The figures can thus be explained either by a seasonal change in focal length, or by a series of discontinuities at the times when either eye end or O.G. were dismounted.

Against the latter explanation there is the fact that no discontinuity was recorded about plate No. 2057, so that the seasonal effect is somewhat more probable.

The first set of plates (Nos. 525-1486) throws no light on the matter, for the most divergent groups in the last column of Table IV. are the first two, which were both taken chiefly in September, so that we must set down their difference to accident. Summarising the results so far, we find—

(a) That the focal position of the plate was essentially different in the two periods before and after the erection of the new dome in 1900.

(B) In the first period the plate was further from the objectglass (A = +36 for the mean of the four quarters), and satisfactorily normal to the line of collimation (values of ▲ nearly same in all quarters).

(7) In the second period the plate was nearer the O.G. (A=0 or less), and the X side (large R.A.'s) much nearer than the x side (small R.A.'s). The values of A for the two sides are about - 25 and ▲ = +25 respectively.

A=

=

Moreover, there are variations in the focal length which may be seasonal, ranging from A= 20 in June-September (plate

nearer O.G.) to A = + 12 in December-April (plate further from O.G.), though these variations may be due to a series of discontinuities.

The result that the plate is further from the O.G. in cold weather and nearer in warm seems a little strange. The telescope tube is no doubt longer in the warm weather; but we are concerned with a differential effect, depending partly on the expansion of the tube and partly on that of the lenses, which may alter their focal length more than the tube alters, and so give an apparently reversed effect. For this reason we cannot look for confirmation (or otherwise) to the observed scale-value of the plate deduced from measures of star images; for this scale value depends on the expansion of tube and plate, perhaps also on that of the reseau.

12. Let us now examine the effect of these differences on the relative number of stars photographed in the different quarterplates. We find that we have in plates 1564-2500 a mass of tolerably homogeneous material with one chief variable, the mean value of A for the whole plate, which probably corresponds to the mean distance of the plate from the O.G. Whether the variations in A are due to a seasonal effect or to accidental discontinuities need not concern us if we take ▲ itself as the independent variable.

TABLE VIII.

Total number of Stars counted on each Quarter-Plate.

TABLE VIII.-continued.

Total number of Stars counted on each Quarter-Plate.

Reducing the numbers to percentages we find for the means of

13. We have now to correct these numbers for the changes in A. Table IV. gives in the last two columns the values of n (the whole number of stars on a quarter-plate) for different values of ▲ on certain assumptions specified in § 6. We may make a small table of the percentage correction to n as follows:

Corrn.

:

A+ 60+ 50+ 40+ 30+ 20+ 10 0-10 20 30 40-50-60 ton{=+75+4°5+2·8 +17+0′9+0*3+0°1 0·0+01+10+2'5 +45 +9'7 Now the mean values of ▲ for the groups of Table IX. and for each quarter-plate, and the consequent corrections to percentage, would be as follows:- :

Subtracting the mean shown in the last column from the separate corrections for each quarter-plate (so as to keep the mean for the whole-plate zero), the numbers of Table IX. corrected would then be as in Table XI.

14. It is clear that we have not removed the chief part of the differences shown in Table IX. by this process, and we must look to other causes for them. One such cause is undoubtedly tilt of

the plate, of which no account has yet been taken in dealing with the number of stars as distinct from the position of best focus. That the tilt seriously affects the number of stars can be seen from elementary geometrical considerations. Let EAOB (fig 2) represent a section of the curved field and DA BC a position of the plate normal to the axis.

Now if the plate be tilted to the position & a By, it is clear that every portion of the half QC or Ry is brought closer to the curve of good focus, and this side of the plate will thus contain more stars all over. On the other side, regarding the plate as first moved parallel to itself to cut the curve at a, and then tilted, we have in what precedes taken account of the movement of translation, but not of the tilt; and the tilt clearly moves every part of this other half-plate further from the curve of good focus, so that we lose stars all over it. The effect of tilt is thus greater than that of translation, because by moving the plate parallel to itself we gain in one part and lose in another, whereas by tilt we either gain or lose all over the half-plate. We must qualify this statement a little when we extend the argument to two dimensions, so as to deal with a curved surface instead of the curved arc shown in the figure. But the general nature of the phenomenon remains; and we see that when a plate is tilted, the side nearer the O.G. gains at the expense of the side more remote.

15. We thus see how parts of the large differences of Tables IX. and XI. probably arise. The quarter-plates XY and Xy, which have ▲ negative, are nearer the O.G. and gain stars

from their opposites xy and Y. But it is not necessary to proceed to a quantitative estimation to see that we cannot in this way explain the whole of the differences. There must be some other contributing cause which makes the Y quadrants 4 per cent. richer in stars than the y quadrants. The difference between the mean ▲ for the Y quadrants and the y quadrants is so slight that tilt cannot explain this considerable excess; and, while reserving the study of tilt for a future paper, we may proceed here briefly to consider the possible causes of an excess of stars in the north half of a plate which cannot be explained (so far as can be seen at present) by the position of the plate.

16. The excess of stars in the N half of the plate may be due to any combination of the following causes :

(a) Optical performance of the O.G., including possible inclination to the line of collimation.

(B) Increased atmospheric absorption for S stars. (y) Actual increase in number of stars as we go northwards. When we combine all the plates in a zone, we eliminate

(to a large extent) variations of uniformity in R.A., but a change with declination may be persistent.

17. It is possible to estimate the approximate magnitude of cause (B). The formula for atmospheric absorption at Oxford of visual rays is given in Mem. R.A.S., vol. xlvii., as o'25 sec. Z.D. We may put Z.D. = 25°; and the variation of sec. Z.D. for 1°, which is the distance between N and S halves of the plate, is about o'oi. The correction is thus about 0025 magnitude. Now Newcomb gives (in his book, The Stars: a Study of the Universe, p. 283) the number of stars of different magnitudes as in the second column of Table XII., whence we get the totals of that magnitude and brighter as in the third column, from which we

form log N and its differences as in the 4th and 5th columns. It appears that at magnitude 11, log N is increasing at the rate