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.

[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 $=0$ 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 $\mu$ concentrated in a point exercises at the distance $\rho$ the magnetic force ${\frac {\mu }{\rho ^{2}}}$, of either repulsion or attraction in the direction $\rho$, according as $\mu$ is positive or negative. Representing by $a$, $b$, $c$, the co-ordinates of $\mu$ in relation to three rectangular axes, and by $x$, $y$, and $z$, the co-ordinates of the point where the force is exercised, so that

$\rho ={\sqrt {}}([x-a)^{2}+(y-b)^{2}+(z-c]^{2})$;

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

${\frac {\mu (x-a)}{\rho ^{3}}},\quad {\frac {\mu (y-b)}{\rho ^{3}}},\quad {\frac {\mu (z-c)}{\rho ^{3}}},$

which, as is easily seen, are equal to the partial differential co-efficients of $-{\frac {\mu }{\rho }}$ relatively to $x$, $y$, and $z$.

If besides $\mu$, there are also in operation other portions of the magnetic fluids $\mu '$, $\mu ''$, $\mu '''$, &c., concentrated in points, of which the distances from the spot where the force is exercised are $\rho '$, $\rho ''$, $\rho '''$, &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

$-\left({\frac {\mu }{\rho }}+{\frac {\mu '}{\rho '}}+{\frac {\mu ''}{\rho ''}}+{\frac {\mu '''}{\rho '''}}+{}{\&}{\mbox{c.}}\right)$,

relatively to $x$, $y$, and $z$.

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 $\operatorname {d} \!\,\mu$, in which the southern fluid is always considered as negative; call $\rho$ the distance of $\operatorname {d} \!\,\mu$ from a point in space, the rectangular co-ordinates of which may be $x$, $y$, $z$; lastly, let $V$ denote the aggregate of ${\frac {\operatorname {d} \!\,\mu }{\rho }}$ comprehending with reversed signs the whole of the magnetic particles of the earth: or say

$V=-\int {\frac {\operatorname {d} \!\,\mu }{\rho }}$.

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

$\xi ={\frac {d\,V}{d\,x}},\quad \eta ={\frac {d\,V}{d\,y}},\quad \zeta ={\frac {d\,V}{d\,z}},\quad \psi ={\sqrt {}}(\xi ^{2}+\eta ^{2}+\zeta ^{2})$.

5.

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

The complete differential of $V$ becomes

${\begin{aligned}d\,V&={\frac {d\,V}{d\,x}}.d\,x+{\frac {d\,V}{d\,y}}.d\,y+{\frac {d\,V}{d\,z}}.d\,z.\\&=\xi d\,x+\eta d\,y+\zeta d\,z\end{aligned}}$.

If we designate by $d\,s$ the distance between the two points to which $V$ and $V+d\,V$ belong, and by $\theta$ the angle which the direction of the magnetic force $\psi$ makes with $d\,s$, we have

$d\,V=\psi \cos \theta .d\,s$,

because as ${\frac {\xi }{\psi }}$, ${\frac {\eta }{\psi }}$, ${\frac {\zeta }{\psi }}$ are the cosines of the angles which the direction of $\psi$ makes with the co-ordinate axes, so ${\frac {d\,x}{d\,s}}$, ${\frac {d\,y}{d\,s}}$, ${\frac {d\,z}{d\,s}}$, are the cosines of the angles between $d\,s$ and the same axes.

Therefore ${\frac {d\,V}{d\,s}}$ is equal to the force resolved in the direction of $d\,s$; the same follows from the equation ${\frac {d\,V}{d\,x}}=\xi$ if we bear in mind that the co-ordinate axes may be arbitrarily chosen.

6.

If two points in space $P^{0}$, $P'$, be connected by an arbitrary line, of which d s represent an indeterminate element, and if, as before, $\theta$ signify the angle between d s and the direction of the magnetic force there existing, and $\psi$ its intensity, then

$\int \psi \cos \theta .d\,s=V'-V^{0}$

if we extend the integration through the whole line, and designate by $V^{0}$, $V'$, the values of $V$ at the extremities.

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

I. The integral $\int \psi \cos .\theta .d\,s$ preserves the same value by whatever path we proceed from $P^{0}$ to $P'$.

II. The integral $\int \psi \cos \theta .d\,s$, extended through the whole length of any re-entering curve, is always $=0$.

III. In a re-entering curve, if $\theta$ is not throughout $=90^{\circ }$, a part of the values of $\theta$ must be greater and a part must be less than $90^{\circ }$.

7.

Those points of space in which $V$ has a value greater than $V^{0}$, are divided from those in which the value of $V$ is less than $V^{0}$, by a surface in all the points of which $V$ has one determinate value $=V^{0}$^[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 $V$ are found. Let $d\,s$ be an infinitely small line perpendicular to the surface, and $V^{0}+d\,V^{0}$ the value of $V$ at its other extremity; then the intensity of the magnetic force will be $={\frac {d\,V^{0}}{d\,s}}$. The series of points for which $V=V^{0}+d\,V^{0}$, 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 $V$ 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 $V$ on the surface of the earth.

