Scientific Memoirs/2/General Theory of Terrestrial Magnetism

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Article V.

General Theory of Terrestrial Magnetism. By Professor Carl Friedrich Gauss, of the University of Göttingen.

[Translated by Mrs. Sabine, and revised by Sir John Herschel, Bart.]

The unwearied zeal with which, in recent times, endeavours have been made to examine the direction and intensity of the magnetic force of the earth, at all parts of its surface, is the more worthy of admiration, as it has been prompted by the pure love of science. Great as is the importance to navigation of the most complete attainable knowledge of the lines of declination, more than this is scarcely required for its purposes. Whilst science delights to render such useful services, her own requisitions have a wider scope, and make it necessary that equal efforts should be devoted to the examination of all the magnetic elements.

It has been customary to represent the results of magnetic observations by three systems of lines, usually termed Isogonic. Isoclinal, and Isodynamic lines. In course of time these lines undergo considerable alterations both in position and in figure, so that a drawing of them represents the phænomena correctly only for the epoch to which it corresponds. Halley's Chart of Declination for 1700 is very different from that of Barlow for 1833; and already Hansteen's Dip Chart for 1780 differs greatly from the present position of the Isoclinal lines. Doubtless, in course of time, similar alterations in the lines of intensity will be manifested; but observations of this nature are altogether too recent to furnish such indications at present.

In all these maps there exist spaces either blank, or in which the lines are but indifferently supported by observation. The inaccessibility of parts of the earth's surface renders perfection in this respect impossible; but a rapid progress towards it may be confidently hoped for.

Viewed from the higher grounds of science even a complete representation of the phænomena after this manner is not itself the final object sought. It is rather analogous to what the astronomer has accomplished, when, for example, he has observed the apparent path of a comet in the heavens. Until the complicated phænomena have been brought in subjection to a common principle, we have only building-stones, not an edifice.

The astronomer, after the comet has disappeared from his view, begins his chief employment, and resting on the laws of gravitation, calculates from the observations the elements of its true path, and is thus enabled to predict its future course. And in like manner the magnetician proposes to himself as the object of his research, as far as the different and in some respects less favourable circumstances permit,—the study of the fundamental causes which produce the phænomena, their magnitude and their mode of operation,—the subjection of the observations, as far as they extend, to those elementary principles,—and the anticipation, with some approximation at least, of their effects, in those regions where observation has not yet penetrated. It is at least well to keep in view this higher object, and to endeavour to prepare the way for it, even though the great imperfection of the data may render its attainment impossible at present.

It is not my purpose here to notice the earlier fruitless attempts to explain the enigma of these phænomena by hypotheses having no physical foundation. A physical foundation can only be allowed to such attempts as have considered the earth as a real magnet, and have employed in the calculation only the demonstrated mode of action of a magnet operating at a distance. All attempts of this nature hithert omade have this in common;—that instead of first examining what the conditions, whether simple or complex, of this great magnet must be to satisfy the phænomena, certain determinate and simple conditions were presupposed, and the subject of inquiry has been the accordance or non-accordance of the phænomena with these presupposed conditions. We see here a repetition of what has often occurred in the early history of astronomy and of other sciences.

The simplest hypothesis of this kind is that which supposes a very small magnet in the centre of the earth; or rather (as it is not likely that any one ever believed in the actual existence of such a magnet) supposes magnetism to be so distributed in the earth, that its collective action at and beyond the surface is equivalent to the action of an imaginary infinitely small magnet; much as gravitation towards a homogeneous sphere is equivalent to the attraction of a sphere of equal mass condensed in its central point. In the supposed case, the magnetic poles are the two points where the prolonged axis of the little central magnet intersects the earth's surface; where the magnetic needle is vertical and the intensity is also greatest. In the great circle midway between these two poles called the magnetic equator, the dip is and the intensity is half as gi-eat as at the poles; between the magnetic equator and either pole, both the dip and the intensity depend on the distance from the said equator (which distance is termed the magnetic latitude) in such manner, that the tangent of the dip is equal to twice the tangent of the magnetic latitude. Lastly, the direction of the horizontal needle must everywhere coincide with the direction of a great circle drawn through the northern magnetic pole.

There is in nature only a rude approximation to all these necessary consequences of the above hypothesis. In reality the line of no dip is not a great circle, but a line of double flexure; equal intensities do not correspond to equal dips; the directions of the horizontal needle are far from all converging to one point; and so on. A very slight consideration is sufficient therefore to show the inadmissibility of this hypothesis.

One of the above propositions is however still employed as an approximation in deducing the line of no dip from observations of dips of small amount made at some little distance from it.

About eighty years ago, Tobias Mayer used a similar hypothesis, but with this modification; that instead of supposing the infinitely small magnet at the centre of the earth, he placed it at about the seventh part of the earth's radius from the centre; at the same time (probably in order to avoid greater complication in the calculations) he retained the wholly arbitrary supposition, of the plane perpendicular to the axis of the magnet passing through the centre of the earth. In this manner, on a comparison of the observed variations and dips, at a very small number of places it is true, he found them agree very well with his calculation. A more extended comparison would have shown that this hypothesis did not afford a much better representation than the first-mentioned one, of the whole phænomena of the dip and declination. No observations of the intensity had been at that time made, at least as far as we know.

Hansteen went a step further, by the endeavour to represent the phænomena on the hypothesis of two infinitely small eccentric magnets of unequal strength. The decisive test of an hypothesis must always be the comparison of its results with those of experiment. Hansteen compared his with observations at forty-eight different places, amongst which however there were only twelve at which the intensity had been determined, and only six complete in the three elements. In these comparisons we find in the dip differences of 13° between calculation and observation[1].

If these differences are greater than are admissible in a satisfactory theory, one cannot avoid drawing the conclusion, that the magnetic conditions of the earth are not such as to admit of representation by means of a concentration in either one or two infinitely small magnets. It is not denied that with a greater number of such fictitious magnets, a sufficient agreement might be ultimately attainable; but how far such a mode of solving the problem might be advisable is quite a different question. The calculations are extremely laborious even with two magnets; with an increased number they would probably present insuperable difficulties. It will be best to abandon entirely this mode of proceeding, which reminds one involuntarily of the attempts to explain the planetary motions by continued accumulation of epicycles.

In the present treatise it is my purpose to develope the general theory of terrestrial magnetism independently of all particular hypotheses as to the distribution of the magnetic fluids in the body of the earth; and to communicate the results which I have obtained from the first application of the method. Imperfect as these results must be, they give an idea of what may be hoped for in future, when trustworthy and complete observations from all parts of the earth shall be obtained, and employed in renewed and more refined attempts.

1.

The force which at each part of the earth imparts a certain direction to a magnetic needle suspended by its centre of gravity, (supposing it free from all extraneous influence, such, for example, as that of another artificial magnet, or the conductor of a galvanic current,) is termed the earth's magnetic force, in so far as the source whence it is derived is to be sought for in the earth itself. It may indeed be doubted, whether the seat of the proximate causes of the regular and irregular changes which are hourly taking place in this force, may not be regarded as external in reference to the earth. We may hope, that from the general attention now directed to these phænomena, much light may shortly be thrown upon their causes. But it should not be forgotten that these changes are comparatively very small, and that there must therefore exist a much more powerful and constantly acting principal force, of which we assume the seat to be in the earth itself. A consequence which follows from this consideration is, that the facts which are to serve as the foundation on which the study of the principal force must be based, ought properly themselves to be first freed from the effects of the anomalous changes. This can only be done by mean values, drawn from numerous and continued observations; and until we shall possess such purified results, from a great number of stations distributed over the whole surface of the globe, the utmost that can be looked for is an approximation, in which there must still remain differences of the order of these anomalies.

2.

The foundation of our researches is the assumption, that the terrestrial magnetic force is the collective action of all the magnetized particles of the earth's mass. We represent to ourselves magnetization as a separation of the magnetic fluids. Admitting this representation, the mode of action of the fluids (repulsion of similar and attraction of dissimilar particles inversely as the square of the distance) belongs to the number of established physical truths. No alteration in the results would be caused by changing this mode of representation for that of Ampère, whereby, instead of magnetic fluids, magnetism is held to consist in constant galvanic currents in the minutest particles of bodies. Nor would it occasion a difference if the terrestrial magnetism were ascribed to a mixed origin, as proceeding partly from the separation of the magnetic fluids in the earth, and partly from galvanic currents in the same; inasmuch as it is known, that for each galvanic current, may be substituted such a given distribution of the magnetic fluids in a surface bounded by the current, as would exercise in each point of external space precisely the same magnetic action as would be produced by the galvanic current itself.

3.

For the measurement of the magnetic fluids we take, as in the Intensitas Vis Magneticæ, &c., for our positive fundamental unity, that quantity of northern fluid which at the unit of distance exercises on an equal quantity of the same fluid a moving force equivalent to what we assume as unity.

When we speak of the magnetic force which in any point of space is produced by the action of the magnetic fluid elsewhere, we always mean to speak of the moving force which is there exercised on the unity of the positive magnetic fluid; therefore in this sense the supposed magnetic fluid concentrated in a point exercises at the distance the magnetic force , of either repulsion or attraction in the direction , according as is positive or negative. Representing by , , , the co-ordinates of in relation to three rectangular axes, and by , , and , the co-ordinates of the point where the force is exercised, so that

;
and resolving the force in parallels to the co-ordinate axes, the components are

which, as is easily seen, are equal to the partial differential co-efficients of relatively to , , and .

If besides , there are also in operation other portions of the magnetic fluids , , , &c., concentrated in points, of which the distances from the spot where the force is exercised are , , , &c., then the components of the whole resulting magnetic force, parallel to the co-ordinate axes, are equal to the partial differential co-efficients of

,

relatively to , , and .

4.

Hence may easily be shown what magnetic force is exercised in each point of space by the earth, however the magnetic fluids may be distributed therein. Imagine the whole volume of the earth, as far as it contains free magnetism (that is to say, separated magnetic fluids), to be divided into infinitely small elements; designate generally the quantity of free magnetic fluid contained in each of these elements by , in which the southern fluid is always considered as negative; call the distance of from a point in space, the rectangular co-ordinates of which may be , , ; lastly, let denote the aggregate of comprehending with reversed signs the whole of the magnetic particles of the earth: or say

.
Thus has in each point of space a determinate value, or it is a function of , , , or of any other three variable magnitudes, whereby we may define points in space. We then obtain, by the following formulæ, the magnetic force in every point of space, and the components of , parallel to the co-ordinate axes, which we shall call , , ,

.


5.

I shall first develope some general propositions which are independent of the form of the function , and are worthy of attention from their simplicity and elegance.

The complete differential of becomes

.


If we designate by the distance between the two points to which and belong, and by the angle which the direction of the magnetic force makes with , we have

,

because as , , are the cosines of the angles which the direction of makes with the co-ordinate axes, so , , , are the cosines of the angles between and the same axes.

Therefore is equal to the force resolved in the direction of ; the same follows from the equation if we bear in mind that the co-ordinate axes may be arbitrarily chosen.

6.

If two points in space , , be connected by an arbitrary line, of which d s represent an indeterminate element, and if, as before, signify the angle between d s and the direction of the magnetic force there existing, and its intensity, then

if we extend the integration through the whole line, and designate by , , the values of at the extremities.

The following corollaries of this fruitful proposition deserve especial notice:—

I. The integral preserves the same value by whatever path we proceed from to .

II. The integral , extended through the whole length of any re-entering curve, is always .

III. In a re-entering curve, if is not throughout , a part of the values of must be greater and a part must be less than .

7.

Those points of space in which has a value greater than , are divided from those in which the value of is less than , by a surface in all the points of which has one determinate value [2].

It follows from the proposition in Art. V., that in each point of this surface the magnetic force has a direction perpendicular to the surface, and towards the side where the higher values of are found. Let be an infinitely small line perpendicular to the surface, and the value of at its other extremity; then the intensity of the magnetic force will be . The series of points for which , form a second surface infinitely near to the first, and at different points in the whole intervening space the intensity of the magnetic force is in the inverse ratio of the distance apart of the two surfaces.

Let alter by infinitely small but equal steps. A system of surfaces will be produced, dividing space into infinitely thin strata, and the inverse ratio of the thickness of the strata to the intensity of the magnetic force will then hold good not only for different points in one and the same stratum, but also for different strata.

8.

We will now take into consideration the values of on the surface of the earth.

At a point of the earth's surface let be the intensity; the direction of the whole magnetic force; the intensity, and the direction of the force projected on the horizontal plane, or the direction of the magnetic meridian, meaning thereby the direction indicated by the north pole of the magnetic needle; the angle between and , or the dip; , , the angles formed by the elementary portion of a line on the surface of the earth and the directions , . Lastly, and correspond to the two extremities of .

We have consequently

And the equation in Art. V. becomes


If two points on the earth's surface and , at which has the value of and , are connected by a line traced on the surface of the earth of which is an indeterminate element, then

,

if the integration be extended through the whole line; and it is plain that three corollaries hold good similar to those in Art. VI., namely,

I. That the integral keeps the same value by whatever path you proceed on the surface of the earth from to .

II. The integral throughout the whole length of a closed line on the surface of the earth is always .

III. In such a closed line, unless throughout its course , a part of the values of must necessarily be acute and a part obtuse.

9.

Propositions I. and II. of the foregoing article (which, properly speaking, are only different modes of expressing the same thing) may be tested, at least approximately, by a reference to observation.

Let be a polygon on the surface of the earth, the sides of which are the shortest lines that can be drawn between their respective extremities, and are therefore portions of great circles, the earth being here considered simply as a sphere. Let , , , &c. be the intensities of the horizontal magnetic force at the points , , , &c.; further, let , , &c. be the declinations reckoned in the usual manner, west of north as positive, east of north as negative; lastly, let be the azimuth of the line at , reckoned in the customary manner, from the south by the west; in like manner the azimuth of the same line taken backwards at , and so on.

