« PreviousContinue »
whole units. Hence it is natural to determine A from P, and the appropriate equation is
P being now measured in days, and not in units of 50 days. The most probable values of A corresponding to different periods are
For P = 100d
16. This single example will suffice to show the method of finding in such cases. There are many other points to be considered, e.g. how to deal with cases where the relation is not linear, or where one of the quantities is not measured but only described. But it seems probable that in astronomy the evaluation of r by the above simple process will be the point most often occurring.
17. But before leaving the more general topic of correlation for the particular application of it we have in view, it is a matter of some interest to return now to the simple example given in § 2, and work it out as below.
This is a very low value, and on referring to Table III. we see that the probable error is greater than o'2: so that the correlation of the quantities A and B may be regarded as unsupported. And yet at first sight there is a fair appearance of relationship. This shows the advantage of having a numerical test.
18. We proceed now to discuss the interpretation of this and similar results. And in the first place it is an obvious suggestion to inquire whether C is also related to P. In a precisely similar manner, which need not be given in detail, we find the correlation for C and P to be
r = '30'10,
and for the probable relationship
where C and P are measured in days. But the value of and its probable error show that the probability of a relationship is here more doubtful. This is quite possibly due to the much greater uncertainty of the values of C, which is more difficult to determine than A, for reasons which those who have discussed variable star observations will readily appreciate.
With this formula the values of C would be:
19. Now, having obtained the suggestion of correlation and two formulæ for A and C, I examined in detail some individual cases which had come under notice and which are given below. It will be seen that there is good confirmatory evidence, so far as it goes, of these relationships.
S Cassiopeia (Ch. No. 432).
2c. In Mem. R.A.S., lv., p. lxix, the Rousdon observations are compared with Chandler's formula in his 3rd Catalogue :
6105E+ 50 sin (10° E+50°).
From observations E= 17 to 21 (about) a period of 630'7 days is derived, differing sensibly from 606 2 given by the formula. It is remarked: "To obtain [630'7 days] from the formula we must alter the periodic term in some way, and especially we must increase the coefficient, for with the coefficient 50 the greatest interval between two consecutive maxima is the greatest value of
which is 6105 + 100 sin 5° or 619'2 days. But we cannot discuss this point without reference to other observations. Chandler used maxima in 1843 and 1863-93, and any new formula must thus
agree with the old one at these periods. Tentatively we may suggest some such modification as
2401590+615E+ 75 sin (12° E+ 90°)."
Chandler's revision gives
610.5E+37 sin (15° E+59°).
The value of the periodic term deduced from the period 610 days by the formulæ found is
47 sin (13°6 E+?).
The coefficient given by the formula is thus in accord with Chandler's first thought, and lies between his revised value and the value suggested very tentatively by the Rousdon observations. The mean of the three different suggested values, 75, 50, and 37, is 54; and the fact that the smallest of these, 37, was used in deducing the formula accounts for part of the difference between the formula value 47 and this mean.
It is perhaps worth noting that if the suggestion of C=75 (made quite independently of the present investigation) is adopted instead of Chandler's C = 37, the value of r is raised from '30 to '43.
The argument 13°6 given by the formula is between the value suggested by the Rousdon observations and Chandler's revision.
R Ursa Majoris (Ch. No. 3825).
21. In Mem. R.A.S., lv., p. lxxvi, the Rousdon observations are compared with Chandler's 3rd Catalogue formula
3021E+15 sin (10° E+ 190°).
It is shown that the Rousdon observations indicate an error of 150° in the periodic term: so that the coefficient of E should be 8° instead of 10°. Chandler's revision gives
3021E+11 sin (8° E+238°).
Pogson's observations of this star have since been very carefully discussed by Miss Blagg; and the outcome of this entirely independent discussion (which will shortly be published in Mem. R.A.S., lviii., in the introduction to Pogson's observations) was to indicate a periodic term
40 sin (8° E+?).
The formula gives for 302 days a term
27 sin (8° E+?),
in which the coefficient is midway between Chandler's revision and the Pogson indication; and the argument in good accordance with both.
T Ursa Majoris (Ch. No. 4511).
22. In Mem. R.A.S., lv., p. lxxviii, the Rousdon observations are compared with Chandler's 3rd Catalogue formula
257 2E+20 sin (9° E+ 90°),
with the conclusion that the agreement is good.
revision gives no change. The correlation formulæ give for a period of 257 days a periodic term 24 sin (7° E+?).
S Ursa Majoris (Ch. No. 4557).
23. On p. lxxx the Rousdon observations are compared with Chandler's 3rd Catalogue formula
226.1E+43 sin (5°•76 E+ 181°·5),
with the conclusion that "there is a fairly satisfactory accordance, though some correction to the formula would improve it." Chandler's revision gives
226.5E+35 sin (5°4 E+ 194°).
The correlation formulæ give for a period of 2265 days a periodic term 22 sin (6°5 E+?), which suggests that Chandler's diminution of the coefficient has not been carried far enough, though in the right direction.
S Cygni (Ch. No. 7220).
24. On p. lxxxix the Rousdon observations are compared with Chandler's formula
322 8E+ 15 sin (12° E+66°).
It is remarked that the "periods at (the Rousdon) epoch would agree better if the coefficient of E in the periodic term were smaller, say 9° instead of 12°."
In Chandler's revision the periodic term is replaced by a secular term
The correlation formulæ give for a period of 323 days
28 sin (8 E+?),
so that the only suggestion made at the time of discussing the Rousdon observations is in the direction of better accordance with the formulæ.
R Sagittæ (Ch. No. 7257).
25. This star might have been included in Table I., but it was decided to draw the line as regards "long-period variables" at 100 days. The elements given in the revision are
70 ̊56E+6·5 sin (2°25 + 47°).
Extrapolating our formulæ, we get C 12 and A=3°9. But if we decide that this star may be regarded as a long-period variable, we should include it in our table, and then the values of C and A would be found closer to those observed, since the star would have great weight. Indeed, the correlation for A and P is raised to r=0.64±07, and the formula becomes
(A — 8°·1) = 1'01 (P — 305)
which gives A=3°3 for P=70.
The value of r for C and P is, however, not much improved, being raised from 30 to 32.
R Cassiopeia (Ch. No. 8600).
26. On p. xcii of the Rousdon Memoir, the observations are compared with Chandler's 3rd Catalogue formula,
4295E+25 sin (15° E+0°),
and it is remarked :
"The corrections to period shown by the different columns of the Rousdon observations are so consistent that it is difficult to believe that the mean result can be so erroneous as the formula would make it. If the formula were altered to
431E+30 sin (12° E+9°)
[there would be a certain improvement]; but it is of course impossible to alter the formula definitively without discussing other observations."
Chandler's revision, which appeared after the above words were in type, gives
4316E+32 sin (9° E+60°).
The correlation formulæ give for the period 431 days a periodic.
35 sin (107 E+?),
showing that the improvements suggested by the Rousdon observations were in the right direction, though not sufficient in magnitude; and that Chandler's revision is in good accord with the formula.
S Delphini (Ch. No. 7431).
27. Chandler gives no indication of a periodic term in his "revision," printing the period as 277'5 days. In discussing Baxendell's observations (before the present correlation work had been undertaken at all), Miss Blagg found clear indications of a periodic term at which a preliminary guess of 9 sin (71° E+?) was made, the 9 being mere guess-work, but the 74 being indicated with