At a point $P$ of the earth's surface let $\psi$ be the intensity; $PM$ the direction of the whole magnetic force; $\omega$ the intensity, and $PN$ the direction of the force projected on the horizontal plane, or $PN$ the direction of the magnetic meridian, meaning thereby the direction indicated by the north pole of the magnetic needle; $i$ the angle between $PM$ and $PN$, or the dip; $\theta$, $t$, the angles formed by the elementary portion $d\,s$ of a line on the surface of the earth and the directions $PM$, $PN$. Lastly, $V$ and $V+d\,V$ correspond to the two extremities of $d\,s$.

We have consequently

$\cos \theta .=\cos i\cos t,\omega =\psi \cos i.$

And the equation in Art. V. becomes

$d\,V=\omega \cos t.d\,s$

If two points on the earth's surface $P^{0}$ and $P'$, at which $V$ has the value of $V^{0}$ and $V'$, are connected by a line traced on the surface of the earth of which $d\,s$ is an indeterminate element, then

$\int \omega \cos t.d\,s=V'-V^{0}$,

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 $\int \omega \cos t.d\,s$ keeps the same value by whatever path you proceed on the surface of the earth from $P^{0}$ to $P'$.

II. The integral $\int \omega \cos t.d\,s$ throughout the whole length of a closed line on the surface of the earth is always $=0$.

III. In such a closed line, unless throughout its course $t=90^{\circ }$, a part of the values of $t$ 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 $P^{0},\,P',\,P''\ldots P^{0}$ 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 $\omega ^{0}$, $\omega '$, $\omega ''$, &c. be the intensities of the horizontal magnetic force at the points $P^{0}$, $P'$, $P''$, &c.; further, let $\delta ^{0}$, $\delta '$, $\delta '',$ &c. be the declinations reckoned in the usual manner, west of north as positive, east of north as negative; lastly, let $(01)$ be the azimuth of the line $P^{0}P'$ at $P^{0}$, reckoned in the customary manner, from the south by the west; in like manner $(10)$ the azimuth of the same line taken backwards at $P'$, and so on.

Let it be observed that $t$ alters continuously in each of the sides of the polygon, but suddenly at the corners, where therefore ithas two different values; for example, at $P$, $t$ has the value $(10)+\delta '$, in consideration that $P'$ is the end of the line $P^{0}P'$; and the value of $18^{rc}+(12)+\delta '$, in regard that it is the beginning of $P'P''$.

We may consider the approximate value of the integral $\int \omega \cos t.d\,s$, extended through $P^{0}P'$, to be

${\frac {1}{2}}(\omega ^{0}\cos t^{0}+\omega '\cos t').P^{0}P'$,

where $t^{0}$ and $t'$ signify the values of $t$ at $P^{0}$ as the beginning, and at $P'$ as the end of $P^{0}P'$. This approximation is all that can be obtained, because we have the values of $\omega$ and t</math> only at the extremities $P^{0}P'$, and is deserving of confidence in proportion to the shortness of the line. The given expression is, in our notation,

$={\frac {1}{2}}(\omega '\cos((10))+\delta ')-\omega ^{0}\cos((01))+\delta ^{0}]).P^{0}P'$.

In like manner, the approximate value of the integral, extended through $P'P''$, is

${\frac {1}{2}}(\omega '\cos((21)+\delta '')-\omega '\cos(12)+\delta ']).P'P''$,

and so on through the whole polygon.

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

${\begin{aligned}&\omega ^{0}(P^{0}P'\cos((01)+\delta ^{0})-P^{0}P''\cos((02)+\delta ^{0}])\\+&\omega '(P'P''\cos((12)+\delta ^{'})-P^{0}P'\cos((10)+\delta '])\\+&\omega ''(P^{0}P''\cos((20)+\delta '')-P'P''\cos((21)+\delta ''])=0.\end{aligned}}$

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	$\delta ^{0}=18^{rc}\;38'$	$i^{0}=67^{\circ }\;56'$	$\psi ^{0}=1.357$
Milan	$\delta '=18\;\;33$	$i'=63\;\,49$	$\psi '=1.294$
Paris	$\delta ''=22\;\;04$	$i''=67\;\;24$	$\psi ''=1.348$

whence it follows that

${\begin{aligned}&\omega ^{0}=0.50980\\&\omega '=0.57094\\&\omega ''=0.51804\end{aligned}}.$

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″	$P^{0}P'$	=	6°	5′	20″
(10)	=	184	35	35
(12)	=	128	47	31	$P'P''$	=	5	41	06
(21)	=	303	48	01
(20)	=	238	20	20	$P^{0}P''$	=	5	32	04
(02)	=	64	10	12

Substituting these values in our equation, and those given above for $\delta ^{0}$, $\delta '$, $\delta ''$, we have

${\begin{aligned}&0=17556\;\omega ^{0}+2774\;\omega '-20377\;\omega '',\\&{\text{or, }}\omega ''=0.86158\;\omega ^{0}+0.13613\;\omega '.\end{aligned}}$

Hence we deduce from the observed horizontal intensities at Göttingen and Milan, that at Paris $\omega ''=0.51696$, agreeing almost exactly with the observed value $0.51804$.