Let it be observed that alters continuously in each of the sides of the polygon, but suddenly at the corners, where therefore ithas two different values; for example, at , has the value , in consideration that is the end of the line ; and the value of , in regard that it is the beginning of .

We may consider the approximate value of the integral , extended through , to be

,
where and signify the values of at as the beginning, and at as the end of . This approximation is all that can be obtained, because we have the values of and t</math> only at the extremities , and is deserving of confidence in proportion to the shortness of the line. The given expression is, in our notation,

.
In like manner, the approximate value of the integral, extended through , is

,

and so on through the whole polygon.

Therefore, for a triangle our proposition gives the approximatively correct equation

It is obvious that in this equation the units of intensity and of distance are arbitrary.

10.

As an example, we will apply the formula to the magnetic elements of

Göttingen
Milan
Paris
whence it follows that


Taking the geographical position of

Göttingen 51° 32′ latitude 58′ longitude from Greenwich
Milan 45 28 9 09
Paris 48 52 2 21

and performing the calculation for a spherical surface only, we find

(01) = 11′ 31″ = 5′ 20″
(10) = 184 35 35
(12) = 128 47 31 = 5 41 06
(21) = 303 48 01
(20) = 238 20 20 = 5 32 04
(02) = 64 10 12
Substituting these values in our equation, and those given above for , , , we have


Hence we deduce from the observed horizontal intensities at Göttingen and Milan, that at Paris , agreeing almost exactly with the observed value .

It is easily seen that if we permit ourselves to take the distances , , &c. instead of their sines, the above formula can be expressed immediately by the geographical longitudes and latitudes of the places.

11.

The line on the earth's surface, in all points of which has the same value , divides generally speaking the parts of the surface in which the value of is greater than , from those in which it is less. The direction of the horizontal magnetic force in each point of this line is obviously perpendicular to it, and towards the side where the greater values of are found. If be an infinitely small line in this direction, and the value of at the other extremity of this line, then is the intensity of the horizontal magnetic force at this place. As here also the series of points corresponding to the value of forms a second line situated infinitely near to the first, and thus marks out on the surface of the earth a zone, within which the values of are between and , and where the horizontal intensity is in an inverse ratio to the breadth of the zone; so by making vary by infinitely small but equal steps from the lowest value on the surface of the earth to the highest, the whole surface of the globe becomes divided into an infinite number of infinitely narrow zones, the direction of the horizontal magnetic force being everywhere perpendicular to the dividing lines, and its intensity being in an inverse ratio to the breadth of the zone at the place in question. The two extreme values of correspond in this point of view to two points, inclosed by the zones, at which the horizontal force is , and where therefore the whole magnetic force can only be vertical: these points are termed the magnetic poles of the earth. The lines dividing the zones are no other than the intersections of the surfaces considered in Article VII. with the surface of the earth, whilst it is only at the poles that they are in contact with it.

12.

The form of the system of lines described in the above article is strictly but the simplest type, which might be subject to many exceptions were we to take into account every possible distribution of magnetism in the earth. We shall not, however, exhaust this subject here, but shall only add a few elucidatory remarks as to the cases of exception. The magnetic condition of the earth, no doubt is such, that the form of the system of lines on its surface corresponds to the description. At least there are certainly no exceptions on the great scale, though probably there may occur local ones. Some philosophers have considered the earth as having two north and two south magnetic poles, but it does not appear that an essential condition was previously fulfilled, by a precise definition being given of what should be understood by a magnetic pole. We intend to apply this denomination to each point of the earth's surface where the horizontal intensity : where therefore, speaking generally, the dip ; but including the singular case, did it exist, where the total intensity . If we were to give the name of magnetic poles to those places where the total intensity is a maximum (i. e. greater than anywhere in the surrounding vicinity), it must not be forgotten that this is something quite different from the above definition; that neither the situation nor the number of these last-named points have any necessary connexion with those of the points first spoken of; and that it tends to confusion when dissimilar things are called by the same name. If we look away from the actual condition of the earth and take the question in its generality, there may certainly exist more than two magnetic poles; but it does not appear to have been noticed that if, for example, two north poles exist, there must necessarily be between them yet a third point, which is likewise a magnetic pole, but is properly neither a north nor a south pole, or is both if that expression be preferred. A consideration of our system of lines will best serve to elucidate this subject. If the function have at a point of the earth's surface a maximum value , and all around smaller values, then a series of progressively decreasing values will correspond to a system of rings, each of which will inclose all the preceding ones, together with the point , and on each of these rings the direction of the horizontal magnetic force, or that of the north pole of the magnetic needle, will be inwards[3].

This is the characteristic mark of a magnetic north[4] pole.

It is clear that the rings may be made so small, or the corresponding values of the function may differ so little from , that any other point may be excluded.

We will designate by the space included by all the points on the surface of the earth at which the value of is greater than a given value , It is clear that may either be one connected surface or several detached spaces, and that , on the bounding lines or lines which separate from other parts where is less than ; by increasing or diminishing , we enlarge or contract the space .

Now let us assume to be a second point of similar properties to so that at it also may have a maximum value . As according to what has been before noticed, may have a value less than , and differing from it by so small an amount that shall fall outside that part of in which is situated; then if we arrange (as we may do) that shall not be less than , it will be greater than , and will necessarily also belong to a part of . Thus and will both be situated in , but in separate portions of it. On the other hand, it is evident that may be taken so small that and shall both be situated in one connected part of ; for by only taking small enough, may be made to embrace the whole surface of the earth.

If then be made to pass progressively through all the values between the first and the second values spoken of, there must be amongst them one which we will call , characterised by being the lowest at which and are still situated in separate portions of , which separate portions will unite whenever is diminished further.

If this union occur at a point , the bounding line on which will have the form of an , crossing at that point; where also we may easily satisfy ourselves that the horizontal intensity must . In fact, the crossing either does or does not take place under an angle of sensible amount.

In the first case, the horizontal force, if it be not , must be directed in the normal to the two different tangents, which is absurd; in the second case, in which the two halves of the touch each other at , or would have the same tangent, the force normal to this tangent could only be directed towards the interior of one half surface of the , which involves a contradiction, as the value of increases towards both sides; therefore is a true magnetic pole according to our definition, but must be considered as a south pole as regards the points nearest to it inside the two openings of the , and as a north pole as regards the points which lie outside. Figure 1. illustrates this form of the system of lines.

If the junction take place at two different points, what has been demonstrated for one point would hold good for the two; and one may easily see that inside the space inclosing and an insular space would be formed, which would gradually contract itself as was diminished, and would necessarily at length resolve itself into a true south pole.

The case is similar when the junction takes place at three or more separate points; but if it take place at once on a whole line, then the horizontal force must disappear on all the points of that line.

It is evident that the assumption of two south poles would in like manner necessitate the existence of a third polar point, which would be neither a south pole nor a north pole, or rather would be both at once.

13.

From what has been developed in the foregoing article, its application to many conceivable exceptions from the simplest type of our system of lines will be readily understood. The whole of the points to which a certain value of corresponds, may be a line consisting of several portions, of which each returns back into itself, but which are quite separate from each other; it may be a line crossing itself; lastly, it maybe a line having on both sides spat es where is greater than on the line, or where it is less.

We may assert that on the earth there are, on the great scale, no deviations of such a nature from the simplest type.

Local deviations, indeed, may well be supposed to exist. Magnetic masses near the surface, though producing no sensible effect at any considerable distance, may obscure and even obliterate the regular progress of the terrestrial magnetic force in their immediate vicinity. In the simplest case the system of lines in such a district might take the form represented in Figure 2.

14.

After this geometrical representation of the relations of the horizontal magnetic force, we proceed to develope the mode of submitting them to calculation. On the surface of the earth becomes a simple function of two variable magnitudes, for which we will take the geographical longitude reckoned eastward from an arbitrary first meridian,—and the distance from the north pole of the earth; we will designate the first of these, or the longitude, by , and the second, or the complement of the geographical latitude, by . Considering the earth as a spheroid of revolution, of which the greater semi-axis , and the lesser semi-axis , an element of the meridian is

;
and an element of the parallel is

Resolving the horizontal magnetic force into two portions, one of which, , acts in the direction of the geographical meridian, and the other, , perpendicularly to that meridian,—and considering as positive when directed towards the north, and as positive when directed towards the west,—then

.
The total horizontal force is then

,
and the tangent of the declination

.
Neglecting the square of the ellipticity, , the expressions become

or, setting the ellipticity quite aside,


.


The data furnished by the observations which we possess are much too scanty, and most of them much too rude, to make it advisable at present to take into account the spheroidal form of the earth. It would not be difficult to do so; but it would complicate the calculations without affording any corresponding advantage. We will therefore adhere to the last-mentioned formula, in a which the earth is considered as a sphere, whose semi-diameter .

15.

If be expressed by a given function of and , can be be deduced from it a priori.

Let the integral , considering as constant in the integration: it is then clear that if we differentiate in a similar manner according; to , ; having a value independent of , or, what is the same thing, constant in all the points of a meridian,—it must hence also be absolutely constant, because all meridians converge and meet at the poles.

If we call the value of at the north pole , then ;
and hence .
This result may also be expressed as follows:


16.

This remarkable proposition, that, if the component of the horizontal magnetic force directed towards the north be given for the whole surface of the earth, then the component directed towards the west (or towards the east) follows of itself, is true, conversely, only with a certain modification. If be expressed by a given function of and , and if represent the indeterminate integral being assumed constant in the integration, then , or has a value independent of , and is, generally speaking, a function of . Thus is such a function; that is to say, the formula gives an imperfect expression for , a part of it containing only remaining undetermined. This want would be supplied if, besides the expression for , we had also that for , for some one given meridian, or to speak generally, for some line extending from the north to the south pole. We see therefore that, if we know the component of the horizontal magnetic force in the direction towards the west for the whole of the earth's surface, and the component in the direction towards the north for all points of some one line extending from the north pole to the south pole, the latter component, for the whole of the earth's surface, follows of itself.

17.

The foregoing investigations apply only to the horizontal portion of the earth's magnetic force. In order to embrace the vertical force also, we must consider the problem in all its generality; therefore must be regarded as a function of three variable magnitudes, expressing the position in space of an undetermined point . We select for the purpose the distance from the centre of the earth, the angle which makes with the northern part of the earth's axis, and the angle , which a plane passing through and the axis of the earth makes with a first meridian, counted as positive towards the east.

Let the function be expanded into a series, decreasing according to the powers of , and to which we give the following form:

, &c.


The co-efficients , , , &c. are here functions of and ; in order to see how they are connected with the distribution of the magnetic fluid in the earth, let be an element of the earth's magnetism, its distance from , and let , , , signify for the same as , , for . We have then extended so as to include every ; further , and if be developed in the series,

then


As , and as according to the fundamental supposition with which we set out, the quantities of positive and of negative fluid are equal in every measureable particle in which they exist, and therefore are equal in the whole earth; that is to say, , it follows that

,

or the first number of our series for goes out.

We see further that has the form

,

where , , . Therefore, according to the explanation laid down in page 13 of the Intensitas Vis Magneticæ, , , are the moments of the earth's magnetism, in relation to three rectangular axes, of which the first is the axis of the earth, and the second and the third are the equatorial radii for longitudes 0 and 90°.

The general formulæ for all co-efficients of the series for may be assumed as known; it is merely necessary for our purpose to remark, that in relation to , , the co-efficients are rational integral functions of , and , and of of the second order, of the third, &c. It is the same as to the co-efficients , , &c.

The series for , and for , converge, so long as is not less than , or rather, not less than the half diameter of a sphere, which includes all the magnetic particles of the earth.

18.

The function being composed of , satisfies the following partial differential equation:

,
which is only transformation of the well-known equation

,
where , , signify the rectangular co-ordinates of . If we substitute the value of ,

, &c.,
it is clear that for the several co-efficients, , , , &c., there will likewise be partial differential equations, of which the general expression is

.


From this equation, combined with the remark in the preceding article, we obtain the general form of . If we represent by the following function of ,

then has the form of an aggregate of parts,

where , , , , &c. are determinate numerical co-efficients.

19.

If the magnetic force at the point be resolved into three forces perpendicular to each other, , , and , of which is directed towards the centre of the earth, and and are tangential to a spherical surface concentric with the earth, passing through , directed northwards in a plane passing through and the axis of the earth, and directed westwards in a plane parallel to the equator of the earth, then

consequently,

On the surface of the earth and are the same horizontal components which we have designated above by those letters; is the vertical component, which is positive when directed downwards.

The expressions for these forces on the surface of the earth are, then,



20.

If we combine, then, with these propositions, the known theorem, that every function of and , which, for all values of , from 0 to 360°, and of , from to 180°, has a determinate finite value, may be developed into a series of the form

the general member of which, satisfies the above partial differential equation,—that such a developement is only possible in one determinate manner,—and that this series always converges,—we obtain the following remarkable propositions.

I. The knowledge of the value of at all points of the earth's surface is sufficient to enable us to deduce the general expression of for all external space, and thus to determine the forces , , , not only on the surface of the earth, but also for all external space.

It is clearly only necessary for this purpose to develope into a series according to the above-mentioned theorem.

In the sequel, therefore, unless it is expressly stated otherwise, the symbol is always to be taken as limited to the surface of the earth, or as that function of , and which follows from the general expression, when is made : thus


II. The knowledge of the value of at all points of the earth's surface is sufficient to obtain all that has been referred to in Prop. I. In fact, according to Art. 15, the integral signifying the value of at the north pole, and the developement of into a series of the form referred to must necessarily be identical with


III. In like manner, and under the considerations in Art. 16, it is clear that the knowledge of on the whole earth, combined with the knowledge of at all points of a line running from one pole of the earth to the other, is sufficient for the foundation of the complete theory of the magnetism of the earth.

