It is worth noticing, however, that the accompanying rise in the column for minima is greater, and that there are indications in some of the variable stars of another term accompanying the main inequality (see the case of S Serpentis in M.N., lxviii. p. 563).

6. To state the case as favourably as possible for the existence of a periodic inequality in the sun-spot period, we may exclude from both columns the values

from A = 14° to A = 30°, and from A = 44° to A = 66°,

and then the mean value for the periodogram for minima becomes 0'46 and for the maximum o'62. The ratio of 3:52 to its own mean is thus nearly 6, and to the mean for both maximum and minimum is nearly 7.

7. Recurring now to the quantities which give us the inequality, if it exists, we find for the formulæ representing the maxima, when A = 53° (which seems the best value),

and for the minima

- 1'4 sin 53° E-02 cos 53° E,

-0.7 sin 53° E- o'i cos 53° E.

But the epochs are not quite the same. The minima begin with 16105, and the maxima with 1616'3, or 5.8 years later, which is 58/11.1352 of a period. The phase of the inequality is thus advanced by 53° ̊ × 52 = 27° for the maxima, and to compare the expressions we must write

-14 sin (53° E-27°)-0'2 cos (53° E-27°)

=

and, as before,

13 sin 53° E+0'4 cos 53° E for maxima,

-07 sin 53° E-o'I cos 53° E for the minima.

8. In Chandler's notation, the sun-spot maxima would therefore be given by some such formula as

const + 4060 E+ 490 sin (53° E+ const),

and the question now arises how far this formula for the Sun accords with those found for the stars and quoted in § 1.

P=4060d in these formulæ, we have

C = 7d·3+0d·064 × 4060 = 267a as against 490

A=2°6+0°018 x 4060 = 76° as against 53°.

Putting

There is, of course, no reason why these formulæ should be strictly linear, so far as we know at present; and if we remark that the value of the coefficient C obtained from the minima is only about half that obtained from the maxima (i.e. in sensible agreement with the star-formula), the accordance is sufficiently good

to suggest further inquiry. One fact emerges from the discussion, viz. that from the available material it is difficult to make sure of the existence of an inequality similar to those shown by the stars. We may take it that the coefficient C is of the right order of magnitude to fit in with the star-formula: and our periodogram shows that in this case it is too small to stand out clearly from the accidental inequalities. It will not be possible to affirm or deny the existence of such an inequality with confidence until the material is improved by extending the series of observations, or possibly by reducing the accidental errors of the older observations by improved discussions of them. One hope of reducing the accidental errors proved vain. It was thought that, since there are two independent series of maxima and minima, they might be used in combination in some way, so that the effect of accidental errors of one series might be reduced by the other. But apparently the two series run together so closely that not much can be gained in this way.

9. The value A = 20° gives a value of C not much smaller than that for A = 53°; and from the material it is not easy to say which of these two possible terms corresponds to the terms found for variable stars. Are there possibly two terms in general? The case of S Serpentis has already been quoted, where the existence of a long-period term had masked the short-period term. In other cases there may be long-period terms affecting the short-period terms to a smaller extent, and this may account for some of the large deviations from the formula. And these two values A = 53° and A = 20° for the Sun may help us, by suggestion, in getting at the facts for the variables. The series of observed maxima for the Sun is much longer and more continuous than those for most of the variables; and it would not be surprising if we got suggestions from it which would help in elucidating the shorter series.

10. Assuming that the value A = 53° corresponds to the terms that have attracted attention for the variables, then the formula

A = 2°6+0°018 x P

does not hold for so large a value of P as P = 4060; for which we should get A = 75°, as remarked in § 7. Can this distant point on the curve be used to improve the formula?

II. Firstly, let us examine the consequence of assuming the formula still linear, and let us determine a and b in the expression

A=a+bP

so as to satisfy the Sun and the mean of the stars; that is, put

It seems doubtful whether our present material is sufficient to enable us to discriminate between these two formulæ, for most of the stars have periods between 200 and 400 days. But we may notice one other supposition, viz.—

12. Secondly, let us adopt the suggestion of a curve of some kind rather than a straight line. The appropriate indices for A and P will be suggested by finding m in the formula

A"= P

for the large values of A and P, i.e. for the case of the Sun. We have

m = log P/log A = log 4060/log 53

= 3.61/172: = 2'1.