It is easily seen that if we permit ourselves to take the distances $P^{0}$, $P''$, &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 $V$ has the same value $=V^{0}$, divides generally speaking the parts of the surface in which the value of $V$ is greater than $V^{0}$, 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 $V$ are found. If $d\,s$ be an infinitely small line in this direction, and $V^{0}+d\,V^{0}$ the value of $V$ at the other extremity of this line, then ${\frac {d\,V^{0}}{d\,s}}$ is the intensity of the horizontal magnetic force at this place. As here also the series of points corresponding to the value of $V=V^{0}+d\,V^{0}$ 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 $V$ are between $V^{0}$ and $V^{0}+d\,V^{0}$, and where the horizontal intensity is in an inverse ratio to the breadth of the zone; so by making $V$ 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 $V$ correspond in this point of view to two points, inclosed by the zones, at which the horizontal force is $=0$, 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 $=0$: where therefore, speaking generally, the dip $=90$; but including the singular case, did it exist, where the total intensity $=0$. 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 $V$ have at a point of the earth's surface $P^{*}$ a maximum value $V^{*}$, 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 $P^{*}$, 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 $V$ may differ so little from $V^{*}$, that any other point may be excluded.

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

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

If then $W$ 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 $=V^{***}$, characterised by being the lowest at which $P^{*}$ and $P^{**}$ are still situated in separate portions of $S$, which separate portions will unite whenever $W$ is diminished further.

If this union occur at a point $P^{***}$, the bounding line on which $V=V^{***}$ will have the form of an $8$, crossing at that point; where also we may easily satisfy ourselves that the horizontal intensity must $=0$. 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 $=0$, 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 $8$ touch each other at $P^{***}$, 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 $8$, which involves a contradiction, as the value of $V$ increases towards both sides; therefore $P^{***}$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 $8$, and as a north pole as regards the points which lie outside. Figure 1. illustrates this form of the system of lines.

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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 $P^{*}$ and $P^{**}$ an insular space would be formed, which would gradually contract itself as $W$ 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 $V$ 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 $V$ 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.

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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 $V$ 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 $\lambda$, and the second, or the complement of the geographical latitude, by $u$. Considering the earth as a spheroid of revolution, of which the greater semi-axis $=R$, and the lesser semi-axis $=(1-\epsilon )R$, an element of the meridian is

$={\frac {(1-\epsilon )^{2}R.d\,u}{(1-(2\epsilon -\epsilon ^{2})\cos u^{2})}}{\frac {3}{2}}$;

and an element of the parallel is

$={\frac {R\sin u.d\,\lambda }{\sqrt {(1-(2\epsilon -\epsilon ^{2})\cos u^{2})}}}$

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

${\begin{aligned}&X=-{\frac {(1-(2\epsilon -\epsilon ^{2})\cos u^{2}){\frac {3}{2}}}{(1-\epsilon )^{2}}}\centerdot {\frac {d\,V}{R\,d\,u}}\\&Y=-{\sqrt {(1-(2\epsilon -\epsilon ^{2})\cos u^{2})}}\centerdot {\frac {d\,V}{R\sin u.d\,\lambda }}\end{aligned}}$.

The total horizontal force is then

$={\sqrt {(X^{2}+Y^{2})}}$,

and the tangent of the declination

$={\frac {Y}{X}}$.

Neglecting the square of the ellipticity, $\epsilon$, the expressions become

${\begin{aligned}&X=-(1+(2-3\cos u^{2})\epsilon )\centerdot {\frac {d\,V}{R\,d\,u}}\\&Y=-(1-\epsilon \cos u^{2})\centerdot {\frac {d\,V}{R\sin u.d\,\lambda }},\end{aligned}}$

or, setting the ellipticity quite aside,

$X=-{\frac {d\,V}{R\,d\,u}}$

$Y=-{\frac {d\,V}{R\sin u.d\,\lambda }}$.

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 $=R$.

15.

If $X$ be expressed by a given function of $u$ and $\lambda$, $Y$ can be be deduced from it a priori.

Let the integral $\int _{0}^{u}X\,d\,u=T$, considering $\lambda$ as constant in the integration: it is then clear that if we differentiate in a similar manner according; to $u$, ${\frac {d\,(V+RT)}{d\,u}}=0$; $V+RT$ having a value independent of $u$, 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 $V$ at the north pole $=V^{*}$, then $T={\frac {V^{*}-V}{R}}$;

and hence

$Y={\frac {d\,T}{\sin u.d\,\lambda }}$.

This result may also be expressed as follows:

$Y={\frac {1}{\sin u}}\int _{0}^{u}{\frac {d\,X}{d\,\lambda }}\centerdot d\,u.$

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 $Y$ be expressed by a given function of $u$ and $\lambda$, and if $U$ represent the indeterminate integral $\int \sin u.Y\,d\,\lambda$ being assumed constant in the integration, then ${\frac {d\,(V+RU)}{d\,\lambda }}=0$, or $V+RU$ has a value independent of $\lambda$, and is, generally speaking, a function of $u$. Thus ${\frac {d\,(V+RU)}{R\,d\,u}}={\frac {d\,U}{d\,u}}-X$ is such a function; that is to say, the formula ${\frac {d\,U}{d\,u}}$ gives an imperfect expression for $X$, a part of it containing $u$ only remaining undetermined. This want would be supplied if, besides the expression for $Y$, we had also that for $X$, 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 $V$ must be regarded as a function of three variable magnitudes, expressing the position in space of an undetermined point $O$. We select for the purpose the distance $r$ from the centre of the earth, the angle $u$ which $r$ makes with the northern part of the earth's axis, and the angle $\lambda$, which a plane passing through $r$ and the axis of the earth makes with a first meridian, counted as positive towards the east.

