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The great preponderance of increasing North Polar Distances in this table clearly indicates that the apex is situated in the Northern Hemisphere, and I may add that its North Declination is considerable, since if it lay near the equator the preponderance of increasing North Polar Distances would not be so great. Next we have to note that there are two regions in which diminishing North Polar Distances ought to be preponderate, viz: between the apex and the North Pole and between the ant-apex and the South Pole. The Southern stars in the catalogue which I am examing are comparatively few, especially at high Southern Declinations, and therefore the table exhibits but faint traces of this second region. The apex clearly lies between the 16th and 21st hours of R. A. and notwithstanding the curious deficiency of diminishing North Polar Distances between the 18th and 19th hours, that interval seems to be the most probable position for the apex. It will be noted too that the region in which diminishing North Polar Distances preponderate is too large to justify us in placing the apex very near the North Pole. A declination of about 45° will, I think, best explain the phenomena on the whole.

I then formed a similar table for the proper motions in Right Ascension adopting the correction already suggested, but this correction has only the effect of converting a relative into an absolute preponderance of increasing Right Ascensions in one part of the sky and hardly affects our conclusions as to the most probable position of the apex.

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Here, if A be the Right Ascension of the apex, there ought to be a preponderance of diminishing Right Ascensions between A and A 180°, and a preponderance of increasing Right Ascensions between A and A+ 180°; while there should be two neutral points corresponding in Right Ascension with the apex and antapex respectively. The first of these neutral points is indicated by the change from a preponderance of increasing to one of diminishing Right Ascensions as we pass from the interval 6h to 7h to the interval 7 to 8h; the second is indicated by an almost equal distribution between 18h and 19h. It is true that there is also a pretty equal distribution between 15" and 16", but this evidently lies in the region where diminishing Right Ascensions preponderate and is due to some accidental cause. The interval 18h to 19h is thus again marked out for the Right Ascension of the apex. The preponderance of diminishing Right Ascensions between 71 and 19h and of increasing Right Ascensions between 19h and 7h is moreover large enough to show that the apex cannot be near the North Pole, for in that case the effect in Right Ascensions would be small. With a larger catalogue we could adopt smaller intervals in Right Ascensions and intervals of Declinations also; and with a larger proportion of Southern stars we could determine the position of the ant-apex as well as that of the apex, and use the former as a check upon the latter; and it is one advantage of this method that the process itself shows the degree in which its results may be relied on. My rough estimate of 280° + 45° may be erroneous to the extent of several degrees in both Right Ascensions and Declination, but I think it will be found nearer to the truth than 264° +25° which Mr. DUNKIN deduced from the same data by applying the method of least squares to the proper motions of the stars as classified by him. The reduction in the sum of the squares of the motions effected by Mr. DUNKIN was moreover very small, whereas the figures given above seem to imply that the apparent proper motions of the stars are largely dependent on the motion of the sun.

But can we by this method ascertain the velocity with which the sun is moving through space? I think we can, as soon as spectroscopic observations have enabled us to estimate the average velocity of the stars. Assuming that the stars are moving indifferently in every direction, one-half of them will be approaching and the other half receding from any point which we may select, and taking the earth as such a point, the spectrocope will

The North

give us the average rate of approach or recession. Pole, for instance, is another such point. Half of the stars will be receding from it and the other half approaching it, in consequence of their own laws; but the effect of the sun's motion is to convert the approach into an apparent retrogression in the case of all stars whose velocity of approach is less than that of the sun. Supposing for example that the sun is approaching the North Pole with the average velocity of an approaching star, one-half of the approaching motions will be changed to apparent recessions and the receding stars will outnumber the approaching in the ratio of three to one. According to the catalogue which I have been considering, the increasing North Polar Distances fall somewhat short of this proportion so that the sun's velocity in the direction of the North Pole is a little less than the average of spectroscopic velocities. How much less could, I believe, with sufficient data be determined exactly, and if we know the velocity with which the sun is approaching the North Pole and the exact position of the apex, we can easily compute the velocity in the direction of the apex. My present rough estimate of this velocity is about twenty miles per second, but this may be erroneous by several miles. A catalogue of the proper motions of not less than 10,000 stars would I think be requisite for any computation whose results could be relied on as fixing the position of the apex within two or three degrees, or as determining the sun's velocity without a considerable percentage of possible error. But the method which I have been advocating requires but little mathematical computation and could be applied to 10,000 stars with less labor than the current methods could be applied to 500.