IV. Finally, it is clear that the complete theory is also deducible from the simple knowledge of the value of on the whole surface of the earth. In fact, if be developed into a series,

&c.
so that the general member satisfies the often-mentioned partial differential equation; must necessarily , and

&c.


21.

On account of the simple nature of the dependence of the several forces , , , on a single function , and the simple relation which they bear to each other, they are far better calculated to serve as a foundation for the theory, than the usual expression of the magnetic force by the three elements, total intensity, inclination, and declination. Or rather, the latter mode, natural as it appears in itself when the question is solely that of comprehending the facts, cannot directly serve for the foundation of the theory (at least not for the first foundation) until it has been translated into the other form.

In this view it would be very desirable that a general graphical representation of the horizontal intensity should be made; partly because it would be more immediately useful for theory than the total intensity; partly because, in far the greater number of cases, the horizontal intensity was originally that which was actually observed, the total intensity having been subsequently deduced from it by means of the dip. It is the more advisable to keep the elements of the horizontal force unmixed, as they can be determined with extreme accuracy with the present instrumental means; at any rate, the observed horizontal intensity should never be suppressed when publishing the deduced total intensity, without at least giving the dip employed in the calculation; so that a person wishing to employ the horizontal intensity for the theory may either have, or be enabled to reproduce, the original observed numbers.

Interesting as it would be to found the theory of terrestrial magnetism on observations of the horizontal needle only, and thus to anticipate the vertical part, or the inclination, it is at present much too soon to do so : the scantiness of the data which we now possess does not allow of our dispensing with the assistance of the vertical part. It is a confirmation of the theory, if we can show the agreement of the different elements when reduced to one principle.

22.

Although we are a priori certain that the series for , , , , converge, nothing can be determined beforehand as to the degree of convergence. If the seats of the magnetic forces be limited to a moderate space around the centre of the earth, or if there were such a distribution of the magnetic fluids in the earth as to be equivalent thereto, the series would converge very rapidly; on the other hand, the further the seats of the magnetic forces extend towards the surface, and the more irregular the distribution, the slower we must be prepared to find the convergence.

In the practical application, absolute exactness is unattainable; we have to desire only a degree of approximation commensurate with the circumstances. The slower the convergence, the greater will be the number of members which must be taken into account to attain a certain degree of accuracy.

Now, contains three members, and requires, therefore, the knowledge of three co-efficients , , ; requires five co-efficients; seven; nine, &c. As we consider , , , &c. as magnitudes of the first, second, and third order, and so on, it is clear that if the calculation is to be pushed to magnitudes of the order inclusive, the values of co-efficients must be determined; therefore, for example, 24 coefficients, if we would go as far as the fourth order.

Every given value of , , or , for given values of and , furnishes an equation between the co-efficients, whilst for each place where the complete elements of the terrestrial magnetic force are known, three equations are given. If we could venture to assume that the members have a sensible influence only as far as the fourth order, complete observations from eight points would be sufficient, theoretically considered, for the determination of all the co-efficients. But such a supposition can hardly be ventured upon, and the accidental errors which beset all observations, together with the neglected members of higher orders, might have a very injurious effect on the results of the elimination[5].

To diminish the unfavourable effect of these circumstances, the number of series of observations from stations well distributed over the whole globe ought to be much greater than that of the unknown values, and these should be derived from the observations by the method of least squares. As all the equations are only linear, the process would, it is true, be uniform; but the extent of the labour, arising from the great number of unknown values and equations, would be such as might well deter the most courageous calculator from undertaking it in this form, especially as the result might be wholly vitiated by the introduction either of defective observations or of accidental errors of calculation.

23.

There is another mode of proceeding, which, as it is free from a part of these difficulties, appears better adapted for a first trial. We shall develope it in this place without omitting to notice objections to which its application may be liable in the present state of the inquiry. This method supposes the knowledge of all three elements at points so grouped on a sufficient number of parallels as to divide them into a sufficient number of equal portions. The numerical values of , , and , are to be first deduced from the given elements of the usual form.

The values of , , , are then brought by the known method in each parallel to the form


We then obtain as many values for each of the co-efficients , , , , &c., as there are parallels of latitude under consideration.

Theory would give in each parallel ; therefore the values of which result from the calculation furnish a kind of measure of the degree of uncertainty which still attaches to the fundamental members.

From the equations

the total number of which is double the number of the parallels, we have to obtain, by the method of least squares, (after substituting in , &c., and in , &c. the corresponding numerical values of ,) as many of the co-efficients , , , &c. as require to be taken into account.

In like manner the equations

the number of which is three times as great as the number of parallels, serve to determine the co-efficients , , , &c. And the following,

determine the coefficients , , , &c. Further, the equations

determine the co-efficients , , , &c.; and we obtain the co-efficients of the succeeding higher numbers in a similar manner.

24.

The chief advantage which this method possesses over that given in Art. 22, consists in the unknown values being broken into groups, each of which is determined by itself, whereby the calculation is greatly facilitated; whereas, in the other mode of proceeding, the intermingling of all the unknown quantities renders their separation extremely difficult. On the other hand, the disadvantage of the second method is, that, instead of being founded on direct observation, it rests on graphical representations, which, in districts where we do possess observations, represent them but rudely, and which, in districts where observations are wanting, are only conjectural, and, to a certain degree, arbitrary, and may therefore differ considerably from the truth. However, we must either postpone all attempts, till such time as we shall be provided with far more complete and trustworthy data than we now possess, or, with our present very scanty means, make a first attempt, from which we are entitled to expect little more than a rough approximation. A close comparison of the results of calculation with those of actual observation in all parts of the earth, furnishes a certain standard by which our success may be estimated. And if this test shall show that the first attempt has not entirely failed, it will powerfully assist suitable preparations for future fresh attempts by either method.

25.

Several years ago I repeatedly began attempts of this kind, from all of which the great inadequacy of the data at my command forced me to desist. I might earlier have concluded such an essay if I had obtained the fulfilment of my often-expressed wish for a general map representing the horizontal intensity. This want could not be supplied by the combination of the imperfect general maps of dip and of total intensity, then existing.

The appearance of Sabine's Map of the Total Intensity (in the Seventh Report of the British Association for the Advancement of Science) has stimulated me to undertake and complete a new attempt, which must be regarded, however, only in the light mentioned in the foregoing article. The data employed in the calculations are for twelve points on seven parallels. They are taken for the intensity from the above-mentioned map; for the declination from Barlow's map (Phil. Trans. 1833); and for the inclination from Horner's map (Physikalisches Wörterbuch, Band vi.). Considerable portions of these maps still remain blank, and we can only fill these up with the greatest uncertainty. It was soon found that the calculation must be pushed at least as far as magnitudes of the fourth order, making the number of co-efficients to be determined amount to twenty-four. In all probability, members of the fifth order will also be found influential; but, in a first attempt, the values of , , , &c. must be still too much charged with errors, arising from the uncertainty of many of the data (and which from their nature these values involve), to permit the introduction of a still greater number of unknown values in the process of elimination.

It should be remarked that the intensities in Sabine's map are expressed according to the arbitrary unity in common use, by which the total intensity in London . In these calculations, and in the tables given in the sequel, this unity has been altered so as to make all the numbers a thousand times greater, the intensity in London on which they rest being made . It is obvious that a unity for the intensity may be taken at pleasure, since the unity for may be considered as arbitrary, and made to accord therewith. If further deductions are desired requiring to be reduced to absolute measure, it will only be necessary to multiply all the co-efficients by the factor which reduces to an absolute measure the intensities expressed according to the arbitrary unity.

26.

The numerical values of the 24 co-efficients obtained by the first calculation, the longitude , being reckoned east from Greenwich, are as follows:

These numbers, which may be considered as the elements of the theory of terrestrial magnetism, are used both here and in the formation of the table to be described in the sequel, just as they were given by calculation, without omitting decimals. To any one conversant with calculation it is superfluous to remark, that these fractional parts have in themselves no value, as we are still far from being able to eliminate with certainty even the integers. But it is important that the observations should be closely compared with one and the same definite system of elements; and, as by leaving out decimals nothing would be gained in point of convenience in computing, there was no reason for altering in any respect the elements given by calculation.

27.

The expression for , developed according to the above numbers, is as follows: for the sake of brevity stands for , and for .


We may here add the completely developed expressions for the three components of the magnetic force.



After these components have been calculated for a given place, we obtain in the following manner the several parts of the determination of the magnetic force, according to the customary form.

Let be the declination, the inclination, the total, and the horizontal intensity. Determine first and by means of the formulæ

,
and then and by means of the following formulæ:

.


28.

As the formulæ for , , , contain 71 members, their immediate calculation is a considerable labour. Its repetition for a great number of places appears the more alarming, considering that we could hardly hope to be secure from the possibility of mistake without going twice over the whole. But little would be gained by suppressing all those members of which the co-efficients are less than an integer, or even less than 10 integers, for the remaining members would still amount to 65. But as the whole value of the work would remain uncertain if not tested by a considerable number of actual observations, I have encountered the labour of calculating a table, by the assistance of which the work will be in the highest degree abridged and facilitated, and at the same time the important object of security against errors of calculation will be materially promoted.

For the construction of the table the values of the coefficients were brought into the following form:


The first table contains those parts of and which are independent of . In the four next tables are found the values of the auxiliary angles , , &c., and the logarithms of , , &c., all for the several degrees of latitude . The table is placed at the end of the memoir.

The calculation for Göttingen is given as an example.

For latitude we find from the tables:

And for longitude , the parts of , , , are found as follows:

X Y Z
+ 500-8 + 1465-2
- 35-71 + 152-89 - 68-99
+ 54-76 + 9-92 - 133-67
- 2-21 + 28-77 + 8-27
- 3-92 + 0-19 + 3-90
X = + 513-72 Y = + 191-77 Z= + 1274-71
The farther calculation then gives:


29.

The following table contains the comparison of our formulæ, with observations at 91 stations in all parts of the earth. As the three maps from which we have taken the data for our calculation are intended to represent the phænomena for the most recent epoch, we have included in our comparison only very recent observations, and we have taken, by preference, observations at those stations where all the three magnetic elements were observed. We are not at present in a condition to demand that the observations should be strictly cotemporaneous, unless we would see our stock reduced to a very small number.