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

$V={\frac {R^{2}P^{0}}{r}}+{\frac {R^{3}P'}{r^{2}}}+{\frac {R^{4}P''}{r^{3}}}+{\frac {R^{5}P'''}{r^{4}}}$, &c.

The co-efficients $P^{0}$, $P'$, $P''$, &c. are here functions of $u$ and $\lambda$; in order to see how they are connected with the distribution of the magnetic fluid in the earth, let $\mathrm {d} \,\mu$ be an element of the earth's magnetism, $\rho$ its distance from $O$, and let $r^{0}$, $u^{0}$, $\lambda ^{0}$, signify for $\mathrm {d} \,\mu$ the same as $r$, $u$, $\lambda$ for $O$. We have then $V=-\int {\frac {\mathrm {d} \,\mu }{\rho }}$ extended so as to include every $\mathrm {d} \,\mu$; further $\rho ={\sqrt {\;}}(r^{2}-2rr^{0})\cos u\cos u^{0}+\sin u\sin u^{0}\cos(\lambda -\lambda ^{0})+r^{0}r^{0}$, and if ${\frac {1}{\rho }}$ be developed in the series,

${\frac {1}{\rho }}={\frac {1}{r}}(T^{0}+T'{\frac {r^{0}}{r}}+T''.{\frac {r^{0}r^{0}}{r^{2}}}+{\&}{\mbox{c.}})$

then

${\begin{aligned}&R^{2}P^{0}=-\int T^{0}\,d\,\mu ,\quad R^{3}P'=-\int T'r^{0}d\,\mu ,\\&R^{4}P''=-\int T''r^{0}r^{0}\mathrm {d} \,\mu ,\;{\&}{\mbox{c.}}\end{aligned}}$

As $T^{0}=1$, 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, $\int \mathrm {d} \,\mu =0$, it follows that

$P^{0}=0$,

or the first number of our series for $V$ goes out.

We see further that $P'$ has the form

$R^{3}P'=\alpha \cos u+\beta \sin u\cos \lambda +\gamma \sin u\sin \lambda$,

where $\alpha =-\int \cos u^{0}r^{0}\,\mathrm {d} \,\mu$, $\beta =-\int \sin u^{0}\cos \lambda ^{0}r^{0}\,\mathrm {d} \,\mu$, $\gamma =-\int \sin u\sin \lambda ^{0}r^{0}\,\mathrm {d} \,\mu$. Therefore, according to the explanation laid down in page 13 of the Intensitas Vis Magneticæ, $-\alpha$, $-\beta$, $-\gamma$ 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 ${\frac {1}{\rho }}$ may be assumed as known; it is merely necessary for our purpose to remark, that in relation to $u$, $\lambda$, the co-efficients are rational integral functions of $\cos u.\sin u\cos \lambda$, and $\sin u\sin \lambda$, and of $T''$ of the second order, $T'''$ of the third, &c. It is the same as to the co-efficients $P''$, $P'''$, &c.

The series for ${\frac {1}{\rho }}$, and for $V$, converge, so long as $r$ is not less than $R$, or rather, not less than the half diameter of a sphere, which includes all the magnetic particles of the earth.

18.

The function $V$ being composed of $-\int {\frac {\mathrm {d} \,\mu }{\rho }}$, satisfies the following partial differential equation:

$0={\frac {r\,d^{2}\,V}{d\,r^{2}}}+{\frac {d^{2}\,V}{d\,u^{2}}}+\cot u.{\frac {d\,V}{d\,u}}+{\frac {1}{\sin u^{2}}}\centerdot {\frac {d^{2}\,V}{d\,\lambda ^{2}}}$,

which is only transformation of the well-known equation

$0={\frac {d^{2}\,V}{d\,x^{2}}}+{\frac {d^{2}\,V}{d\,y^{2}}}+{\frac {d^{2}\,V}{d\,z^{2}}}$,

where $x$, $y$, $z$ signify the rectangular co-ordinates of $O$. If we substitute the value of $V$,

$V={\frac {R^{3}P'}{r^{2}}}+{\frac {R^{4}P''}{r^{3}}}+{\frac {R^{5}P'''}{r^{4}}}+$, &c.,

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

$0=n(n+1)P^{(n)}+{\frac {d^{2}\,P^{(n)}}{d\,u^{2}}}+\cot u{\frac {d\,P^{(n)}}{d\,u}}+{\frac {1}{\sin u^{2}}}\centerdot {\frac {d^{2}\,P^{(n)}}{d\,\lambda ^{2}}}$.

From this equation, combined with the remark in the preceding article, we obtain the general form of $P^{(n)}$. If we represent by $P^{n,m}$ the following function of $u$,

${\Big (}\cos u^{n-m}-{\frac {(n-m)(n-m+1)}{2(2n-1)}}\cos u^{n-m-2}+{\frac {(n-m)(n-m-1)(n-m-2)(n-m-3)}{2\centerdot 4(2n-1)(2n-3)}}\cos u^{n-m-4}-{\&}{\mbox{c.,}}{\Big )}\sin u^{m}$

then $P^{(n)}$ has the form of an aggregate of $2n+1$ parts,

$P^{(n)}=g^{n,0}P^{n,0}+(g^{n,1}\cos \lambda +h^{n,1}\sin \lambda )P^{n,1}+(g^{n,2}\cos 2\lambda +h^{n,2}\sin 2\lambda )P^{n,2}+{\mbox{,}}{\&}{\mbox{c.}}+(g^{n,n}\cos n\lambda +h^{n,n}\sin n\lambda )P^{n,n},$

where $g^{n,0}$, $g^{n,1}$, $h^{n,1}$, $g^{n,2}$, &c. are determinate numerical co-efficients.

