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for the time when some measure shall be devised for evading the errors which such extraordinary changes of the magnetie declination entail on the mean values of the regular variations. If hereafter the dependence of the magnetic declination upon the hour of the solar day shall be so accurately discovered as to be reduced to a formula, we may be able, by the help of that portion of the curve which is undisturbed, to calculate the remainder. At present this formula must be an empirical one, derived from the faulty observations themselves, and in its defective state is available only in a partial degree, for purifying these observations. Our chief resource now lies in levelling, as far as possible, the excessive excursions by the influence of undisturbed days with which they are combined ; though this can be done only by sacrificing, in part, the more perfect observations. The case in hand teaches us that this method will not always be effectual in bringing out approximate results. The irregularities may be so great as to over-rule the regular law. This is less likely to happen in proportion to the number of days that can be observed in each month ; and hence, again, the necessity of deducing our means from as numerous observations as can be obtained.
The dependence of the diurnal magnetic changes on solar time rests upon the evidence of a large number of observations, collected from remote sources. But there is a difficulty in conceiving of the exact manner in which this connection is sustained. Perhaps it will always be a hopeless task to attempt to trace the intricate path by which the heat deposited at one moment in the centre of our system, arrives at its final result of causing a deviation in the direction of the magnetic meridian. And while this is the case, it will be impossible to enter upon the mathematical analysis of the problem, and deduce formula which can be used for detecting the errors of theory, or correcting or supplying the deficiencies of observation, according to the well-known relation subsisting between these different methods of investigation. But the artifices of analysis will frequently take hold of cases which cannot be approached by any direct process. The observations allow us to proceed upon the ground that the declination, or the ordinate of the diurnal curve of declination, is a function of the solar day. It may, then, like any other periodic function, be supposed to be expressed in a series of terms, arranged according to the sines and cosines of the time and its integral multiples.* Thus, if
t = the time expressed in parts of a day as its unit,
* It was according to this mathematical developement that Professor Peirce calculated the empirical curves.
and if S denote the sum of the terms which correspond to the different values of n, we have for the general form :
D= A + S.C. sin. 2 in (t + cm). The values of A, C, and c, are readily determined by the following formula. Let observations be taken at equal intervals for several whole days, and let
h = time of observation, counted from the beginning of each
magnetic day, in parts of a day as unity : Dhe . = the mean of the observations taken at the time h of each day.
Then, if S' denote the sum of all the terms which correspond to the different values of h, we have 1. mA= S'D
m representing the number of intervals on each day. 2. m C. cos. 2 m ca = 2 S D, sin. 2 = m h. 3. m C. sin. 2 an cn = 2 S' Dh cos. 2 anh.
There is no known periodic function which does not admit of developement according to the sines and cosines of the time and its integral multiples, and, in the absence of positive evidence, the same thing may be assumed in regard to that under present consideration. The constant A, being equal to is the mean of all the partial results obtained from observation for the several intervals into which the day is distributed for this purpose. By substituting different values for n, we obtain an indefinite number of terms out of the general one C. sin. 2 + n (t + cp). It appears, however, from the calculation, that the series rapidly converges, so that the first four or five terms are sufficient to give the declination within a degree of exactness corresponding to the accuracy of the observations themselves. Dividing the 2nd equation by the 3rd, we have the value of the tang. 2 in Cn, and multiplying equation 2nd by cos. 2 an cn, and equation 3rd by sin. 2 An Cn, and adding them together, we readily find the value of Cn. Thus, if the numbers 1, 2, 3, be successively taken for n, we shall have the following equation for finding the approximate declination, or the empirical magnetic
D= A + C, sin. 2 * (t + cu + Csin. 4 . (t + ca) + Cz sin. 6 *
(t + c).
The empirical thermometric curve is calculated on the same princi-
(t + dz).