This suggests some formula such as either

A2= a(P+p)

or (A + a)2=bP.

13. Determining the constants from the two cases of the Sun and mean of the stars, we find for the two suppositions

A2 073 (P-220)

(A + 9°)2=0·94 P.

Of these, the former gives impossible values of A for periods below 220 days, and is thus unsuitable. The latter gives values of A for different values of P, as below.

P= 100d 200d 300d 400d 5ood

Ag=

A1 =

=

600d

0'7 47 78 104
78 104 127 148 (new formula).

5.6 6.8 8.0 9'2 10'4

116 (linear formula).

- 4'9 2 I -0°2 +12 +23 +32.

These differences are larger than the former, and it seems probable that we can discriminate even now in favour of the original formula.

SUMMARY.

The sun-spot maxima (and minima) occur on the average at intervals of 11.125 years. But the individual maxima (and minima) show discordances which have been tabulated for 26 periods by Wolfer. Analysing these by the periodogram method of Professor Schuster, there are indications of two periodicities, one of which the phase advances 53° per period of 11 years, the other of which the phase advances only 20°. The cycles are completed in about 75 and 200 years respectively. The amplitude of each inequality is about a year, but the accidental errors are so large that either or both of these inequalities may be spurious.

The quicker moving inequality (53) can be brought into line with similar inequalities for the long-period variables; the best formula connecting A (advance of phase in degrees per period) with P (the period in days) being the simple linear formula

A=4°4+0 ̊012 P.

The slower moving inequality (20°) may quite possibly have analogies in the stars, but as yet the material is not sufficient to declare.

Researches on the Solar Constant and the Temperature
of the Sun. By Dr. J. Scheiner, Assoc. R.A.S.

In No. 55 of the Publications of the Astrophysical Observatory, Potsdam, I have published an extended paper on this subject, and I should like to give a short report of the results to the readers of the Monthly Notices.

The measures of the Sun's radiation were made with the Ångström Electric Compensation Pyrheliometer, to which I had given a modified exterior form and a parallactic motion with clockwork. On eleven days in June and July 1903 I made a long series of observations on the top of the Gorner Grat in Canton Wallis (Switzerland), from which I could derive the radiation of the Sun outside our atmosphere. This part of the problem is the most difficult one, and, according to my view, it cannot be solved from measurements of the solar radiation alone. From such observations a portion only of the real solar constant can be obtained, because only that portion of the loss by absorption in our atmosphere can be calculated which is based upon the continuous increase of absorption with growing thickness of the atmospheric layer traversed by the radiation. With carbon dioxide and water vapour there exists a nearly sudden absorption in the highest thin layers of the atmosphere, which must be treated as a constant to be added to the

result from the radiation-curves. Therefore this latter result is not the solar constant as generally supposed, and I have chosen for it the term "Strahlungsconstante or "Constant of Radiation." From my observations on the Gorner Grat it amounts to 195-2'02 gr. cal. The remaining constant, which must be added to it for obtaining the Solar Constant, can be found only from experimental researches in the laboratory. To this part of the problem I have devoted much labour in measuring the absorption of carbon dioxide and superheated water vapour with varying depth of layer.

This very complicated research cannot be described in a short abstract, and I must therefore refer to the original paper. The result is that for reducing the Radiation Constant to the Solar Constant there must be added for carbon dioxide 1%, for water vapour 7%, and for the ultra violet absorption 1%, whence the Solar Constant for the unit of distance is found to be 2'22-2°29 gr. cal., with a probable error of 2%.

The constant of the Stefan law, which is necessary for calculating the effective temperature of the Sun from the Solar Constant, I have ascertained by different methods, and with the same pyrheliometer, thus eliminating the constant error of the apparatus which cannot exceed 1 per cent. The "black radiation" of known temperature was measured from black platinum, rendered incandescent by an electric current, from light flames of different thicknesses, and from the artificial "black body." The latter results were the most exact and the effective temperature of the Sun based upon them was found to be 6196°-6252°.

Further on, I have endeavoured to calculate the real temperature of the solar photosphere from the effective temperature by employing the known data on the absorption of the solar atmosphere. Of course this research cannot be of the same exactness as the foregoing one, especially because the photosphere has no definite temperature, consisting as it does of layers of very different temperature.

Neglecting the errors arising from our ignorance of the structure of the photosphere, its average temperature comes out as 7065°.