19.

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

$X=-{\frac {d\,V}{r\,d\,u}},\quad Y=-{\frac {d\,V}{r\sin u\,d\,\lambda }},\quad Z=-{\frac {d\,V}{d\,r}};$

consequently,

${\begin{aligned}&X=-{\frac {R^{3}}{r^{3}}}\left({\frac {d\,P'}{d\,u}}+{\frac {R}{r}}\centerdot {\frac {d\,P''}{d\,u}}+{\frac {R^{2}}{r^{2}}}\centerdot {\frac {d\,P'''}{d\,u}},{\&}{\mbox{c.}}\right)\\&Y=-{\frac {R^{3}}{r^{3}\sin u}}\left({\frac {d\,P'}{d\,\lambda }}+{\frac {R}{r}}\centerdot {\frac {d\,P''}{d\,\lambda }}+{\frac {R^{2}}{r^{2}}}\centerdot {\frac {d\,P'''}{d\,\lambda }},{\&}{\mbox{c.}}\right)\\&Z={\frac {R^{3}}{r^{3}}}\left(2P'+{\frac {3RP''}{r}}+{\frac {4R^{2}P}{r^{2}}},{\&}{\mbox{c.}}\right)\end{aligned}}$

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

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

$X=-\left({\frac {d\,P'}{d\,u}}+{\frac {d\,P''}{d\,u}}+{\frac {d\,P'''}{d\,u}}+{},{\&}{\mbox{c.}}\right)$

${\begin{aligned}&Y=-{\frac {1}{\sin u}}\left({\frac {d\,P'}{d\,\lambda }}+{\frac {d\,P''}{d\,\lambda }}+{\frac {d\,P'''}{d\,\lambda }}+{},{\&}{\mbox{c.}}\right)\\&Z=2P'+3P''+4P'''+{},{\&}{\mbox{c.}}\end{aligned}}$

20.

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

$P^{0}+P'+P''+P'''+{},{\&}{\mbox{c.}}$

the general member of which, $P^{n,}$ 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 $V$ at all points of the earth's surface is sufficient to enable us to deduce the general expression of $V$ for all external space, and thus to determine the forces $X$, $Y$, $Z$, not only on the surface of the earth, but also for all external space.

It is clearly only necessary for this purpose to develope ${\frac {V}{R}}$into a series according to the above-mentioned theorem.

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

$V=R(P'+P''+P'''+{},{\&}{\mbox{c.}})$

II. The knowledge of the value of $X$ 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 $\int _{0}^{u}X\,d\,u={\frac {V^{0}-V}{R}}$ signifying the value of $V$ at the north pole, and the developement of $\int _{0}^{u}X\,d\,u$ into a series of the form referred to must necessarily be identical with

$V^{0}-P'-P''-P''',{\&}{\mbox{c.}}$

III. In like manner, and under the considerations in Art. 16, it is clear that the knowledge of $Y$ on the whole earth, combined with the knowledge of $X$ 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 $Z$ on the whole surface of the earth. In fact, if $Z$ be developed into a series,

$Z=Q^{0}+Q'+Q''+Q'''+,$ &c.

so that the general member satisfies the often-mentioned partial differential equation; $Q^{0}$ must necessarily $=0$, and

$P'={\frac {1}{2}}Q',\quad P''={\frac {1}{3}}Q'',\quad P'''={\frac {1}{4}}Q''',$ &c.

21.

On account of the simple nature of the dependence of the several forces $X$, $Y$, $Z$, on a single function $V$, 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 $V$, $X$, $Y$, $Z$, 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, $P'$ contains three members, and requires, therefore, the knowledge of three co-efficients $g^{1\centerdot 0}$, $g^{1\centerdot 1}$, $h^{1\centerdot 1}$; $P''$ requires five co-efficients; $P'''$ seven; $P^{IV}$ nine, &c. As we consider $P'$, $P''$, $P'''$, &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 $n$ inclusive, the values of $n^{2}+2n$ co-efficients must be determined; therefore, for example, 24 coefficients, if we would go as far as the fourth order.

Every given value of $X$, $Y$, or $Z$, for given values of $u$ and $\lambda$, 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 $X$, $Y$, and $Z$, are to be first deduced from the given elements of the usual form.