Latitude. Longitude. Declination.
Computed. Observed. Difference.
° ′ ° ′ ° ′ ° ′ ° ′
1 Spitzbergen + 79 50 11 40 + 26 31 +25 12 + 1 19
2 Hammerfest 70 40 23 46 + 12 23 + 10 50 + 1 33
3 Mag. Pole of Ross. 70 5 263 14 − 22 23
4 Reikiavik 64 8 338 5 + 40 12 + 43 14 − 3 2
5 Jakutsk 62 1 129 45 + 0 5 + 5 50 − 5 45
6 Porotowsk 62 1 131 50 + 0 4 + 4 46 − 4 42
7 Nochinsk 61 57 134 57 − 0 3 + 2 11 − 2 14
8 Tschernoljes 61 31 136 23 0 0 + 3 30 − 3 30
9 Petersburg 59 56 30 19 + 6 47 + 6 44 + 0 3
10 Christiania 59 54 10 44 + 19 55 + 19 50 + 0 5
11 Ochotsk 59 21 143 11 − 0 18 + 2 18 2 36
12 Tobolsk 58 11 68 16 − 7 19 − 10 29 + 3 10
13 Tigil River 58 1 158 15 − 4 20 − 4 6 − 0 14
14 Sitka 57 3 224 35 − 28 45 − 28 19 − 0 26
15 Tara 56 54 74 4 − 7 44 − 9 36 + 1 52
16 Catharinenburg 56 51 60 34 − 5 20 − 6 18 + 0 58
17 Tomsk 56 30 85 9 − 7 21 − 8 64 + 1 13
18 Nishny Novogorod 56 19 43 57 + 1 10 − 0 27 + 1 37
19 Krasnojarsk 56 1 92 57 − 5 49 − 6 40 + 0 51
20 Kasan 55 48 49 7 − 1 7 − 2 22 + 1 15
21 Moscow 55 46 37 37 + 4 26 + 3 2 +1 24
22 Königsberg 54 43 20 30 + 14 15 + 13 22 + 0 53
23 Barnaul 53 20 83 56 − 7 0 − 7 25 + 0 25
24 Uststretensk 53 20 121 51 + 1 29 + 4 21 − 2 52
25 Gorbizkoi 53 6 119 9 + 1 5 + 2 54 − 1 49
26 Petropaulowsk 53 0 158 40 − 3 34 − 4 6 + 0 32
27 Uriupina 52 47 120 4 + 1 16 + 4 4 − 2 48
28 Berlin 52 30 13 24 + 18 31 + 17 5 + 1 26
29 Pogromnoi 52 30 111 3 − 0 38 + 0 18 − 0 56
30 Irkutsk 52 17 104 17 − 2 27 − 1 38 − 0 49
31 Stretensk 52 15 117 40 + 0 54 + 2 52 − 1 58
32 Stepnoi 52 10 106 21 − 1 52 − 1 8 − 0 44
33 Tschitanskoi 52 1 113 27 0 0 + 1 13 − 1 13
34 Nertschinsk 51 56 116 31 + 0 42 + 2 53 − 2 11
35 Werchneudinsk 51 50 107 46 − 1 26 − 0 24 − 1 2
36 Orenburg 51 45 55 6 − 2 48 − 3 22 + 0 34
37 Argunskoi 51 33 119 56 + 1 22 + 3 44 − 2 22
38 Göttingen 51 32 9 56 + 20 28 + 18 38 + 1 50
39 London 51 31 359 50 + 25 37 + 24 0 + 1 37
40 Nertschinsk Mine 51 19 119 37 + 1 20 + 4 6 −2 46
41 Tschindant 50 34 115 32 + 0 34 + 2 14 − 1 40
42 Charazaiska 50 29 104 44 − 2 9 − 2 27 + 0 18
43 Zuruchaitu 50 23 119 3 + 1 18 + 3 11 − 1 53
44 Troizkosawsk 50 21 106 45 − 1 34 − 0 12 − 1 22
45 Abagaitujewskoi 49 35 117 50 + 1 8 + 2 54 − 1 46
46 Altanskoi 49 28 111 30 − 0 16 + 0 48 − 1 4
47 Mendschinskoi 49 26 108 55 − 0 56 + 0 12 − 1 8
48 Paris 48 52 2 21 + 24 6 + 22 4 + 2 2
49 Chunzal 48 13 106 27 − 1 30 − 1 6 − 0 24
50 Urga 47 55 106 42 − 1 26 − 1 16 − 0 10
Inclination. Intensity.
Computed. Observed. Difference. Computed. Observed. Difference.
° ′ ° ′ ° ′
1 + 82 1 + 81 11 + 0 50 1·599 1·562 + 0·037
2 77 19 77 15 + 0 4 1·545 1·506 + 0·039
3 88 48 90 0 − 1 12 1·717
4 80 40 77 0 + 3 40 1·527
5 74 36 74 18 + 0 18 1·661 1·697 − 0·036
6 74 27 74 0 + 0 27 1·658 1·721 − 0·063
7 74 12 73 37 + 0 35 1·653 1·713 − 0.060
8 73 48 73 8 + 0 40 1·648 1·700 — 0·052
9 70 25 71 3 — 0 38 1·469 1·410 + 0·059
10 72 4 72 7 — 0 3 1·456 1·419 + 0·037
11 71 36 70 41 + 0 55 1·621 1·615 + 0·006
12 70 13 71 1 — 0 48 1·575 1·557 + 0·018
13 69 55 68 28 + 1 27 1·583 1·577 + 0·006
14 76 30 75 51 + 0 39 1·697 1·731 — 0·034
15 69 46 70 28 — 0 42 1·586 1·575 + 0·011
16 68 24 69 16 — 0 52 1·535 1·523 + 0·012
17 70 33 70 55 — 0 22 1·613 1·619 — 0·006
18 67 9 68 41 — 1 32 1.469 1·442 + 0·027
19 70 24 71 0 — 0 36 1·638 1·657 — 0·019
20 67 13 68 25 — 1 12 1·477 1·433 + 0·044
21 66 45 68 57 — 2 12 1·446 1·404 + 0·042
22 67 19 69 26 — 2 7 1·410 1·365 + 0·045
23 67 50 68 10 — 0 20 1·591 1·605 — 0·014
24 68 32 68 11 + 0 21 1·609 1·656 — 0·047
25 68 32 68 22 + 0 10 1·611 1·660 — 0·049
26 65 31 63 50 + 1 41 1·521 1·489 + 0·032
27 68 17 67 53 + 024 1·612 1·667 — 0·055
28 66 45 68 8 — 1 22 1·391 1·367 + 0·024
29 68 25 68 8 + 0 17 1·616 1·640 — 0·024
30 68 17 68 14 + 0 3 1·616 1·647 — 0·031
31 67 55 67 38 + 0 17 1·606 1·649 — 0·043
32 68 12 68 10 + 0 0 1·615 1·663 — 0·048
33 67 56 67 42 + 0 14 1·609 1·668 — 0·059
34 67 43 67 11 + 0 32 1·604 1·635 — 0·031
35 67 55 68 6 — 0 11 1·612 1·657 — 0·045
36 63 14 64 44 — 1 30 1·461 1·432 + 0·029
37 67 10 66 54 + 0 16 1·595 1·655 — 0·060
38 66 43 67 56 — 1 13 1·388 1·357 + 0·031
39 68 54 69 17 — 0 23 1·410 1·372 + 0·038
40 66 59 66 33 + 0 26 1·593 1·617 — 0·024
41 66 35 66 32 — 0 3 1·592 1·650 ­— 0·058
42 66 45 66 56 — 0 11 1·599 1·643 — 0·044
43 66 12 66 13 — 0 1 1·584 1·626 — 0·042
44 66 38 66 19 + 0 19 1·597 1·642 — 0·045
45 65 33 64 48 + 0 45 1·577 1·583 — 0·006
46 65 46 65 20 + 0 26 1·585 1·619 — 0·034
47 65 48 65 31 + 0 17 1·587 1·630 — 0·043
48 66 45 67 24 — 0 39 1·389 1·348 + 0·041
49 64 42 64 29 + 0 13 1·574 1·612 — 0·038
50 64 25 64 4 + 0 21 1·571 1·583 — 0·012
Latitude. Longitude. Declination.
Computed. Observed. Difference.
° ′ ° ′ ° ′ ° ′ ° ′
51 Astrachan + 46 20 48 0 + 1 40 + 1 12 + 0 28
52 Chologur 46 0 110 34 — 0 20 + 0 49 — 1 9
53 Ergi 45 32 111 25 — 0 6 + 1 7 — 1 13
54 Milan 45 28 9 9 + 20 56 + 18 33 + 2 23
55 Sendschi 44 45 110 26 — 0 20 + 0 30 — 0 50
56 Batchay 44 21 112 55 + 0 16 + 0 59 — 0 43
57 Scharabudurguna 43 13 114 6 + 0 32 + 0 46 — 0 14
58 Naples 40 52 14 6 + 18 53 + 15 20 + 3 33
59 Chalgan 40 49 114 58 + 0 42 + 1 13 — 0 31
60 Pekin 39 54 116 26 + 0 58 + 1 48 — 0 50
61 Terceira 38 39 332 47 + 25 17 + 24 18 + 0 59
62 San Francisco 37 49 237 35 − 16 22 − 14 55 − 1 27
63 Port Praya 14 54 336 30 + 16 17 + 16 30 − 0 13
64 Madras 13 4 80 17 − 4 1
65 Galapagos Island − 0 50 270 23 − 8 57 − 9 30 + 0 33
66 Ascension 7 56 345 36 + 14 37 + 13 30 + 1 7
67 Pernambuco 8 4 325 9 + 5 58 + 5 54 + 0 4
68 Callao 12 4 285 46 − 9 6 − 10 0 + 0 54
69 Keeling's Islands 12 5 96 55 + 0 23 + 1 12 − 0 49
70 Bahia 12 59 321 30 + 3 12 + 4 18 − 1 6
71 St. Helena 15 55 354 17 + 18 48 + 18 0 + 0 48
72 Otaheite 17 29 210 30 − 5 45 − 7 34 + 1 49
73 Mauritius 20 9 57 31 + 11 9 + 11 18 − 0 9
74 Rio de Janeiro 22 5 316 51 − 1 11 − 2 8 + 0 57
75 Valparaiso 33 2 288 19 − 13 45 − 15 18 + 1 33
76 Sydney 33 51 151 17 − 7 51 − 10 24 + 2 33
77 Cape of Good Hope 34 11 18 26 + 27 24 + 28 30 − 1 6
78 Monte Vid 34 53 303 47 − 11 23 + 5 36 − 0 24
79 K. George Sound 35 2 117 56 + 5 12 + 5 30 − 0 24
80 New Zealand 35 16 174 0 − 11 10 − 14 0 + 2 50
81 Concepcion 36 42 286 50 − 14 43 − 16 48 + 2 5
82 Blanco Bay 38 57 298 1 − 12 57 − 15 0 + 2 3
83 Valdivia 39 53 286 31 − 16 13 − 17 30 + 1 17
84 Chiloe 41 51 286 4 − 16 56 − 18 0 + 1 4
85 Hobarttown 42 53 147 24 − 5 51 − 11 6 + 5 15
86 Port Low 43 48 285 58 − 17 32 − 19 48 + 2 16
87 Port San Andres 46 35 284 25 − 19 4 − 20 48 + 1 44
88 Port Desire 47 45 294 5 − 16 52 − 20 12 + 3 20
89 R. Santa Cruz 50 7 291 36 − 18 23 − 20 54 + 2 31
90 Falkland Islands 51 32 301 53 − 15 16 − 19 0 + 3 44
91 Port Famine 53 38 289 2 − 20 28 − 23 0 + 2 32
Inclination. Intensity
Computed. Observed. Difference. Computed. Observed. Difference.
° ′ ° ′ ° ′
51 + 56 59 + 59 58 — 2 59 1·358 1·334 + 0·024
52 62 31 61 54 + 0 37 1·545 1·580 — 0·035
53 61 58 61 22 + 0 36 1·539 1·559 — 0·020
54 62 13 63 48 — 1 35 1·331 1·294 + 0·037
55 61 15 60 42 + 0 33 1·529 1·530 — 0·001
56 60 46 60 18 + 0 28 1·520 1·553 — 0·033
57 59 32 59 3 + 0 29 1·502 1·538 — 0·036
58 56 26 58 53 — 2 27 1·271 1·271
59 56 51 56 17 + 0 34 1·465 1·459 + 0·006
60 55 43 54 49 + 0 54 1·448 1·453 — 0·005
61 68 34 68 6 + 0 28 1·469 1·457 + 0·012
62 64 14 62 38 + 1 36 1·592 1·591 + 0·001
63 45 51 46 3 - 0 12 1·168 1·156 + 0·012
64 4 14 6 52 - 2 38 1·038 1·031 + 0·007
65 13 24 9 29 + 3 55 1·085 1·069 + 0·016
66 5 32 1 39 + 3 53 0·813 0·873 - 0·060
67 13 2 13 13 - 0 11 0·909 0·914 - 0·005
68 - 3 23 - 7 3 + 3 40 0·994
69 - 39 19 - 38 33 - 0 46 1·161
70 + 3 59 + 5 24 - 1 25 0·883 0·871 + 0·012
71 - 14 55 -18 1 + 3 6 0·808 0·836 - 0·028
72 - 27 26 - 30 26 + 3 0 1·113 1·094 + 0·019
73 - 54 8 - 54 1 - 0 7 1·060 1·144 - 0·084
74 - 14 49 - 13 30 - 1 19 0·879 0·878 + 0·001
75 - 37 56 - 39 7 + 1 11 1·094 1·176 - 0·082
76 — 58 11 — 62 49 + 4 38 1·667 1·685 — 0·018
77 — 51 4 — 52 35 + 1 31 0·981 1·014 — 0·033
78 — 35 34 — 35 40 + 0 6 1·022 1·060 — 0·038
79 — 62 39 — 64 41 + 2 2 1·658 1·709 — 0·051
80 — 54 46 — 59 32 + 4 46 1·616 1·591 + 0·025
81 — 42 49 — 44 13 + 1 24 1·147 1·218 — 0·071
82 — 42 1 — 41 54 — 0 7 1·103 1·113 — 0·010
83 — 46 13 — 46 47 + 0 34 1·145 1·238 — 0·093
84 — 48 14 — 49 26 + 1 12 1·227 1·313 — 0·086
85 ­— 66 57 — 70 35 + 3 38 1·894 1·817 + 0·077
86 — 50 4 — 51 20 + 1 16 1·257 1·326 — 0·069
87 — 53 0 — 54 14 + 1 14 1·310
88 — 51 22 — 52 43 + 1 21 1·263 1·359 — 0·096
89 — 53 49 — 55 16 + 1 27 1·321 1·425 — 0·104
90 — 52 46 — 53 25 + 0 39 1·276 1·367 — 0·091
91 — 57 38 — 59 53 + 2 15 1·424 1·532 — 0·108

I add the following notices concerning the observations used in this comparison.

The determinations of the intensity are taken for the most part from Sabine's report on the Variations of the Magnetic Intensity in the above-mentioned Seventh Report of the British Association for the Advancement of Science.

We are indebted for the great number of magnetic observations in the Russian Empire, and in the neighbouring parts of China, to

Hansteen. (Poggendorff's Annals.)

Erman. (Reise um die Erde, and manuscript communications.)

Von Humboldt. (Voyage aux régions équinoxiales, T. 13.)

Fuss. (Mémoires de l'Académie des Sciences de St. Petersbourg, Sixième Série.)

Fedor. (Communicated in manuscript, through Struve).

Reinke. (Observations Météorologiques et Magnétiques, faites dans l'étendue de l'Empire de Russie, redigées par A. T. Kupffer, Nr. II.)

At the following places a mean has been taken of the determinations of several observers. The differences between them are sometimes greater than can be attributed to yearly changes.

12. Tobolsk.
Declination. Hansteen, 1828, − 9° 58′
Erman, 1828, − 9 47
Fuss, 1830, − 11 52
Fedor, 1833, − 10 20
Inclination. Erman, 1828, 71 7
Von Humboldt, 1829, 70 56
Fuss, 1830, 71 1
Fedor, 1833, 71 2
16. Catharinenburg.
Declination. Hansteen, 1828, − 6° 27′
Erman, 1828, − 7 23
Reinke, 1836, − 5 5
Inclination. Erman, 1828, 69 24
Von Humboldt, 1829, 69 6
Fuss, 1830, 69 19
Fedor, 1832, 69 15
17. Tomsk.
Declination. Hansteen, 1828, − 8° 32′
Erman, 1829, − 8 36
Inclination. Erman, 1829, 70 59
Fuss, 1830, 70 51
18. Nishny Novogorod.
Declination. Erman, 1828, − 0° 46′
Fuss, 1830, − 0 8
19. Krasnojarsk.
Declination. Hansteen, 1829, − 6° 43′
Erman, 1829, − 6 37
Fedor, 1835, − 7 26
Inclination. Erman, 1829, 70 53
Fedor, 1835, 71 8
20. Kasan.
Inclination. Erman, 1828, 68° 21′
Von Humboldt, 1829, 68 27
Fuss, 1830, 68 26
21. Moscow.
Declination. Hansteen, 1828, + 3° 3′
Erman, 1828, + 3 1
Inclination. Erman, 1828, 68 58
Von Humboldt, 1829, 68 57
30. Irkutsk.
Declination. Hansteen, 1829, − 1° 37′
Erman, 1829, − 1 52
Fuss, 1830, − 1 25
Inclination. Erman, 1829, 68 7
Fuss, 1830, 68 15
Fuss, 1832, 68 20
36. Orenburg.
Inclination. Von Humboldt, 1829, 64° 41′
Fedor, 1832, 64 47
44. Troizkosawsk.
Declination. Hansteen, 1829, + 0° 5′
Erman, 1829, + 0 33
Fuss, 1830, − 0 1
Inclination. Erman, 1829, 66 14
Fuss, 1830, 66 24

Most of the determinations in the southern hemisphere are supplied by Captains King and Fitz Roy, and are taken from a little work by Sabine, (Magnetic Observations made during the Voyages of H. B. M.'s Ships Adventure and Beagle, 1826–1836.)