Plates IV, and VI, will show how rapidly the series of both formulæ converge, and the limit of error incurred by dropping all the terms after the 5th. In the formulæ for October, the 5th term of the declination cannot exceed ,034 of a minute, and the 5th term in the value of the temperature cannot be greater than ,4 of a degree of Fahrenheit. From the nature of an empirical curve, our confidence in it must bear some proportion to the accuracy of the observations. If these observations are exposed to errors from any cause, as we have seen that they are, the empirical curve will suffer, though in a less degree, on their account. The error which in a single diurnal curve is left in its naked state, is of course diminished in the mean curve of several days by the levelling influence which all the days exercise upon any single one. But this process reduces : it does not extinguish the error. The passage
from the mean of the observed curves to the empirical curve, carries us one step further towards the true expression of the actual phenomena of magnetism. For a considerable mean error arising from irregular disturbances, which in the first is concentrated upon a single moment, will be in the second curve distributed over the whole day, and may therefore disfigure the general character of the day, though it does not distort extremely any particular part. Moreover, it is easy, in calculating the values of the constants in the empirical formula, to omit observations of an extraordinary character, and which are notoriously burdened with strange anomalies. This we see on Plate V, in the instance of the September days, and to a less extent in October. The whole character of the curve for the former is changed from what we have reason to believe is the real diurnal curve; although it has escaped those large and prominent excursions which appear three or four times in the mean of the observed curves. In seasons of great disturbance it would be more safe to rely on the empirical curve than the observed curve; but in quiet times, as the empirical curve borrows all its truth and expression from these observations, the latter have more claim to consideration than the calculated places. It is obvious, from the principle on which the empirical curve rests, and the manner in which the constants are deduced, that they will answer only for one curve, and must be calculated separately for every new curve that is required. As the form of these equations, and the time, which is the only variable, are the same for each curve, whatever changes exist in the diurnal curve from one month to another in the year, must be indicated by a corresponding change in the independent constants. And moreover, if there be, as the comparison of recent and old observations lead us to believe, secular periods for the magnetic declination, they will betray themselves by slow variations in the mean yearly values of these same constants. It becomes, then, an object of curious inquiry to ascertain what are the values of A, C1, C2, Cz; C1, C2, C3, &c., for every month in the year; and after this their mean values from one year to another. It is possible that
the laws of the secular changes may be better studied from the variations of these constants than from immediate observations. Four of these formulæ are here given, with the names of the months to which they belong, and the number of days employed in calculating them; t = the time from of Gott. M. T. June, 10 days. Declination* = 90°17',3---3,853 sin.(t-16-2124)-1',537 sin. 2(t-9531"24')
-0',948 sin. 3 (t+0 27"9*)—0'644 sin. 4(t—422"9"). August, 4 days. Declination =9°13',9—3,907 sin. (t—15412m47')—2'009 sin. 2 (1—9646"584)
0'878 sin. 3 (1+0'51"15'). September, 5 days. Declination =9°21',9--2',932 sin. (t–.9"34"18")—1',530 sin. 2(t-8*12*8')—
0',494 sin. 3 (t+1932-29")—1',090 sin. 4 (t–0-29"58"). October, 5 days. Declination =9°18',7—1',575 sin.(t-1360"42 )-2,379 sin. 2(t-10*40*584)
0',508 sin. 3 (t–04"58*)-0',034 sin. 4 (t+0612"32"). Here we close our investigation of the diurnal magnetic curve. The existence of such a curve, regularly formed every day, cannot be doubted; its general uniformity is also very observable. The limits of the times of maximum and minimum declination in different longitudes, show conclusively that it is in some way connected with local solar time. Developing the declination according to the most general form of periodic functions, we have obtained the preceding formulæ, from which the empirical curves drawn on Plate V were calculated. These calculated curves stand there side by side with the mean observed curves, by which the constants of the formulæ were determined. The calculated curve, as we might expect, is less broken than the mean curve; still, the two agree in a striking manner, and the greatest deviations are in those months which suffered most from magnetic perturbations. In June and August the empirical curve and the mean curve keep close together, and these were periods of unusual magnetic repose ; for in the mean of the latter month the disturbed term day was omitted. June was most quiet of the two, and shows it by a superior agreement between its mean and empirical curve. If there were no permanent change of declination, but only the daily oscillation uninterrupted by disorderly fluctuations, the meridian would swing, day after day, through the same arc; and a few observations would be sufficient to establish a rigorous formula, which would evolve an
The first term in the value of the declination is obtained directly in parts of the scale, and is afterwards reduced to absolute numbers in the usual way of deriving the real declination from the reading of the scale. This process will be soon explained.
empirical curve strictly coincident with the observed curve. The want of this uniformity is felt in the variation of the constants of the formulæ already given. This simplicity does not exist in the motions of the heavenly bodies any more than in the magnetic movements. But the analysis is different. In astronomy we know the cause of the disturbance, and allow for it at once, without deranging the general analytical expression. In the other case we have no theory, no hypothesis ; and the mathematical form must vary with the observations. Hence the difference between the constants in the formulæ for the four months. They are no greater than might be expected from the known change of absolute declination from day to day, the limits of the times of maxima and minima, and the longer and more irregular derangements which beset the diurnal movement. The mean curves of many months, drawn from the most abundant materials, are requisite for investigating the law by which these constants vary, and rendering them available for calculating the secular periods of the earth's magnetism. We think it is apparent from all that has been adduced, that the diurnal magnetic curve is as clearly a function of solar time as the daily thermometric curve. We are not to expect any greater uniformity in the effect than in the cause. If the thermometric curve is sometimes imperfectly formed, the same thing may happen to the magnetic curve without destroying our belief in its connexion with the sun. The change of constants in one class of formulæ appears likewise in the other, as the three following thermometric formulæ make manifest:
August, 5 days. Temperature = 67°,6+8°,8 sin. (t—15457" 28") + 0°,9 sin. 2 (t-434" 56') +
1°,1 sin. 3 (1+0 44" 56'). September, 5 days. Temperature = 50°,2-10°,0 sin. (1-13438" 54)—3°, 2 sin. 2 (1-931" 4*)
0°,3 sin. 3 (1-21)-0,8 sin. 4-439" 48*). October, 5 days. Temperature = 47°,7—30,8 sin. (t-14" 21" 44")-0°,8 sin. 2 (1-9"17" 36")—
0°,4 sin. 3 (t-635" 25)-0°,4 sin. 4 (t-038"12").
It still remains to discuss briefly those disturbances of the magnetic meridian which have no apparent law. We have occasionally alluded to them as irregular purturbations which produce perplexity in ascertaining the true diurnal curve. We are to inquire whether even they must be regarded as wholly inexplicable, or whether they cannot be connected in coincidence of time at least with other wellknown phenomena of nature. There are few days in the year when strange fluctuations of greater or less amount are not exhibited; but there are some periods distinguished above all others by