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

${\begin{aligned}&X=k+&k'\cos \lambda +K'\sin \lambda +k''\cos 2\lambda +K''\sin 2\lambda \\&&+k'''\cos 3\lambda +K'''\sin 3\lambda +\mathrm {\&c.} \\&Y=l+&l'\cos \lambda +L'\sin \lambda +l''\cos 2\lambda +L''\sin 2\lambda \\&&+l'''\cos 3\lambda +L'''\sin 3\lambda +,\,\mathrm {\&c.} \\&Z=m+&m'\cos \lambda +M'\sin \lambda +m''\cos 2\lambda +M''\sin 2\lambda \\&&+m'''\cos 3\lambda +M'''\sin 3\lambda +\mathrm {,\&c.} \end{aligned}}$

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

Theory would give in each parallel $l=0$; therefore the values of $l$ 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

${\begin{aligned}&k=-g^{1\centerdot 0}{\frac {d\,P^{1\centerdot 0}}{d\,u}}-g^{2\centerdot 0}{\frac {d\,P^{2\centerdot 0}}{d\,u}}-g^{3\centerdot 0}{\frac {d\,P^{3\centerdot 0}}{d\,u}}-,\;\mathrm {\&c.} \\&m=2g^{1\centerdot 0}P^{1\centerdot 0}+3g^{2\centerdot 0}P^{2\centerdot 0}+4g^{3\centerdot 0}\rho ^{3\centerdot 0}+,\;\mathrm {\&c.} {\mbox{,}}\end{aligned}}$

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 ${\frac {d\,P^{1\centerdot 0}}{d\,u}}$, &c., and in $P^{1\centerdot 0}$, &c. the corresponding numerical values of $u$,) as many of the co-efficients $g^{1\centerdot 0}$, $g^{2\centerdot 0}$, $g^{3\centerdot 0}$, &c. as require to be taken into account.

In like manner the equations

${\begin{aligned}-&k'=g^{1\centerdot 1}{\frac {d\,P^{1\centerdot 1}}{d\,u}}+g^{2\centerdot 1}{\frac {d\,P^{2\centerdot 1}}{d\,u}}+g^{3\centerdot 1}{\frac {d\,P^{3\centerdot 1}}{d\,u}}+,\;\mathrm {\&c.} \\&L'=g^{1\centerdot 1}{\frac {P^{1\centerdot 1}}{\sin u}}+g^{2\centerdot 1}{\frac {P^{2\centerdot 1}}{\sin u}}+g^{3\centerdot 1}{\frac {P^{3\centerdot 1}}{\sin u}}+,\;\mathrm {\&c.} \\&m'=2g^{1\centerdot 1}P^{1\centerdot 1}+3g^{2\centerdot 1}P^{2\centerdot 1}+4g^{3\centerdot 1}P^{3\centerdot 1}+,\;\mathrm {\&c.} {\mbox{,}}\end{aligned}}$

the number of which is three times as great as the number of parallels, serve to determine the co-efficients $g^{1\centerdot 1}$, $g^{2\centerdot 1}$, $g^{3\centerdot 1}$, &c. And the following,

${\begin{aligned}-&K'=h^{1\centerdot 1}{\frac {d\,P^{1\centerdot 1}}{d\,u}}+h^{2\centerdot 1}{\frac {d\,P^{2\centerdot 1}}{d\,u}}+h^{3\centerdot 1}{\frac {d\,P^{3\centerdot 1}}{d\,u}}+,\;\mathrm {\&c.} \\-&l'=h^{1\centerdot 1}{\frac {P^{1\centerdot 1}}{\sin u}}+h^{2\centerdot 1}{\frac {P^{2\centerdot 1}}{\sin u}}+h^{3\centerdot 1}{\frac {P^{3\centerdot 1}}{\sin u}}+,\;\mathrm {\&c.} \\&M'=2g^{1\centerdot 1}P^{1\centerdot 1}+3h^{2\centerdot 1}P^{2\centerdot 1}+4h^{3\centerdot 1}P^{3\centerdot 1}+,\;\mathrm {\&c.} \end{aligned}}$

determine the coefficients $h^{1\centerdot 1}$, $h^{2\centerdot 1}$, $h^{3\centerdot 1}$, &c. Further, the equations

${\begin{aligned}-&k''=g^{2\centerdot 2}{\frac {d\,P^{2\centerdot 2}}{d\,u}}+g^{3\centerdot 2}{\frac {d\,P^{3\centerdot 2}}{d\,u}}+g^{4\centerdot 2}{\frac {d\,P^{4\centerdot 2}}{d\,u}}+,\;\mathrm {\&c.} \\&L''=2g^{2\centerdot 2}{\frac {P^{2\centerdot 2}}{\sin u}}+2g^{3\centerdot 2}{\frac {P^{3\centerdot 2}}{\sin u}}+2g^{4\centerdot 2}{\frac {P^{4\centerdot 2}}{\sin u}}+,\;\mathrm {\&c.} \\&m''=3g^{2\centerdot 2}P^{2\centerdot 2}+4g^{3\centerdot 2}P^{3\centerdot 2}+5g^{4\centerdot 2}P^{4\centerdot 2}+,\;\mathrm {\&c.} \end{aligned}}$

determine the co-efficients $g^{2\centerdot 2}$, $g^{3\centerdot 2}$, $g^{4\centerdot 2}$, &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 $k$, $m$, $k'$, &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 $=1\centerdot 372$. 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 $=1372$. It is obvious that a unity for the intensity may be taken at pleasure, since the unity for $\mu$ may be considered as arbitrary, and made to accord therewith. If further deductions are desired requiring $\mu$ 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 $\lambda$, being reckoned east from Greenwich, are as follows:

