be deduced from as large a number of days as can be conveniently observed; every day in the month would not be superfluous. This could easily be done at Cambridge, if provision were made for conducting the two hourly observations on every day according to the English plan. But these daily observations of the magnetometer have not been attempted, and would be impossible with the present resources of the Observatory.* It is not surprising that what occurred once in six months' observations on the temperature, should have exhibited itself also in the diurnal curves of the declination magnetometer. This brings up the case of the September magnetic curve, which we have excluded from any part in determining the mean quantities, because it was calculated to injure the just average values which we are endeavouring to reach. The curve in question may be traced out on Plate V, by following the directions there given, and a single glance will show how the times of its extreme elongations differ from those of the other three mean curves, which are on the same plate. The general appearance of these three curves indicates a law to which the fourth curve must be a palpable exception and transgression. An attention to the separate diurnal curves for the five days in September, from which the mean curve is reduced, will afford an explanation of this violation of what we may regard a principle of the earth's magnetism. On September 20-1, (magnetic day, commencing, as in all cases, at 10 P.M., Gott. M.T.,) we have these three maxima:

The whole range of the magnetometer during this day is 45'.5. The smallest elongation from the meridian is at 8h 26′ P.M., and the greatest at 4h 56′ A.M., as the lowest number of the scale indicates greatest declination. But these, like the rest of the maxima and minima of this day, are mere lawless excursions caused by sudden derangements, and have no connexion with the regular diurnal

A Director, with three assistants, all of whose time should be devoted to the work, have been considered, elsewhere, as the full personnel of such an establish

ment.

magnetic curve. Between the third and fourth minima the bar moves to 109', or 14' in arc, in the space of 20 minutes, and then falls back eastward again. So between the fifth and sixth minma the bar moves to 98'.5 of the scale, or 18' forward, and back again in 1 8'. The time occupied in going from the second or greatest maximum to the seventh minimum is only 1 10', although the space is 36', or three times the ordinary daily range of the magnetic meridian for this month. If these perturbations were less frequent and not spread over the whole 24 hours, means might be devised of shutting out their influence, and interpolating the true daily curve in the gap left by their removal. But on this day no safe way of making the reduction presents itself, and the times of maximum and minimum are both left indeterminate.

On September 22-3, the magnetometer was more quiet; occasional oscillations of considerable extent occur, but the maximum point is very regularly formed at 11 36' A.M., and the minimum, though less definite, is placed at 6h 06′ A.M. On September 24th, westerly disturbances take place between 4 36′ and 7 36′ A.M., the time when the minimum generally shows itself; the effect is to bring the mean curve at this period too far to the west, or to make the apparent mean time of maximum earlier than it is in fact, or would appear if the observations were free from irregular variations. These derangements do not cease till nearly the close of the third magnetic day, so as to throw uncertainty on the time of maximum of this day also.

September 24-5, (Magnetic day.) Disturbances break out again in strange forms. This day was affected by wholly unprecedented motions; distinguished not so much for their extent as for their number and the rapidity with which they succeeded one another. A faint idea of them is conveyed by looking at those which occurred between 7 20' A.M., and 7 40' A.M., Gott. M.T., of October 22nd, represented on plate IV. The direction of the motion changed 60 times or more, from east to west and back, between 1 and 2 P.M., Gott. M.T. The whole sweep of the needle for this day is only 26', an area much less than is often traversed in the diurnal motion; but the number and frequency of the oscillations is unparalleled. The greatest declination occurs at 3h 60′ A. M. Cambridge M. T., and the least at 3 36' P. M.; but another minimum, more nearly resembling the regular daily minimum, appears at 5 36′ A.M.

September 25-6, (Magnetic day.) The magnetic storm has subsided. The curve for this day is as quiet as that of the 4th day on Plate IV. The eye readily perceives, that now we have only regular diurnal changes, and that the times of greatest and least elongation from the astronomical meridian can be trusted. minimum is at 6 26' A.M., and the maximum at 0h 16' P.M., Cambridge M.T.

The

We have here ample and abundant explanation of that singular figure assumed by the mean daily curve for September. We look

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,

d the ordinate of the diurnal curve for the time t,

n = the ratio of the circumference to the diameter,

*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 n (t + c2).

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:

Dh 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

m representing the number of intervals on each day.

2. m C cos. 2 π n c1 = 2 S' D sin. 2 πnh. 3. m С sin. 2 π n c1 = 2 S′' Dâ cos. 2 π n h.

S Dh

m

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 + c). 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 n c, and multiplying equation 2nd by cos. 2

nc, and equation 3rd by sin. 2 n c1, and adding them together, we readily find the value of C. 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

curve:

DAC, sin. 2 (t + c1) + C2 sin. 4 (t+ca) + C3 sin. 6 (t + C3).

The empirical thermometric curve is calculated on the same principle by this formula :—

π

D=B + D, sin. 2 π (t + d1) D2 sin. 4 TM (t + d2) + D3 sin. 6 π

(t + ds).

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, C3; 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