Page images
PDF
EPUB

been found by the use of mean months, the mean temperature of the 29th day of February can be interpolated with as much accuracy as that of any other day whatever. The time elapsed from the beginning of the year to the middle of the intercalary day is 31+28++ of 0·2422-59·1211 days, and the corresponding abscissa is found to be 58° 16'.

The monthly means of temperature at New Haven, as given in the Transactions of the Connecticut Academy of Arts and Sciences, from 86 years' observations, are

[blocks in formation]

From these data I have obtained the equation of mean daily. temperatures throughout the year, in the way already stated in this Journal, xli, 373, except that instead of finally reducing it from the usual form,

y=a+a, sin(x+E1)+a2 sin(2x+E2)+a ̧ sin(3x+E3)+ &c., into a form where the signs before the terms are sometimes plus and sometimes minus, I have reduced it to

y=a+a, sin(x-e1)+a2 sin2(x-e2)+a ̧ sin3(x—ez)+ &c.,

in accordance with the formula

[blocks in formation]

This prevents confusion of signs, and at the same time preserves the significance of the arc e,, making it measure the time elapsed from the beginning of the year to the first ascending node of the term in which it occurs.

The New Haven equation of temperatures then is

y=4911222.902 sin(x-110° 39′ 22′′)+289 sin 2(x-20° 56′)
+443 sin 3(x-57° 42')+022 sin 4(x-75° 22')
+402 sin 5(x-3° 53')+093 sin 6.x.

An equation of this kind, to be perfect, ought to express accurately all the facts implied in the observed series of monthly means, so that the mean for any one of the calendar months may be derived from it with precision, by integrating yda between the proper limits for the beginning and end of the month, and dividing by the arc which measures its length. Let the general form,

3

y=a+a, sin(x-e ̧)+a2 sin 2(x—e2)+a ̧ sin 3(x-e3)+ &c., be treated in this way between the limits x' and ' correspond

ing to the beginning and end of a month whose mean is m, and let us make

[ocr errors][ocr errors][merged small]

sin a

sin 2α

sin3a

sin(B-e,)+az sin2(B-2)+α3 sin3 (6-3)+ &c.

m=a+a,

α

sin a sin 2α

The values of

&c., depend only on the length of the

α

month, and their logarithms are given in the subjoined table, for months of all the different lengths.

[blocks in formation]

The values of the arc ß, which measures the time from the beginning of the year to the middle of a month, are for the calendar months

[blocks in formation]

and for mean months they are 15°, 45°, 75°, &c.

344 43 21

Now in the expression for the monthly mean m, let the constants a, a,, e,, a,, e,, &c., take those values which have been found for them in the New Haven equation, and the following monthly means for the calendar months may be obtained,

[blocks in formation]

The errors of these computed values as compared with the monthly means actually observed are

[blocks in formation]

This example shows the degree of accuracy with which an equation obtained by the method of mean months may be expected to represent any observed series of means for calendar

months. The reason why the computed values do not agree exactly with the observed ones is, that the curve of the form

y=a+a, sin(x-e1),

by means of which my system of twelve equations was obtained, is only an approximation to the true curve for any month, and is not the same as the computed curve whose equation contains twelve constants. The two curves approach each other closely, and intersect at several points, but they do not coincide. They both include the same monthly mean for the mean month, but not for the calendar month. It is probable, however, that no other method of reduction of equal simplicity will give an equation which expresses the means for the calendar months so accurately as this.

It should be noticed that when monthly means of rain-fall are to be corrected for the inequality of the months by my system of equations, the correction must be applied not to the mean total amount of rain for any month, but to the mean daily amount for that month. Take, for instance, the results of 24 years' observation at Albany, from 1826 to 1849 inclusive, given by F. B. Hough in the "New York Meteorology." The mean total amounts of rain and melted snow and hail for the calendar months, in inches of depth, are

[blocks in formation]

Dividing each of these by the number of days in the month, we have the following values of the mean daily rain-fall, for calendar months, in decimals of an inch:

[blocks in formation]

Now applying the correction, we obtain the mean daily rain-fall

for mean months,

[blocks in formation]

and the equation of the curve is found to be

y=1123+0202 sin(x-106° 43')+0106 sin 2(x-107° 10′)
+0112 sin 3(x-14° 11')+0031 sin 4(x-51° 7')
0024 sin 5(x-56° 40′)+0015 sin 6x.