$g^{1\centerdot 0}=+\;925\centerdot 782\qquad \qquad$	$g^{2\centerdot 2}=+\;\;\;0\centerdot 493$
$g^{2\centerdot 0}=-\;\;\;22\centerdot 059$	$g^{3\centerdot 2}=-\;73\centerdot 193$
$g^{3\centerdot 0}=-\;\;\;18\centerdot 868$	$g^{4\centerdot 2}=-\;45\centerdot 791$
$g^{4\centerdot 0}=-\;108\centerdot 855$	$h^{2\centerdot 2}=-\;39\centerdot 010$
$g^{1\centerdot 1}=+\;\;\;89\centerdot 024$	$h^{3\centerdot 2}=-\;22\centerdot 766$
$g^{2\centerdot 1}=-\;144\centerdot 913$	$h^{4\centerdot 2}=+\;42\centerdot 573$
$g^{3\centerdot 1}=+\;122\centerdot 936$	$g^{3\centerdot 3}=+\;\;\;1\centerdot 396$
$g^{4\centerdot 1}=-\;152\centerdot 589$	$g^{4\centerdot 3}=+\;19\centerdot 774$
$h^{1\centerdot 1}=-\;178\centerdot 744$	$h^{3\centerdot 3}=-\;18\centerdot 750$
$h^{2\centerdot 1}=-\;\;\;\;\;6\centerdot 030$	$h^{4\centerdot 3}=-\;\;\;0\centerdot 178$
$h^{3\centerdot 1}=+\;\;\;47\centerdot 794$	$g^{4\centerdot 4}=+\;\;\;4\centerdot 127$
$h^{4\centerdot 1}=+\;\;\;64\centerdot 112$	$h^{4\centerdot 4}=+\;\;\;3\centerdot 175$

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 $V$, developed according to the above numbers, is as follows: for the sake of brevity $e$ stands for $\cos u$, and $f$ for $\sin u$.

${\begin{aligned}{\frac {V}{R}}&=-1\centerdot 977+937\centerdot 103e+71\centerdot 245e^{2}-18\centerdot 868e^{3}\\&\qquad \qquad \qquad \qquad \qquad \qquad -108\centerdot 855e^{4}\\&+(64\centerdot 437-79\centerdot 518e+122\centerdot 936e^{2}+152\centerdot 589e^{3})f\cos \lambda \\&+(-188\centerdot 303-33\centerdot 507e+47\centerdot 794e^{2}+64\centerdot 112e^{3})f\sin \lambda \\&+(7\centerdot 035-73\centerdot 193e-45\centerdot 791e^{2})f^{2}\cos 2\lambda \\&+(-45\centerdot 092-22\centerdot 766e-42\centerdot 573e^{2})f^{2}\sin 2\lambda \\&+(1\centerdot 396+19\centerdot 774e)f^{3}\cos 3\lambda \\&+(-18\centerdot 750-0\centerdot 178e)f^{3}\sin 3\lambda \\&+4\centerdot 127f^{4}\cos 4\lambda \\&+3\centerdot 175f^{4}\sin 4\lambda \mathrm {.} \end{aligned}}$

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

${\begin{aligned}X&=(937\centerdot 103+142\centerdot 490e-56\centerdot 603e^{2}-435\centerdot 420e^{3})f\\&+(-79\centerdot 518+181\centerdot 435e-298\centerdot 732e^{2}-368\centerdot 808e^{3}\\&\qquad \qquad \qquad \qquad \qquad \qquad +610\centerdot 357e^{4})\cos \lambda \\&+(-33\centerdot 507+283\centerdot 892e+259\centerdot 349e^{2}-143\centerdot 383e^{3}\\&\qquad \qquad \qquad \qquad \qquad \qquad -256\centerdot 448e^{4})\sin \lambda \\&+(-73\centerdot 193-105\centerdot 652e+219\centerdot 579e^{2}+183\centerdot 164e^{3})f\cos 2\lambda \\&+(-22\centerdot 766+175\centerdot 330e+68\centerdot 098e^{2}-170\centerdot 292e^{3})f\sin 2\lambda \\&+(19\centerdot 774-4\centerdot 188e-79\centerdot 096e^{2})f^{2}\cos 3\lambda \\&+(-0\centerdot 178+56\centerdot 250e+0\centerdot 716e^{2})f^{2}\sin 3\lambda \\&-16\centerdot 508ef^{3}\cos 4\lambda \\&-12\centerdot 701ef^{3}\sin 4\lambda \\Y&=(188\centerdot 303+33\centerdot 507e-47\centerdot 794e^{2}-64\centerdot 112e^{3})\cos \lambda \end{aligned}}$

${\begin{aligned}&+(64\centerdot 437-79\centerdot 518e+122\centerdot 936e^{2}-152\centerdot 589e^{3})\sin \lambda \\&+(90\centerdot 184+45\centerdot 532e-185\centerdot 46e^{2})f\cos 2\lambda \\&+(14\centerdot 070-146\centerdot 386e-91\centerdot 582e^{2})f\sin 2\lambda \\&+(56\centerdot 250+0\centerdot 534e)f^{2}\cos 3\lambda \\&+(4\centerdot 188+59\centerdot 322e)f^{2}\sin 3\lambda \\&-12\centerdot 701f^{3}\cos 4\lambda \\&+16\centerdot 508f^{3}\sin 4\lambda \\Z&=-24\centerdot 593+1896\centerdot 847e+400\centerdot 343e^{2}-75\centerdot 471e^{4}\\&\qquad \qquad \qquad \qquad \qquad -544\centerdot 275e^{4}\\&+(79\centerdot 700-107\centerdot 763e+491\centerdot 744e^{2}-762\centerdot 946e^{3})f\cos \lambda \\&+(-395\centerdot 724-155\centerdot 473e+191\centerdot 176e^{2}+320\centerdot 560e^{3})f\sin \lambda \\&+(34\centerdot 187-292\centerdot 772e-228\centerdot 955e^{2})f^{2}\cos 2\lambda \\&+(-147\centerdot 439-91\centerdot 064e+212\centerdot 865e^{2})f^{2}\sin 2\lambda \\&+(5\centerdot 584+98\centerdot 870e)f^{3}\cos 3\lambda \\&+(-75\centerdot 000-0\centerdot 890e)f^{3}\sin 3\lambda \\&+20\centerdot 635f^{4}\cos 4\lambda \\&+15\centerdot 876f^{4}\sin 4\lambda {\mbox{.}}\end{aligned}}$