The determinations for the several other stations are taken partly from the above-named sources, and partly from the following:

1. Spitzbergen. Observer, Sabine, 1823. (From his Account of Experiments to determine the Figure of the Earth.)

2. Hammerfest. The declination and inclination are the means of the determinations of Sabine, 1823 (Pendulum Experiments); and of Parry, 1827. (Narrative of an Attempt to reach the North Pole.)

3. Magnetic Pole, from Captain James Ross, 1831. (Phil. Trans. 1834.)

4. Reikiavik, from observations by Lottin, 1836, (Voyage en Islande.)

28. Berlin, from Encke, 1836. (Astronomisches Jahrbuch, 1839.)

38. Göttingen. The declination is for October 1, 1835 (Resultate für 1836, page 39); the inclination is reduced to the same epoch by interpolation between von Humboldt's observation in 1826, and Forbes' in 1837.

39. London, from observations communicated in manuscript. The declination, by Captain James Ross, for the mean epoch, April, 1838; and the inclination by Phillips, Fox, Ross, Johnson, and Sabine, for the mean epoch of May, 1838.

48. Paris, for 1835, from the Annuaire for 1836.

54. Milan, 1837, by Kreil. Communicated by him in manuscript.

58. Naples, from observations by Sartorius and Listing. The intensity, which was determined according to absolute measure, has been reduced to the common unity, by the application of the factor given in Article 31.

64. Madras, 1837, from observations by Taylor, taken from the Journal of the Asiatic Society of Bengal, May, 1837.

30

In judging of the differences between calculation and observation, as shown in the foregoing tabular comparison, it must be remembered, on the one hand, that almost all the observations are charged both with the errors of observation, and with the influence of the accidental anomalies of the magnetic force itself, and that they do not correspond to the same year[6]; and, on the other hand, that our formulæ do not include members beyond the fourth order, whereas those of the following order may still be very sensible. When due weight is allowed to these circumstances, the agreement between calculation and experiment appears to be as satisfactory as we are entitled to expect from a first attempt.

As our expression for may therefore be safely regarded as coming near the truth, at least in its more important members, it has appeared worth while to form a graphical representation of the course of the numerical values of this function. This has been done in a map drawn by Dr. Goldschmidt, in three parts, the first on Mercator's projection, passing round the globe, and including all the parallels between 70° north, and 70° south lat.; the other two being polar projections, extending to lat. 65°. The corrections and additions which will arise from a fresh calculation resting on more perfect data, may, doubtless, cause material alterations of position in these lines, particularly in the high southern latitudes; but no important change in the whole form of the system of lines can be supposed without such alterations in the expression for as would destroy the agreement with existing observations. We are thus led to the important result, that the system of lines of equal values of , on the surface of the earth, is actually comprehended by the simplest type described in Art. 13, and that consequently there are on the earth only two magnetic poles, apart from the possible case of local exception spoken of in Art. 13.

The exact computation of the places of these two poles, according to our elements, gives them as follows:

1. In 73° 35′ north lat., 264° 21′ long, east from Greenwich, the value of the total intensity being = 1·701 in the unity in common use.

2. In 72° 35′ south lat., 152° 30′ long., the total intensity = 2·253.

At the first of these points has its greatest value, = + 895·86; at the second its smallest value, = — 1030·24.

According to Captain James Ross's observation the north magnetic pole falls 3° 30′ to the south of its position according to our calculation, which gives at that place a direction of the magnetic force, differing 1° 12′ from observation, as may be seen in the table of comparisons. We must expect a considerably greater displacement of the position of the southern pole. At Hobart Town, which is the nearest station to this pole, calculation gives too low a dip by 3° 38′, as far as the observation can be depended upon. It seems probable, therefore, that the actual south magnetic pole is considerably north of the position given by our calculation, and that it may be looked for in about 66° lat., and 146° long.

31.

The two points on the earth's surface where the horizontal force vanishes, and which are called magnetic poles, may, it is true, be allowed a certain significancy on account of their relation to the form of the phenomena of the horizontal force all over the earth; but we must be careful not to give them undue consideration. The chord which unites these two points has no significancy, and it would be a gross mistake to call it the magnetic axis of the earth. The only mode of giving a generally valid signification to the idea of the magnetic axis of a body is laid down in the 5th Article of the Intensitas Vis Magneticæ, where it is understood to mean the straight line in which the moment of the free magnetism contained in the body is a maximum. In order to determine both the position of the magnetic axis of the earth in this sense, and the moment of the earth's magnetism in relation to this same axis, we only require, as noticed in Art. 17, a knowledge of the members of the first order of . According to our elements. Art. 26, , and , , are the moments of terrestrial magnetism with respect to the axis of the earth, and to the two radii for longitudes 0 and 90. In speaking of the earth's axis, the direction towards the north pole is to be understood, and the negative sign of the corresponding moment shows that the magnetic axis makes with it an obtuse angle, or that its magnetic north pole is turned towards the south.

The direction hence found for the magnetic axis is parallel to that diameter of the earth which is from 77° 50′ north lat., and 296° 29′ lon., to 77° 50′ south lat,, 116° 29′ lon.; and the magnetic moment in relation to this axis is . It must be remembered that in our elements the unity of intensity employed is a thousandth part of the unity in common use. In order to obtain the reduction to the absolute unity established in the Intensitas Vis Magneticæ, we must remark that in that work the horizontal intensity at Göttingen for the 19th of July, 1834, was found = 1·7748, which, combined with the dip 68° 1′, gives the total intensity = 4·7414. The total intensity, according to the unity employed above, was 1357. Thus the reducing factor is = 0·0034941, and the magnetic moment of the earth, expressed according to the absolute unity,


As the millimetre is the unit of length employed in the above absolute unity for the earth's magnetic force, must also be given in millimetres; and, as the ellipticity of the earth need not be taken into account, it will be sufficient to consider as the radius of a circle 40000 millions of millimetres in circumference. Hence the above magnetic moment will be expressed by a number of which the logarithm = 29,93136, or by 853800 quadrillions. By experiments made in the year 1832 (Intensitas, Art. 21) the magnetic moment of a magnet bar, of a pound weight, was found to be, according to the same absolute unity, = 100877000. The magnetic moment of the earth is therefore 8464 trillion times greater. Thus 8464 trillions of such magnet bars, with parallel magnetic axes, would be required to replace in external space the magnetic influence of the earth. Supposing the magnetism of the earth to be uniformly distributed throughout its volume, it would hence be equal to eight such bars (more exactly 7·831) for every cubic metre. This result thus enounced preserves its meaning even, if instead of considering the earth as an actual magnet, we should prefer to ascribe terrestrial magnetism simply to constant galvanic currents in the earth. But if we consider the earth as an actual magnet, we are obliged to ascribe to each of its portions, of the size of the eighth of a cubic metre, on an average, at least[7] as great a force of magnetism as that contained in one of the above-mentioned bars. Such a result will be an unexpected one to philosophers.

32.

The manner of the actual distribution of the magnetic fluids in the earth necessarily remains undetermined. In fact, according to a general theorem which has been already mentioned in the 2nd article of the Intensitas, and will be treated of in greater detail at a future opportunity, we may substitute for any supposed distribution of the magnetic fluids in the interior of a body occupying space, a distribution on the surface of the same space, which shall leave the effect on every point of external space precisely the same. It may be easily concluded from hence, that one and the same action on all external space may be deduced from an infinite number of different distributions of the magnetic fluids in the interior.

We are enabled to assign on the other hand that fictitious distribution on the surface of the earth, which shall be perfectly equivalent to the actual distribution in the interior, as regards the external resultant of the forces; and the spherical form of the earth allows us to do so in a very simple manner.

We may express the density of the magnetic fluid in each point of the earth's surface, i. e. the quantum of the fliud which corresponds to the unit of surface, by the formula

or by

The result of this formula will be hereafter exhibited by a graphical representation. We shall only notice here that it is negative in the northern and positive in the southern parts of the earth, but in such manner that the dividing line cuts the equator twice (in longitudes 6° and 186°); its points most distant from the equator being in about 15° north and 15° south latitude: and further that in the northern hemisphere there are two minima, but in the southern hemisphere but one maximum. According to a cursory computation, these minima and this maximum are

− 209·1 in 55° N. lat., 263° lon.
− 200·0 in 71° N. lat., 116° lon.
+ 277·7 in 70° S. lat., 154° lon.

These values are founded on the unity of our elements, and must therefore be multiplied by 0·0034941 to obtain their expression in absolute measure.

33.

It has been already said that our elements are to be regarded only as a first approximation. So considered, their agreement with the observations in Art. 29 is sufficiently satisfactory. It cannot be doubted that a much greater agreement would be obtained by an improved calculation, even with these observations. The only difficulty of such a calculation is its length, which would be still alarmingly great, even supposing it abridged by the introduction of such skilful methods as have been employed by astronomers in correcting the elements of the planetary and cometary paths. Although this difficulty might be easily surmounted by dividing the work amongst a number of computers, it does not appear advisable to undertake such an amended calculation at present, when there is still so little certainty in the data from a great number of places which it would be important to employ. It will be preferable in the first place to pursue further the comparison of the elements with observations, whence the means will be afforded of giving much greater certainty to the general maps, than has been hitherto possible by the exclusively empirical mode.

We may be allowed to give a few glances at the future progress of the theory, the perfect realization of which may indeed be far distant.

34.

For the satisfactory refinement and completion of the elements, it will be requisite to make much higher demands than have been hitherto complied with, as to the data furnished by obsenation. Their accuracy at all the points employed ought to be such as has hitherto been obtained at a very few only; they should be cleared from the effect of irregular changes; they should be all for the same epoch. It will probably be long before such demands are satisfied.

Next to this the chief desideratum is to obtain complete observations (i. e. including all three elements) from points in those large parts of the earth's surface where such observations are still wholly wanting. Every new station will have for the general theory an importance proportionate in great measure to its distance from those we already possess.

After a sufficient interval of time shall have elapsed, the elements may be determined afresh for a second epoch, and their secular changes may be thence deduced. Manifestly it will be essential for this purpose to reject altogether the present measure of the intensities, and to substitute for it an absolute measure.

In the course of the present century these alterations will no longer appear uniform, and the examination of the course and progress of the elements will offer to men of science inexhaustible materials for research.

35.

Conclusions as to interesting points of theory may also be expected in future.

In our theory it is assumed that every determinate magnetized particle of the earth contains precisely equal quantities of positive and negative fluid. Supposing the magnetic fluids to have no reality, but to be merely a fictitious substitute for galvanic currents in the smallest particles of the earth, this equality is necessarily part of the substitution; but if we attribute to the magnetic fluids an actual existence, there might without absurdity be a doubt as to the perfect equality of the quantities of the two fluids.

In regard to detached magnetic bodies (natural or artificial magnets), the question as to whether they do or do not contain a sensible excess of either magnetic fluid might easily be decided by very exact and delicate experiments.

In case of the existence of any such excess in a body of this nature, a plumb-line to which it should be attached should deviate from the true vertical position in the direction of the magnetic meridian.

If experiments of this kind, made with a great number of artificial magnets and in a locality sufficiently distant from iron, never showed the slightest deviation, (which we should rather expect,) the equality of the two fluids might with the highest degree of probability be inferred for the whole earth; though without wholly excluding the possibility of some inequality.

The only difference which the existence of such an inequality would occasion in our theory would be, that (Art. 17) would no longer be . The consequence of this would be, that for all external space it would be necessary to add to the expression for the member ; so that on the surface of the earth the (constant) member must be added, but and would be in no respect affected. When there shall exist in future times a much more extensive collection of accurate observations than we at present possess, it may be examined whether a vanishing value of is or is not required for their accurate representation. With our present data such an undertaking would be wholly useless.

36.

Another part of our theory on which there may exist a doubt is, the supposition that the agents of the terrestrial magnetic force are situated exclusively in the interior of the earth. If we seek for their immediate causes, partly or wholly, without the earth, and confine ourselves to known scientific grounds, we can only think of galvanic currents. But the atmosphere is no conductor of such currents, neither is vacant space; thus, in seeking in the upper regions for a vehicle of galvanic currents we go beyond our knowledge. ·But our ignorance gives us no right absolutely to deny the possibility of such currents; we are forbidden to do so by the enigmatical phenomena of the Aurora Borealis, in which there is every appearance that electricity in motion performs a principal part. It will therefore still be interesting to examine what form magnetic action arising from such currents would assume on the surface of the earth.

37.

Let us, then, assume the existence of constant galvanic currents in a concave sphere, , surrounding the earth, and call all the space included by , and all the space external to . Whatever may be configuration of the galvanic currents, we can always substitute for them a fictitious distribution of the magnetic fluids in the space , the magnetic action of which, in all other spaces and , will be exactly similar to that of the currents.

This important proposition, which has been already mentioned (Art. 3.), rests on the following grounds: first, that these currents may be resolved into an infinite number of elementary currents (i. e. such as may be considered linear); secondly, the well-known theorem, first demonstrated, I believe, by Ampère, that in place of each linear current bounding an arbitrary surface, we may substitute a distribution of the magnetic fluids on both sides of this surface, at immeasurably small distances from it, with the same action; thirdly, the evident possibility of assigning for every re-entering line inside , a surface bounded by it and situated wholly inside .