If we assign to x the value appropriate for any given day in the year, the resulting value of y will be the average depth of rain-fall at Albany for that day, expressed in decimals of an inch.

After an equation has been obtained, there ought to be some

check to show whether it is free from errors of computation. This may be secured by deriving from it the mean for any one of the mean months. When the constants in the Albany equation are transferred to the general expression for the monthly mean, and a and take the values appropriate for the third mean month for instance, the result is m=0975; the agreement of this with the daily mean for the third mean month as previously found, is evidence that the equation of the curve has been computed correctly.

January 11th, 1867.

ART. XXXVI.-Researches on Solar Physics; by WARREN DE LA RUE, Esq., Pres. R.A.S., BALFOUR STEWART, Esq., Super intendent of the Kew Observatory, and BENJAMIN LOEWY, Esq., Observer and Computer to the Kew Observatory.

Second Series (in continuation of First Series).* 'Area-measurement of the Sun-spots observed by Carrington during the seven years from 1854-1860 inclusive, and deduction therefrom.

34. In our first paper (Art. 13) we stated that Mr. Carrington had very kindly placed at our disposal all his original drawings of sun-spots. Our first step was to arrive at some estimate of the accuracy of these sketches, and we requested Dr. von Bose, who assisted Mr. Carrington in the greater part of his observations, to give us a short outline of the method employed in obtaining them.

From his account, it would appear that the sun's disk was thrown upon a screen, and that each group as represented on the screen was separately drawn on a sheet of paper. The groups on paper were then each separately compared with those on the screen and modified where faulty; and this process was continued until the paper sketches agreed as nearly as possible with the groups on the screen. It would thus appear that very great care was taken with these sketches. [Engravings of several of Carrington's sketches alongside of those of corresponding groups as taken by the Kew Heliograph are given in the original memoir, showing that Carrington has obtained by the method above described a very great accuracy of delineation.]

35. The trustworthiness of Carrington's sun-pictures being thus established, it seemed to us that the labor of measuring for each group the amount of spotted area would be well bestowed, inasmuch as the method hitherto employed, namely, the mere statement of the number of sun-spots occurring at any pe* From a memoir printed for private circulation; tables and plates, and many paragraphs omitted. For First Series, see p. 179.

riod, can only be supposed to afford very approximate means of estimating the extent of solar activity at that period; while, again, if we wish to study the behavior with respect to size of each group as it passes over the visible disk, this can only be done accurately by the laborious but sure method of meas

urement.

36. Method adopted in measuring Carrington's groups. In order to accomplish this task, the following method was adopted :-In the first place, in order to obtain the apparent area of any group, a piece of plate glass had a number of lines etched upon it, by means of which it was cut up into squares, the side of each square being th of an inch. In order to facilitate reading, each fifth line was painted red.

This piece of glass was then applied (the engraved face toward the drawing) to the group whose apparent area it was desired to measure, and the number of squares and fractional parts of a square occupied by the umbra, the penumbra, and the whole spot was separately reckoned and noted down. If it was found that the number of squares reckoned for the whole spot was equal to the sum of those reckoned for the umbra and penumbra together, it was concluded that the measurement was correct. This method of checking the accuracy of the measurement had the further advantage of giving separately the areas of the umbra and penumbra, thus affording determinations which may be made use of in advancing our knowledge of the subject, although not used by us in our present research.

37. But it is evident that after the apparent area of a group has thus been correctly estimated, this apparent area will not indicate the real size of the group, unless allowance is made for the foreshortening occasioned by its angular distance from the visual center of the disk.

[The practical methods by which this allowance for foreshortening was made are given in detail. The final results of the measurements form an extensive table and give the material for a graphical representation of the observed spotted area for each clear day from the beginning of 1854 to the end of 1860.]

40. Distribution of Spotted Area over Disk.-Our next inquiry has reference to the relative distribution of spotted area over different parts of the solar disk. We use the word disk in contradistinction to surface, because it is evident that, on account of the sun's rotation, the center of his visible disk on one day does not represent the same portion of the solar surface as on another day; indeed from this cause it is well known that sun-spots travel over the visible disk from left to right. It is therefore one inquiry to study from day to day the relative distribution of spotted area over different parts of the sun's actual surface, and another to study the same from day to day over different

« PreviousContinue »