repère stars, and be on the same part of the plate, whereas the same star would be determined elsewhere on a different centre.

After some trial it appeared that substantial justice would be done if Greenwich and Paris were given a weight double that of the other observatories, and if the weight of each observatory's contribution, so far as it depended upou the number of observations contributed by each, were on the same scale as was used in making the system P.Ph. of § 4.

Two Catania plates and one San Fernando were rejected for unexplained systematic discordance of large amount; a certain number of obvious misprints in the Circulars were corrected, and a few discordant observations, probably misprints, were excluded.

Finally, stars whose weighted mean had weight 4 or more were passed into a first class, and these form the photographic standard system discussed in what follows. In the region covered

by the planet between 1900 Oct. 1 and Dec. 31 the catalogue contains 1300 first class stars. And in addition there are about 2500 stars in a second class, whose weight is less than 4, which are not for the present included in the standard system.

$10. Test of the Standard System.-As a test of the homogeneity of the contributions which the different observatories make to the standard system, I have discussed the differences, Standard minus Individual contribution to it, from several points of view.

A. The differences, taken with regard to sign, have been grouped (1) in magnitude groups, to see if any residual magnitude equation remained; and (2) in date groups, to see if there were any discontinuity along the path of the planet which might gradually distort the standard system, and damage an ultimate determination of the mass of the Moou.

B. The differences, taken without regard to sign, have been grouped likewise (1) in magnitude groups, to see how far the interagreement of the photographic places is dependent on magnitude; and (2) in date groups, to see if the same standard of internal agreement is maintained along the whole system.

The magnitude groups are those used in S.P.P. No. 5, Tables III., IV., and V.

The date groups are broader than those used in Table VII. (loc. cit.).

Group I. covers Sept. 15-Oct. 23

II.

Oct. 24-Nov. 24

III.

Nov. 25-Dec. 15

IV.

Dec. 15-Dec. 31

There is no special significance in this particular division, which arose more or less accidentally, and has been retained for convenience.

§11. Search for Magnitude Equation in R.A.-The division. into magnitude groups of the differences Standard minus Individual contribution was made separately for the four date groups. But there is no evidence in the results of any definite change of magnitude equation with the date; if any such change exists, it is obscured by the accidental disco dances. Consequently I give only the mean results for the whole period Oct.-Dec., except in the case of San Fernando, to which a correction for magnitude equation has been applied. The separate results for each San Fernando date group are given.

The figures in brackets are the numbers of observatory means contributing to the mean discordances which follow.

The results of this table confirm the conclusion of S.P.P. No. 5, that the photographic series which contribute to the standard system are free from relative magnitude equation, with the exception of San Fernando.

The latter observations have been corrected, and we can now see how far this correction has been successful. The correction applied was decidedly non-linear; the above comparison gives some indication that a slight non-linear correction of opposite curvature is now required; that is to say, that the curvature of the original determination was a little excessive. (The material used was not altogether the same.) The magnitude equation results from the separate date groups are not, however, accordant enough to make it probable that any magnitude equation correction for San Fernando is sharply determinable; and we may conclude that the attempt to correct these R. A.'s has been as successful as the material allows.

The Upsala and Minneapolis results were not published in time to be included in the first edition of the standard system. But they show no magnitude equation when compared with it, and may therefore be included safely on a revision.

§ 12. Search for Magnitude Equation in Declination.—The declinations have been treated exactly as the right ascensions. Again there is no trace of any definite change of magnitude equation with the date, and the general means are free from it. The corrections applied to the declinations of San Fernando and Toulouse seem to have been successful in eliminating the large errors originally found in them. Table IV. gives the results for the date groups in which the elimination has been least successful, and for the

mean.

The lately published declinations of Upsala and Minneapolis show no magnitude equation relative to the standard system.

§ 13. Conclusion as to Photographic Magnitude Equation.—As the result of our operations, we have a system of stars derived from photographs made at nine different observatories; and the comparison of each series with the mean shows no relative maguitude equation.

Further, two additional series, not originally included, show no magnitude equation relative to the system.

It is extremely unlikely that these eleven series should be affected by magnitude equation of like sign and amount.

I conclude that our photographic standard system is sensibly free, not only from relative, but from absolute magnitude equation.

§14. Search for Progressive or Uniform Discordances irrespective of Magnitude. With the fear of magnitude equation removed, we may examine the mean divergences in the date groups, to see if there is any evidence of progressive or general discordance. The tables which follow are self-explanatory.