II. THE SUN'S MOTION IN SPACE.

By W. H. S. MONCK.

In a paper already communicated to this Society I expressed my opinion that the apparent proper motions of the fixed stars are much more largely dependent on the sun's motion in space than is commonly supposed by astronomers, and that the contrary conclusion was chiefly deduced from erroneous assumptions as to the magnitudes and distances of the stars whose proper motions were under consideration. I have since examined M. BOSSERT'S

list of stars having a proper motion of o".5 or upwards annually in the Bulletin Astronomique for March, 1890, and the result has been to confirm my previous conclusions. M. BOSSERT's list seems to me to contain some duplicates as well as some binary stars, which I think ought to be struck out, but I have in the present paper dealt with his list as I found it. Though the large proper motion of some stars is no doubt due to their vicinity to us, I think it may be fairly assumed that the stars in M. BOSSERT'S list possess more than the ordinary amount of true proper motion, and that consequently if the effect of the sun's motion is plainly evident in the case of these stars we may conclude that the proper motions of other stars are affected by it in a still higher degree. The list contains 269 stars. Of these 10 have no motion in North Polar Distance while of the remainder the motion of 176 is in increasing N. P. D. and only 83 in diminishing N. P. D. Again I remarked that assuming the R. A. of the apex of the sun's way to be 270°, the effect of the sun's motion would be to increase the R. A. for all stars between 270° and 90° R. A. and to diminish it for all stars between 90° and 270°.

M. BOSSERT's list contains 139 stars whose R. A. lies between 270° and 90°, of which 101 have an increasing R. A., 35 a diminishing R. A. and 3 are neutral; while of 130 stars between 90° and 270°, 33 have an increasing R. A., 96 a diminishing R. A. and I is neutral. There is another test which affords equally decisive results. The effect of the sun's motion on that of the stars is insignificant in the case of star's situated near the apex and antapex of the sun's way. Hence, if the apparent proper motions are largely influenced by the sun's motion, stars with large proper motion will exhibit two minima near these points. The average number of stars to each hour of R. A. in M. BOSSERT's list is a little over 11, but between 6" and 7h there are only 5 and between 18h and 19h only 3. There are some irregularities in the list but

nowhere does the number fall so low as in the two hours which I have indicated. Of 80 stars whose proper motion amounts to I" or upwards not a single one occurs in either of these hours of R. A.

M. BOSSERT's list also confirms a remark which I previously made as to the necessity of dealing with Sirian and Solar stars separately. On comparing it with the DRAPER Catalogue and taking the stars common to both, the Solar stars outnumber the Sirians in the proportion of eight to one. Two explanations

may be offered of this fact. One is that the Solar stars are really moving faster through space than the Sirians; the other is that the Solar stars are about eight times as numerous as the Sirians, but that the Sirian stars, owing to their greater brilliancy, are visible at distances considerably exceeding those at which Solar stars of equal mass can be seen, in consequence of which their number has been over-estimated.

Should the old methods of mathematical computation be still adopted, the data for them should at all events be improved. Besides separating Sirian from Solar stars, magnitudes (when they enter into the computation) should in all cases be determined photometrically. When this has been done, assuming the distribution of the stars to be uniform and no light lost in transmission, the proper motions can be reduced to a common basis by multiplying the observed proper motion of each star by 10%, where m is the magnitude of the star. The unit in this case is a star whose photometric magnitude is o. If the stars are classified not according to their magnitude but according to their proper motions, a method of successive approximation might be adopted. Thus, suppose that we determine the sun's motion from the class of stars whose proper motions lie between o".08 and o". 12 annually. Correct these proper motions by the result thus obtained, rejecting those which are considerably increased or diminished and inserting in place of them stars whose apparent proper motion is above o′′. 12 or below o".08 but which would fall within these limits on correction, and then repeat the computation with the new class of stars. Previous determinations of the solar motions might suffice for a rough correction at the outset. Such a method would no doubt be laborious but great labor has already been bestowed on such computations. Unfortunately the results hitherto attained can hardly be regarded as repaying this labor. If two or three successive approximations gave almost identical results we might consider the problem solved unless the stars employed in the computation had some systematic motion of their

own.

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