If we designate by the aggregate of all the quotients produced by dividing all the elements of the imaginary magnetic fluid by the distance of an indeterminate point, in or ; of course it is understood that the elements of the southern fluid are to be considered as negative. Then will the partial differential quotients of , (just like those of in our theory) express the components of the magnetic force which the galvanic currents produce at .

38.

Although we must defer to another opportunity the detailed developement of the theory from which the proposition employed in the last article is taken, yet there is an important point relating to it which deserves to be noticed here. If we construct two different surfaces, and , each bounded by the same linear current ,—and (taking the simplest case for the sake of brevity) having no other point in common,—they will include a portion of space. Now, if be situated without this space, we obtain for that constant portion of which belongs to , one and the same value, whether we assign the magnetic fluids to or ; and this value is equal to the product of the intensity of the galvanic current (measured by an appropriate unity) multiplied by the solid angle, the summit of which is at , and which is included by straight lines, drawn from to the points of ; or, which is the same thing, multiplied by that portion of the spherical surface described with radius round , which is the common projection of both and .

If, on the other hand, be situated inside the space included by and , it is true that the two values of the part of in question will not be the same, whether we assign the magnetic fluids to or to , because different parts of the spherical surface alluded to correspond to them,—which parts, taken together, make up the whole spherical surface. But as the galvanic current has opposite directions towards and , opposite signs must be applied in the two cases to the intensity of the current, in the multiplication into the parts of the spherical surface. The consequence is, that the algebraic difference between the values of the part of in question is equal to the product of the intensity of the current multiplied by the whole spherical surface, or by .

Hence it may easily be deduced, that if is situated in , the value of remains independent of the choice of the connecting surface; that if, on the other hand, is situated in , the absolute value of does indeed depend on that choice, but the differential of does not.

The highly fruitful theorem here touched upon,—according to which, in relation to the magnetic action of a linear galvanic current, the product of the intensity of that current, into the portion of spherical surface which is bounded by the line of current from outwards, has the same import in regard to attracting or repelling forces, as the parts of the mass divided by the distance from ,—still requires in its generality many fuller explanations, which must be reserved for a detailed treatment of this subject.

39.

The value of , which in general is a function of , , and , passes on the surface of the earth into a function of and , and

are the horizontal components of the magnetic force proceeding from the galvanic currents, directed respectively towards the north and west. It is manifest that the remarkable propositions mentioned in Art. 15. and 16. hold good likewise in this case. But as to the third component, the vertical magnetic force, the case will be somewhat different, if the agents are situated above, from what it would be supposing them to be situated in the interior. To eliminate the vertical magnetic force resulting from the former supposition, must first be considered as a function of both , , and ; it must be differentiated according to , and then must be substituted.

But for the inner space , to which the surface of the earth belongs, can only be developed in a series according to ascending powers of . If we make

is a constant magnitude, namely, the value of at the centre of the earth; , , , &c., on the other hand, are functions of , and , which satisfy the same partial differential equations as , , , above.

Hence it follows, in a similar manner to Art. 20, that the knowledge of the value of at every point of the earth's surface is sufficient to enable us to deduce therefrom the general expresssion for the space ; that we may arrive at the knowledge of this value with the exception of a constant part,—or, which is the same thing, at the knowledge of the co-efficients , , , &c.,—by the knowledge of the horizontal forces on the surface of the earth; but that the value of the vertical force on the surface of the earth is not

as it would be if the forces acted outwards from the interior of the earth, but is


Now, as our numerical elements (Art. 26.), determined under the supposition of the first formula, give a very satisfactory representation of the phenomena generally, whereas, the phenomena would be wholly incompatible with the second formula, the fallacy of the hypothesis, which places the causes of terrestrial magnetism in space external to the earth, may be looked upon as proved.

40.

At the same time, this must not be looked upon as decidedly disproving the possibility of a part, though comparatively a very small part, of the terrestrial magnetic force proceeding from the upper regions: a far more full and far more accurate knowledge of the phenomena may in future throw light on this important point of theory. If, under the supposition of mixed causes, we attach the same meaning as before to the signs , , , , &c., , , , , applying the former to the causes acting from within, and the latter to the causes acting from without; and if we further put , , , , &c., then on the surface of the earth,

where satisfies the same partial differential equation as (Art. 18.); and the two components of the horizontal magnetic force there existing are expressed by


The propositions mentioned in Articles 15. and 16. retain therefore their validity in this case, and we can determine the magnitudes , , , &c. simply from the knowledge of the horizontal forces, but without being able in any degree to conclude from hence only as to the existence of mixed causes. But if we consider the vertical force by itself, and bring it into the form

so that satisfies the above-mentioned partial differences, then

and, consequently,

Thus, by the combination of the horizontal force with the vertical, we obtain the means of dividing into its constituent parts and , and thus of learning whether a sensible value must be assigned to the latter. Only the constant part of , namely, , is left wholly undetermined by the observations, the reason of which is plain from Art. 38.

Hence it appears important, in this interesting point of view likewise, to consider the horizontal magnetic force by itself, and we see in this an additional reason for the recommendations in Art. 21.

41.

Sufficient data for the investigation above alluded to will probably long be wanting. But it is worthy of notice, that the variations of the magnetic force, which manifest themselves simultaneously at different points of the earth's surface, are susceptible of a perfectly similar treatment. This is the case both with the regular changes corresponding to the periods of the day and of the year, and with the irregular changes. Perhaps in this way, the necessary materials may be much earlier collected. It may be well to subjoin some general remarks concerning these future researches.

After bringing the observed simultaneous changes for each place into the form of alterations of the components of the magnetic force , , , it must first be examined whether the alterations of the two horizontal components comport themselves in correspondence with our theory, according to which and , must be values of the partial differential quotients of a function of and according to these variables. If this is found to be the case, the conclusion will be, that the causes either are actual galvanic currents, or at least act in the same manner as such currents, or as separated magnetic fluids. In the opposite case, it would be proved that the causes cannot be galvanic currents.

We see that highly important conclusions may be derived even from the knowledge of the changes in the horizontal force only, supposing the determinations sufficiently accurate, numerous, and extensive. But if we add thereto the simultaneous changes in the vertical force, then, supposing the first case, the method in the preceding article will inform us whether the causes are situated above or below the surface of the earth; and further, as they are probably situated in a stratum of small thickness compared to the whole body of the earth, it may be possible to determine the mode of their propagation, at least approximatively.

As regards the second case spoken of above as possible, it certainly appears to me but little probable as concerns the regular changes in the terrestrial magnetic force depending on the time of the year or of the day. In regard to the irregular changes occurring in short intervals, I should hardly venture to pronounce a conjecture at present. If these irregular changes arise from great electric movements above the atmosphere, it would be difficult to place these in the category of galvanic currents; for although everything seems to indicate that we should regard galvanic currents as electricity in motion, yet every movement of electricity is not a galvanic current—it is so only when the movement forms a circle returning back into itself. As it is only under this condition that it is allowable to make the often-mentioned substitution of separated magnetic fluids instead of the galvanic current, then, in the hypothesis mentioned, our relations between the components would no longer apply; that is to say, the second case would actually present itself. Even the certain establishment of this important circumstance would be in itself of great interest, and by that time we may hope to possess such extensive and accurate observations as may make it possible to trace both the source and the nature of the causes.

G.

