Scientific Memoirs/2/General Theory of Terrestrial Magnetism
Article V.
General Theory of Terrestrial Magnetism. By Professor Carl Friedrich Gauss, of the University of Göttingen.
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 buildingstones, 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 nonaccordance 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 gieat 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 firstmentioned 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 fortyeight 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 coordinates of in relation to three rectangular axes, and by , , and , the coordinates of the point where the force is exercised, so that; 
which, as is easily seen, are equal to the partial differential coefficients 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 coordinate axes, are equal to the partial differential coefficients of, 
relatively to , , and .
4.
. 
. 
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. 
, 
because as , , are the cosines of the angles which the direction of makes with the coordinate 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 coordinate axes may be arbitrarily chosen.
6.
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 reentering curve, is always .
III. In a reentering 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, 
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, 
. 
, 
and so on through the whole polygon.
Therefore, for a triangle our proposition gives the approximatively correct equationIt 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 
Taking the geographical position of
Göttingen  51°  32′  latitude  9°  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)  =  5°  11′  31″  =  6°  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 
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 lastnamed 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.
; 
. 
, 
. 
. 
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 lastmentioned formula, in a which the earth is considered as a sphere, whose semidiameter .
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  . 
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. 
, 
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 coefficients of the series for may be assumed as known; it is merely necessary for our purpose to remark, that in relation to , , the coefficients are rational integral functions of , and , and of of the second order, of the third, &c. It is the same as to the coefficients , , &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.
, 
, 
, &c., 
. 
where , , , , &c. are determinate numerical coefficients.
19.
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.
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 abovementioned 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 : thusIII. 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. 
&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 coefficients , , ; requires five coefficients; 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 coefficients 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 coefficients, 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 coefficients. 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 formWe then obtain as many values for each of the coefficients , , , , &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 equationsthe 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 coefficients , , , &c. as require to be taken into account.
In like manner the equationsdetermine the coefficients , , , &c.; and we obtain the coefficients 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 oftenexpressed 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 abovementioned 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 coefficients to be determined amount to twentyfour. 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 coefficients by the factor which reduces to an absolute measure the intensities expressed according to the arbitrary unity.
26.
The numerical values of the 24 coefficients 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.
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, 
. 
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 coefficients 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 
+ 5008  + 14652  
 3571  + 15289   6899 
+ 5476  + 992   13367 
 221  + 2877  + 827 
 392  + 019  + 390 
X = + 51372  Y = + 19177  Z= + 127471 
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.

 



 

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 abovementioned 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 abovenamed 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 abovementioned 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 formulaor 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 plumbline 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 wellknown 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 reentering 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.
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 makeis 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 coefficients , , , &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 notNow, 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.
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 oftenmentioned 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 