TABLE I. TABLE I.
° °
+ 90  +   0·0   + 1652·9    + 45    + 605·0   + 1354·1  
89  10·3 1652·8 44 620·7 1334·2
88  20·5 1652·7 43 636·2 1313·6
87  30·8 1625·4 42 651·5 1292·1
86  41·2 1652·1 41 666·6 1270·0
85  51·6 1651·7 40 681·5 1247·1
84  62·1 1651·1 39 696·2 1223·5
83  72·8 1650·5 38 710·6 1199·2
82  83·5 1649·7 37 724·7 1174·1
81  94·3 1648·8 36 738·5 1148·5
80 105·3 1647·7 35 752·0 1122·0
79 116·5 1646·4 34 765·2 1094·9
78 127·8 1645·0 33 777·9 1067·2
77 139·3 1643·3 32 790·3 1038·9
76 151·0 1641·4 31 802·3 1009·9
75 162·9 1639·3 30 813·9 980·5
74 175·0 1637·0 29 825·0 950·4
73 187·4 1634·3 28 835·7 919·9
72 199·9 1631·3 27 845·9 888·9
71 212·6 1628·0 26 855·7 857·4
70 225·6 1624·4 25 864·9 825·5
69 238·9 1620·3 24 873·7 793·2
68 252·3 1615·9 23 882·0 760·5
67 266·0 1611·0 22 889·8 727·5
66 279·9 1605·7 21 897·0 694·1
65 294·0 1600·0 20 903·8 660·5
64 308·3 1593·7 19 910·0 626·7
63 322·8 1586·9 18 915·8 592·6
62 337·6 1579·6 17 921·0 558·4
61 352·5 1571·7 16 925·7 523·9
60 367·6 1563·2 15 929·8 489·4
59 382·9 1554·1 14 933·5 454·8
58 398·3 1544·4 13 936·7 420·1
57 413·9 1534·0 12 939·4 385·4
56 429·6 1523·0 11 941·6 350·7
55 445·4 1511·2 10 943·3 316·0
54 461·3 1498·9  9 944·6 281·3
53 477·2 1485·8  8 945·4 246·7
52 493·3 1471·9  7 945·7 212·3
51 509·3 1457·4  6 945·7 177·9
50 525·4 1422·1  5 945·2 143·7
49 541·4 1426·0  4 944·3 109·6
48 557·4 1409·2  3 943·0  75·8
47 573·4 1391·6  2 941·4  42·1
46 589·2 1373·2 + 1  939·4 +   8·6  
45 605·0 1354·1  0 937·1 −  24·6  
TABLE I. TABLE I.
° °
0 + 937·1   − 24·6  − 45   + 680·2   − 1275·1  
− 1  934·5  57·6 46 672·0 1299·5
2 931·5  90·3 47 663·5 1323·9
3 928·3 122·8 48 645·8 1348·1
4 924·8 154·9 49 645·9 1372·3
5 921·0 186·9 50 636·7 1396·2
6 917·0 218·5 51 627·2 1420·0
7 912·8 249·8 52 617·3 1443·7
8 908·4 280·8 53 607·2 1467·1
9 903·8 311·6 54 596·8 1490·3
10 899·1 342·0 55 586·0 1513·2
11 894·1 372·1 56 574·9 1536·1
12 889·1 402·0 57 563·5 1558·6
13 883·9 431·6 58 551·7 1580·8
14 878·6 460·8 59 539·6 1602·7
15 873·2 489·8 60 527·0 1624·2
16 867·7 518·6 61 514·1 1645·4
17 862·1 547·0 62 500·9 1666·1
18 856·4 575·3 63 487·2 1686·5
19 850·7 603·2 64 473·2 1706·4
20 844·9 631·0 65 458·8 1725·9
21 839·1 658·5 66 444·0 1744·9
22 833·2 685·7 67 428·9 1763·3
23 827·3 712·8 68 413·3 1781·2
24 821·4 739·7 69 397·4 1798·6
25 815·4 766·4 70 381·2 1815·3
26 809·3 792·9 71 364·6 1831·4
27 803·2 819·3 72 347·6 1846·9
28 797·1 845·5 73 330·3 1861·6
29 790·9 871·6 74 312·7 1875·7
30 784·7 897·5 75 294·8 1889·1
31 778·5 923·3 76 276·6 1901·7
32 772·1 949·0 77 258·1 1913·5
33 765·7 974·6 78 239·3 1924·6
34 759·3 1000·1 79 220·3 1934·8
35 752·7 1025·5 80 201·0 1944·2
36 746·1 1050·9 81 181·6 1952·8
37 739·3 1076·1 82 161·9 1960·5
38 732·5 1101·2 83 142·1 1967·3
39 725·5 1126·3 84 122·1 1973·3
40 718·4 1151·3 85 101·9 1978·3
41 711·1 1176·2 86 81·7 1982·5
42 703·7 1201·0 87 61·3 1985·7
43 696·0 1225·8 88 40·9 1988·0
44 688·2 1250·5 89 20·5 1989·5
45 680·2 1275·1 90 0 1989·9
TABLE II.
° ° ′ ° ′ ° ′
+ 90 292 9 2·07430 22 29 2·07430 172 29
89 292 4 2·07444 22 7 2·07437 172 27 0·72139
88 291 50 2·07488 22 2 2·07458 172 20 1·02153
87 291 26 2·07653 21 54 2·07493 172 8 1·19615
86 290 52 2·07669 21 43 2·07543 171 51 1·31904
85 290 10 2·07811 21 29 2·07607 171 30 1·41333
84 289 19 2·07990 21 11 2·07686 171 3 1·48952
83 288 20 2·08211 20 51 2·07781 170 31 1·55192
82 287 14 2·08477 20 28 2·07891 169 54 1·60623
81 286 0 2·08791 20 2 2·08017 169 11 1·65259
80 284 41 2·09156 19 33 2·08160 168 22 1·69305
79 283 16 2·09573 19 2 2·08320 167 28 1·72868
78 281 46 2·10046 18 28 2·08498 166 27 1·76027
77 280 13 2·10574 17 52 2·08693 165 20 1·78844
76 278 37 2·11157 17 14 2·08906 164 6 1·81369
75 276 59 2·11794 16 34 2·09138 162 45 1·83641
74 275 20 2·12481 15 52 2·09388 161 16 1·85697
73 273 41 2·13215 15 9 2·09658 159 41 1·87567
72 272 3 2·13991 14 24 2·09945 157 57 1·89278
71 270 25 2·14803 13 37 2·10252 156 6 1·90856
70 268 50 2·15646 12 50 2·10577 154 6 1·92325
69 267 17 2·16512 12 2 2·10920 151 59 1·93709
68 265 46 2·17394 .11 13 2·11280 149 44 1·95028
67 264 19 2·18288 10 24 2·11658 147 21 1·96304
66 262 56 2·19183 9 34 2·12052 144 51 1·97558
65 261 36 2·20074 8 44 2·12461 142 15 1·98809
64 260 19 2·20954 7 55 2·12885 139 33 2·00074
63 259 7 2·21816 7 5 2·13322 136 46 2·01369
62 257 58 2·22656 6 15 2·13772 133 55 2·02708
61 256 53 2·23468 5 26 2·14232 131 2 2·04101
60 255 52 2·24246 4 38 2·14703 128 8 2·05556
59 254 55 2·24986 3 50 2·15183 125 15 2·07077
58 254 1 2·25686 3 3 2·15669 122 22 2·08665
57 253 11 2·26339 2 17 2·16162 119 33 2·10318
56 252 24 2·26944 1 32 2·16659 116 48 2·12032
55 251 40 2·27497 0 48 2·17159 114 8 2·13799
54 250 59 2·27996 0 5 2·17661 111 35 2·15610
53 250 21 2·28439 359 23 2·18164 109 7 2·17456
52 249 46 2·28822 358 43 2·18666 106 47 2·19326
51 249 13 2·29145 358 3 2·19166 104 34 2·21210
50 248 43 2·29406 357 25 2·19662 102 29 2·23098
49 248 15 2·29603 356 49 2·20155 100 32 2·24979
48 247 49 2·29734 356 13 2·20641 98 42 2·26848
47 247 25 2·29799 355 39 2·21121 96 59 2·28692
46 247 3 2·39796 355 6 2·21593 95 24 2·30508
45 246 43 2·29724 354 34 2·22057 93 56 2·32288
TABLE II.
° ° ′ ° ′ ° ′
+ 45 246 43 2·29724 354 34 2·22057 93 56 2·32288
44 246 24 2·29581 345 4 2·22512 92 34 2·34027
43 246 6 2·29367 353 35 2·22956 91 18 2·35721
42 245 49 2·29080 353 7 2·23389 90 9 2·37367
41 245 34 2·28719 352 40 2·23811 89 5 2·38961
40 245 19 2·28282 352 14 2·24221 88 6 2·40502
39 245 5 2·27770 351 50 2·24618 87 12 2·41988
38 244 52 2·27179 351 26 2·25002 86 23 2·43417
37 244 39 2·26510 351 4 2·25372 85 39 2·44789
36 244 25 2·25760 350 43 2·25728 84 58 2·46103
35 244 12 2·24928 350 22 2·26071 84 22 2·47360
34 243 58 2·24012 350 3 2·26398 83 48 2·48558
33 243 44 2·23010 349 44 2·26711 83 19 2·49699
32 243 28 2·21920 349 27 2·27009 82 52 2·50782
31 243 10 2·20742 349 10 2·27292 82 28 2·51808
30 242 51 2·19471 348 54 2·27560 82 7 2·52779
29 242 30 2·18107 348 38 2·27813 81 48 2·53693
28 242 5 2·16647 348 23 2·28052 81 32 2·54554
27 241 37 2·15089 348 9 2·28275 81 18 2·55360
26 241 4 2·13431 347 55 2·28483 81 6 2·56113
25 240 26 2·11671 347 41 2·28677 80 55 2·56815
24 239 41 2·09807 347 28 2·28856 80 47 2·57465
23 238 49 2·07839 347 15 2·29021 80 39 2·58066
22 237 49 2·05768 347 3 2·29171 80 33 2·58618
21 236 37 2·03595 346 50 2·29309 80 29 2·59121
20 235 13 2·01326 346 38 2·29433 80 25 2·59578
19 233 35 1·98970 346 26 2·29544 80 22 2·59991
18 231 39 1·96540 346 14 2·29642 80 20 2·60356
17 229 23 1·94057 346 2 2·29728 80 19 2·60679
16 226 45 1·91553 345 49 2·29802 80 18 2·60959
15 223 41 1·89072 345 36 2·29865 80 17 2·61198
14 220 9 1·86675 345 23 2·29917 80 16 2·61397
13 216 7 1·84438 345 10 2·29958 80 15 2·61556
12 211 35 1·82457 344 56 2·29990 80 15 2·61677
11 206 34 1·80835 344 42 2·30014 80 13 2·61761
10 201 12 1·79678 344 27 2·30028 80 11 2·61809
9 195 33 1·79064 344 11 2·30035 80 9 2·61822
8 189 50 1·79046 343 55 2·30035 80 5 2·61802
7 184 15 1·79621 343 37 2·30029 80 0 2·61750
6 178 56 1·80737 343 19 2·30018 79 54 2·61667
5 174 3 1·82310 343 0 2·30002 79 46 2·61554
4 169 39 1·84235 342 40 2·29983 79 37 2·61414
3 165 47 1·86409 342 18 2·29961 79 25 2·61246
2 162 26 1·88741 341 56 2·29938 79 12 2·61054
1 159 34 1·91156 341 32 2·29914 78 56 2·60839
0 157 9 1·93596 341 7 2·29890 78 37 2·60603
TABLE II.
° ° ′ ° ′ ° ′
0 157 9 1·93596 341 7 2·29890 78 37 2·60603
− 1 155 7 1·96018 340 40 2·29869 78 15 2·60347
2 153 26 1·98393 340 12 2·29850 77 50 2·60075
3 152 3 2·00702 339 42 2·29836 77 22 2·59789
4 150 55 2·02930 339 11 2·29827 76 50 2·59491
5 150 0 2·05070 338 38 2·29824 76 14 2·59185
6 149 16 2·07116 338 3 2·29830 75 34 2·58874
7 148 41 2·09068 337 27 2·29846 74 50 2·58562
8 148 14 2·10923 336 49 2·29873 74 1 2·58252
9 147 54 2·12683 336 10 2·29912 73 8 2·57949
10 147 39 2·14348 335 29 2·29965 72 11 2·57658
11 147 28 2·15919 334 46 2·30033 71 8 2·57383
12 147 22 2·17398 334 1 2·30118 70 1 2·57129
13 147 18 2·18785 333 15 2·30222 68 49 2·56902
14 147 16 2·20083 332 27 2·30345 67 32 2·56707
15 147 16 2·21292 331 37 2·30489 66 11 2·56549
16 147 18 2·22413 330 47 2·30655 64 45 2·56435
17 147 19 2·23446 329 54 2·30845 63 15 2·56368
18 147 22 2·24391 329 1 2·31059 61 42 2·56354
19 147 24 2·25250 328 6 2·31298 60 5 2·56397
20 147 25 2·26022 327 11 2·31564 58 26 2·56499
21 147 26 2·26706 326 14 2·31856 56 44 2·56664
22 147 25 2·27302 325 16 2·32176 55 1 2·56893
23 147 23 2·27809 324 18 2·32523 53 17 2·57187
24 147 19 2·28227 323 20 2·32899 51 32 2·57546
25 147 13 2·28554 322 21 2·33302 49 47 2·57966
26 147 4 2·28790 321 22 2·33733 48 3 2·58447
27 146 52 2·28932 320 22 2·34191 46 20 2·58984
28 146 37 2·28978 319 23 2·34675 44 39 2·59572
29 146 18 2·28928 318 24 2·35186 43 0 2·60207
30 145 55 2·28780 317 25 2·35722 41 24 2·60883
31 145 27 2·28530 316 27 2·36281 39 51 2·61593
32 144 54 2·28177 315 30 2·36863 38 21 2·62331
33 144 15 2·27720 314 33 2·37467 36 55 2·63090
34 143 30 2·27156 313 37 2·38091 35 32 2·63864
35 142 37 2·26483 312 42 2·38733 34 13 2·64646
36 141 36 2·25701 311 48 2·39392 32 58 2·65430
37 140 25 2·24809 310 56 2·40066 31 46 2·66210
38 139 4 2·23808 310 4 2·40754 30 38 2·66980
39 137 30 2·22701 309 14 2·41454 29 34 2·67736
40 135 43 2·21492 308 25 2·42163 28 33 2·68471
41 133 40 2·20190 307 37 2·42882 27 36 2·69181
42 131 20 2·18809 306 51 2·43606 26 42 2·69862
43 128 39 2·17367 306 6 2·44336 25 52 2·70510
44 125 37 2·15891 305 23 2·45069 25 4 2·71121
45 122 10 2·14420 304 41 2·45804 24 19 2·71691
TABLE II.
° ° ′ ° ′ ° ′
−45 122 10 2·14420 304 41 2·45804 24 19 2·71691
46 118 16 2·13005 304 1 2·46539 23 37 2·72218
47 113 56 2·11708 303 22 2·47272 22 58 2·72698
48 109 7 2·10605 302 44 2·48003 22 21 2·73129
49 103 53 2·09781 302 8 2·48730 21 47 2·73508
50 98 16 2·09320 301 33 2·49451 21 14 2·73833
51 92 24 2·09289 301 0 2·50166 20 44 2·74100
52 86 25 2·09739 300 28 2·50873 20 16 2·74307
53 80 27 2·10679 299 57 2·51571 19 49 2·74543
54 74 40 2·12081 299 28 2·52260 19 25 2·74453
55 69 11 2·13887 299 0 2·52937 19 1 2·74550
56 64 5 2·16018 298 33 2·53603 18 40 2·74495
57 59 25 2·18391 298 7 2·54256 18 20 2·74370
58 55 12 2·20923 297 43 2·54895 18 1 2·74169
59 51 25 2·23544 297 20 2·55521 17 43 2·73892
60 48 4 2·26198 296 57 2·56131 17 26 2·73535
61 45 4 2·28840 296 36 2·56727 17 11 2·73094
62 42 26 2·31436 296 16 2·57306 16 57 2·72566
63 40 5 2·33963 295 57 2·57868 16 43 2·71948
64 38 1 2·36405 295 39 2·58413 16 31 2·71235
65 36 10 2·38751 295 22 2·58941 16 19 2·70421
66 34 32 2·40996 295 5 2·59451 16 8 2·69503
67 33 5 2·43134 294 50 2·59942 15 58 2·68474
68 31 47 2·45165 294 35 2·60415 15 49 2·67328
69 30 37 2·47088 294 22 2·60868 15 40 2·66056
70 29 35 2·48904 294 9 2·61302 15 32 2·64650
71 28 40 2·50615 293 57 2·61716 15 24 2·63100
72 27 50 2·52223 293 45 2·62111 15 17 2·61395
73 27 5 2·53729 293 35 2·62485 15 11 2·59520
74 26 25 2·55136 293 25 2·62839 15 5 2·57459
75 25 49 2·56447 293 16 2·63172 14 59 2·55193
76 25 17 2·57662 293 7 2·63484 14 54 2·52699
77 24 48 2·58784 292 59 2·63776 14 50 2·49948
78 24 23 2·59816 292 52 2·64046 14 45 2·46904
79 24 0 2·60758 292 45 2·64296 14 42 2·43523
80 23 40 2·61613 292 39 2·64524 14 38 2·39746
81 23 22 2·62382 292 34 2·64730 14 35 2·35498
82 23 7 2·63067 292 29 2·64915 14 32 2·30676
83 22 53 2·63668 292 25 2·65079 14 30 2·25136
84 22 42 2·64187 292 21 2·65220 14 28 2·18665
85 22 32 2·64624 292 18 2·65340 14 26 2·10937
86 22 25 2·64981 292 16 2·65439 14 25 2·01401
87 22 19 2·65258 292 14 2·65515 14 24 1·89028
88 22 15 2·65456 292 13 2·65570 14 23 1·71505
89 22 12 2·65574 292 12 2·65603 14 23 1·41453
90 22 11 2·65614 292 11 2·65614 14 23
TABLE III.
° ° ° °
+90 347 16 77 16 176 59
89 347 15 0·60246 77 16 0·60263 176 59 9·17222
88 347 13 0·90273 77 15 0·90333 176 58 9·77385
87 347 8 1·07753 77 12 1·07889 176 56 0·12532
86 347 2 1·20066 77 9 1·20311 176 53 0·37419
85 346 54 1·29525 77 5 1·29903 176 49 0·56672
84 346 44 1·37159 77 0 1·37704 176 45 0·72351
83 346 32 1·43517 76 55 1·44260 176 40 0·83554
82 346 19 1·48927 76 48 1·49899 176 34 0·96937
81 346 3 1·53601 76 40 1·54833 176 27 1·06923
80 345 45 1·57682 76 32 1·59206 176 19 1·15802
79 345 25 1·61273 76 22 1·63121 176 10 1·23779
78 345 3 1·64451 76 12 1·66655 176 1 1·31006
77 344 39 1·67272 76 0 1·69865 175 50 1·37599
76 344 13 1·69780 75 48 1·72795 175 39 1·43647
75 343 43 1·72012 75 35 1·75483 175 27 1·49222
74 343 12 1·73995 75 20 1·77955 175 14 1·54381
73 342 38 1·75753 75 5 1·80237 175 0 1·59171
72 342 1 1·77302 74 49 1·82347 174 45 1·63630
71 341 20 1·78662 74 31 1·84301 174 29 1·67772
70 340 37 1·79844 74 13 1·86114 174 12 1·71684
69 339 51 1·80860 73 53 1·87798 173 54 1·75329
68 339 1 1·81720 73 32 1·89362 173 35 1·78747
67 338 7 1·82433 73 11 1·90815 173 14 1·81956
66 337 9 1·83005 72 48 1·92165 172 53 1·84971
65 336 6 1·83444 72 24 1·93420 172 31 1·87806
64 334 59 1·83756 71 58 1·94584 172 7 1·90472
63 333 48 1·83947 71 32 1·95663 171 42 1·92979
62 332 30 1·84022 71 4 1·96663 171 16 1·95338
61 331 7 1·83986 70 35 1·97585 170 48 1·97557
60 329 38 1·83845 70 4 1·98440 170 20 1·99642
59 328 3 1·83604 69 33 1·99224 169 50 2·01601
58 326 20 1·83270 69 0 1·99944 169 18 2·03440
57 324 29 1·82850 68 25 2·00602 168 45 2·05165
56 322 30 1·82350 67 49 2·01200 168 10 2·06780
55 320 23 1·81779 67 12 2·01743 167 34 2·08291
54 318 6 1·81148 66 33 2·02232 166 56 2·09694
53 315 39 1·80465 65 52 2·02669 166 17 2·11015
52 313 2 1·79747 65 10 2·03056 165 35 2·12237
51 310 14 1·79005 64 26 2·03396 164 52 2·13370
50 307 14 1·78257 63 41 2·03690 164 7 2·14417
49 304 4 1·77522 62 54 2·03941 163 20 2·15372
48 300 42 1·76818 62 5 2·04151 162 31 2·16267
47 297 8 1·76168 61 14 2·04320 161 40 2·17076
46 293 25 1·75593 60 22 2·04451 160 47 2·17810
45 289 31 1·75115 59 27 2·04545 159 51 2·18474
TABLE III.
° ° ′ ° ′ ° ′
+ 45 289 31 1.75115 59 27 2.04545 159 51 2.81474
44 285 30 1.74752 58 31 2.04065 158 53 2.19069
43 281 22 1.74521 57 33 2.04632 157 53 2.20468
42 277 9 1.74436 56 33 2.04627 156 50 2.20064
41 272 54 1.74504 55 30 2.04592 155 44 2.20468
40 268 38 1.74726 54 26 2.04530 154 36 2.20815
39 264 24 1.75098 53 20 2.04441 153 25 2.21106
38 260 15 1.75611 52 12 2.04328 152 11 2.21343
37 256 10 1.76251 51 1 2.04191 150 55 2.21531
36 252 13 1.77000 49 49 2.04034 149 35 2.21671
35 248 23 1.77838 48 34 2.03857 148 12 2.21766
34 244 43 1.78746 47 17 2.03662 146 46 2.21819
33 241 11 1.79704 45 28 2.03452 145 16 2.21834
32 237 49 1.80692 44 37 2.03228 143 44 2.21813
31 234 36 1.81694 43 14 2.02991 142 8 2.21759
30 231 32 1.82693 41 49 2.02744 140 29 2.21677
29 228 35 1.83676 40 22 2.02488 138 47 2.21568
28 225 47 1.84632 38 53 2.02226 137 1 2.21438
27 223 6 1.85551 37 22 2.01958 135 12 2.21287
26 220 31 1.86452 35 50 2.01686 133 20 2.21123
25 218 2 1.87248 34 15 2.01413 131 25 2.20947
24 215 38 1.88014 32 39 2.01139 129 26 2.20762
23 213 18 1.88721 31 1 2.00866 127 25 2.20572
22 211 3 1.89364 29 22 2.00595 125 21 2.20380
21 208 51 1.89942 27 41 2.00328 123 15 2.20189
20 206 42 1.90455 26 0 2.00065 131 25 2.20947
19 204 35 1.90900 24 17 1.99808 118 56 2.19821
18 202 30 1.91277 22 33 1.99557 116 43 2.19649
17 200 26 1.91588 20 48 1.99313 114 29 2.19487
16 198 23 1.91832 19 3 1.99077 112 14 2.19337
15 196 21 1.92011 17 17 1.98848 109 58 2.19199
14 194 18 1.92126 15 31 1.98626 107 41 2.19075
13 192 15 1.92179 11 57 1.98413 105 23 2.18963
12 190 12 1.92170 11 57 1.98207 103 6 2.18864
11 188 7 1.92104 10 11 1.98007 100 49 2.18776
10 186 1 1.91982 8 24 1.97815 98 33 2.18699
9 183 53 1.91806 6 38 1.97629 96 17 2.18630
8 181 43 1.91581 4 52 1.97446 94 2 2.18568
7 179 31 1.91309 3 7 1.97208 91 48 2.18510
6 177 16 1.90995 1 22 1.97092 89 36 2.18454
5 174 59 1.90641 359 37 1.96919 87 25 2.18397
4 172 38 1.90253 357 54 1.96746 85 16 2.18336
3 170 15 1.89835 356 11 1.96573 83 8 2.18269
2 167 48 1.89392 354 29 1.96397 81 3 2.18191
1 165 17 1.88929 352 48 1.96218 78 59 2.18103
0 162 43 1.88452 351 8 1.90635 76 57 2.17998
TABLE III.
° ° ′ ° ′ ° ′
0 162 43 1.88452 351 8 1.96035 76 57 2.17998
– 1 160 6 1.87966 349 29 1.95846 74 56 2.17876
2 157 25 1.87476 347 50 1.95649 72 58 2.17733
3 154 41 1.86989 346 13 1.95444 71 1 2.17566
4 151 54 1.86509 344 36 1.95228 69 6 2.17374
5 149 4 1.86042 343 1 1.95002 67 12 2.17154
6 146 11 1.85592 341 26 1.94764 65 20 2.16905
7 143 17 1.85164 339 53 1.94512 63 29 2.16623
8 140 20 1.84762 338 20 1.94246 61 39 2.16309
9 137 22 1.84338 336 47 1.93964 59 50 2.15959
10 134 23 1.84045 335 16 1.93667 58 2 2.15573
11 131 23 1.83733 333 45 1.93352 56 15 2.15150
12 128 24 1.83452 332 14 1.93020 54 29 2.14689
13 125 25 1.83203 330 45 1.92669 52 43 2.14188
14 122 27 1.82983 329 15 1.92299 50 57 2.13648
15 119 31 1.82790 327 47 1.91910 49 12 2.13067
16 116 36 1.82621 326 18 1.91501 47 26 2.12446
17 113 44 1.82470 324 50 1.91071 45 41 2.11785
18 110 54 1.82335 323 22 1.90621 43 55 2.11083
19 108 7 1.82211 321 54 1.90150 42 9 2.10341
20 105 23 1.82091 320 26 1.89658 40 22 2.09559
21 102 43 1.81971 318 58 1.89145 38 34 2.08737
22 100 5 1.81846 317 30 1.88612 36 45 2.07878
23 97 30 1.81710 316 2 1.88057 34 56 2.06981
24 94 59 1.81560 314 34 1.87483 33 5 2.06047
25 92 31 1.81388 313 5 1.86887 31 13 2.05078
26 90 5 1.81193 311 37 1.86272 29 20 2.04076
27 87 43 1.80968 310 8 1.85637 27 26 2.03041
28 85 23 1.80711 308 38 1.84983 25 29 2.01975
29 83 5 1.80419 307 8 1.84311 23 32 2.00881
30 80 50 1.80087 305 38 1.83621 21 33 1.99760
31 78 36 1.79714 304 7 1.82913 19 32 1968614
32 76 25 1.79296 302 35 1.82188 17 30 1.97445
33 74 14 1.78834 301 3 1.81447 15 26 1.96255
34 72 5 1.78323 299 31 1.80690 13 20 1.95047
35 69 57 1.77765 297 58 1.79919 11 14 1.93821
36 67 49 1.77157 296 25 1.79134 9 6 1.92581
37 65 42 1.76499 294 51 1.78335 6 57 1.91327
38 63 35 1.75791 293 16 1.77524 4 47 1.90061
39 61 27 1.75034 291 41 1.76701 2 37 1.88785
40 59 19 1.74228 290 6 1.75866 0 26 1.87498
41 57 10 1.73373 288 31 1.75020 358 14 1.86202
42 55 0 1.72472 286 55 1.74163 356 3 1.84896
43 52 49 1.71526 285 19 1.73297 353 52 1.83580
44 50 37 1.70537 283 43 1.72420 351 42 1.82252
45 48 23 1.69506 282 7 1.71533 349 33 1.80912
TABLE III.
° ° ′ ° ′ ° ′
– 45 48 23 1.69506 282 7 1.71533 349 33 1.80912
46 46 7 1.68438 280 31 1.70636 347 25 1.79558
47 43 49 1.67335 278 56 1.69726 345 18 1.78186
48 41 29 1.66199 277 21 1.68810 343 13 1.76793
49 39 7 1.65036 275 47 1.67880 341 10 1.75376
50 36 42 1.63848 274 13 1.66937 339 10 1.73931
51 34 16 1.62640 272 40 1.65981 337 12 1.72452
52 31 47 1.61415 271 8 1.65009 335 17 1.70935
53 29 17 1.60177 269 37 1.64021 333 25 1.69375
54 26 45 1.58929 268 7 1.63013 331 35 1.67764
55 24 11 1.57675 266 30 1.61985 329 50 1.66098
56 21 37 1.56416 265 13 1.60933 328 7 1.64368
57 19 2 1.55158 263 47 1.59855 326 28 1.62568
58 16 26 1.53898 262 23 1.58747 324 52 1.60691
59 13 51 1.52638 261 2 1.57607 323 21 1.58728
60 11 17 1.51376 259 42 1.56430 321 52 1.56672
61 8 44 1.50111 258 25 1.55212 320 27 1.54513
62 6 13 1.48839 257 9 1.53949 319 6 1.52242
63 3 45 1.47556 255 56 1.52635 317 48 1.49850
64 1 20 1.46254 254 46 1.51265 316 34 1.47326
65 358 58 1.44928 253 37 1.49834 315 24 1.44658
66 356 40 1.43567 252 31 1.48335 314 17 1.41834
67 354 27 1.42163 251 28 1.46760 313 13 1.38840
68 352 19 1.40704 250 27 1.45101 312 12 1.35661
69 350 15 1.39176 249 29 1.43351 311 15 1.32281
70 348 18 1.37567 248 34 1.41498 310 21 1.28680
71 346 25 1.35860 247 41 1.39531 309 30 1.24837
72 344 39 1.34039 246 51 1.37437 308 42 1.20727
73 342 59 1.32084 246 3 1.35202 307 57 1.16322
74 341 25 1.29975 245 18 1.32808 307 16 1.11588
75 339 56 1.27687 244 36 1.30235 306 37 1.06485
76 338 34 1.25192 243 57 1.27458 306 0 1.00966
77 337 18 1.22457 243 21 1.24448 305 27 0.94972
78 336 8 1.19443 242 47 1.21167 304 56 0.88472
79 335 4 1.16100 242 16 1.17572 304 28 0.81256
80 334 5 1.12370 241 47 1.13602 304 3 0.73327
81 333 13 1.08172 241 22 1.09181 303 40 0.64493
82 332 26 1.03401 240 59 1.04207 303 19 0.54547
83 331 45 0.97911 240 39 0.98533 303 1 0.43201
84 331 10 0.91487 240 21 0.91948 302 46 0.30031
85 330 40 0.83802 240 6 0.84123 302 33 0.04380
86 330 16 0.74302 239 54 0.74509 302 22 9.95118
87 329 57 0.61958 239 45 0.62075 302 14 9.70281
88 329 44 0.44456 239 38 0.44509 302 8 9.35148
89 329 35 0.14417 239 34 0.14432 302 5 8.74992
90 329 33 − ∞ 239 33 − ∞ 302 3 − ∞
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  1. In the declination there is even a difference in one instance of 29 degrees; but it is proper to estimate the error of the calculation, not by the number of degrees of declination, but by the true angular difference between the calculated and observed directions, which in the case in question is 11½ degrees.
  2. If the function could have any arbitrarily chosen form, then in particular cases a maximum or a minimum value of might correspond to an insulated point, or to an insulated line, around which only greater or only less values might be found, or it might correspond to a surface on both sides of which there might be greater or on both less values. But the conditions to which the function is subjected do not allow the occurrence of such excepted cases. A full development of this subject, as it is unnecessary for our present object, must be reserved for another occasion.
  3. These rings, themselves assumed as infinitely small, are not necessarily circular, but generally speaking oval, so that the normal direction of the magnetic needle in reference to them only coincides with the direction towards at four points of each ring. Great error may be involved, therefore, if without further precaution, the intersection of the prolongations of two compass directions at considerable distances is assumed to be .
  4. We conform here to the mode of speaking in common usage, according to which the point established by Captain James Ross is so designated, although properly speaking it is a south pole, when the earth itself is considered as a magnet.
  5. In such a mode of determination, the effect of these circumstances would be least injurious if the eight points were distributed symmetrically on the surface of the earth; that is to say, if they coincided, or nearly so, with the corners of a cube inscribed in the globe.
  6. The last article presents instances of discordances between different observers at one and the same place; I will notice some others, which are much greater than can with any degree of probability be attributed to yearly changes. The dip at Valparaiso was, in 1829, according to King, 40° 11′; in 1835, according to Fitz Roy, 38° 3′. In Mauritius the intensity was 1·096 in 1818, according to Freycinet, and 1·192 in 1836, according to Fitz Roy. The difference is still greater at Otaheite, where Erman's intensity = 1·172 in 1830, and Fitz Roy's, in 1835, = 1·017. Otaheite is a station of the highest importance for the future improvement of the elements: the difference between the two determinations made there by different observers, considerably exceeds the greatest difference between the computed and observed intensities in our eighty-six comparisons.
  7. In as far as we are not prepared to assume the magnetic axes of all the magnetized particles of the earth to be everywhere parallel to each other,—the more imperfect this parallelism, the greater must be the average force of magnetism in the parts to produce the same total magnetic moment.

 This work is a translation and has a separate copyright status to the applicable copyright protections of the original content.

Original:

This work was published before January 1, 1929, and is in the public domain worldwide because the author died at least 100 years ago.

Public domainPublic domainfalsefalse

Translation:

This work was published before January 1, 1929, and is in the public domain worldwide because the author died at least 100 years ago.

Public domainPublic domainfalsefalse