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 $\delta$ be the declination, $i$ the inclination, $\psi$ the total, and $\omega$ the horizontal intensity. Determine first $\delta$ and $\omega$ by means of the formulæ

$X=\omega \cos \delta ,\;Y=\omega \sin \delta$,

and then $i$ and $\psi$ by means of the following formulæ:

$\omega =\psi \cos i,\;Z=\psi \sin i$.

28.

As the formulæ for $X$, $Y$, $Z$, 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:

${\begin{aligned}&X=a^{0}+a'\cos(\lambda +A')+a''\cos(2\lambda +A'')+\\&\qquad \qquad a'''\cos(3\lambda +A''')+a^{\mathrm {IV} }\cos(4\lambda +A^{\mathrm {IV} })\\&Y=b'\cos(\lambda +B')+b''\cos(2\lambda +B'')+\\&\qquad \qquad b'''\cos(3\lambda +B''')+b^{\mathrm {IV} }\cos(4\lambda +B^{\mathrm {IV} })\\&Z=c^{0}+c'\cos(\lambda +C')+c''\cos(2\lambda +C'')+\\&\qquad \qquad c'''\cos(3\lambda +C''')+c^{\mathrm {IV} }\cos(4\lambda +C^{\mathrm {IV} }).\end{aligned}}$

The first table contains those parts of $X$ and $Z$ which are independent of $X$. In the four next tables are found the values of the auxiliary angles $A'$, $A''$, &c., and the logarithms of $a'$, $a''$, &c., all for the several degrees of latitude $\phi =90^{\circ }-u$. The table is placed at the end of the memoir.

The calculation for Göttingen is given as an example.

For latitude $51^{\circ }\;32'$ we find from the tables:

$a^{0}=+500.8$		$c^{0}=+1465\centerdot 2$
$\log a'=2.28980$	$\log b'=2.18900$	$\log c'=2.20204$
$\log a''=1.79403$	$\log b''=2.03220$	$\log c''=2.12777$
$\log a'''=1.32522$	$\log b'''=1.46845$	$\log c'''=1.43199$
$\log a^{\mathrm {IV} }=0.59391$	$\log b^{\mathrm {IV} }=0.70016$	$\log c^{\mathrm {IV} }=0.59091$
$A'=249^{\circ }30'$	$B'=358^{\circ }24'$	$C'=105^{\circ }44'$
$A''=311\;\;45$	$B''=64\;\;50$	$C''=165\;\;15$
$A'''=234\;\;10$	$B'''=318\;\;13$	$C''=42\;\;22$
$A^{\mathrm {IV} }=142\;\;26$	$B^{\mathrm {IV} }=232\;\;26$	$C^{\mathrm {IV} }=322\;\;26$

And for longitude $9^{\circ }\;56'{\frac {1}{2}}$, the parts of $X$, $Y$, $Z$, 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:

${\begin{aligned}\delta &=\,+\;\,20^{\circ }\;28'\quad \log \omega =2.73907.\\i&=\,+\;\,66\;\;\,43\\\psi &=1387.6{\mbox{, or, in the unity commonly employed,}}\\\psi &=1.3876{\mbox{.}}\end{aligned}}$

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	0·
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 ${\frac {V}{R}}$ 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 ${\frac {V}{R}}$ 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 $V$, 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 ${\frac {v}{R}}$ 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 $V$. According to our elements. Art. 26, $P'=+925\centerdot 782\cos u+89\centerdot 024\sin u$ $\cos \lambda -178\centerdot 744\sin u\sin \lambda$ , and $-925\centerdot 782R^{3}$, $-89\centerdot 024R^{3}$, $+178\centerdot 744R^{3}$ 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 $=947\centerdot 08R^{3}$. 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,

$=3\centerdot 3092R^{3}.$

As the millimetre is the unit of length employed in the above absolute unity for the earth's magnetic force, $R$ 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 $R$ 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

${\frac {1}{4\pi }}\left({\frac {V}{R}}-2Z\right),$

or by $-{\frac {1}{4\pi }}(3P'+5P''+7P'',\,{}+9P^{\mathrm {IV} },\,\mathrm {\&c.} )$

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 $P^{0}$ (Art. 17) would no longer be $=0$. The consequence of this would be, that for all external space it would be necessary to add to the expression for $Z$ the member ${\frac {R^{2}P^{0}}{r^{2}}}$; so that on the surface of the earth the (constant) member $P^{0}$ must be added, but $X$ and $Y$ 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 $P^{0}$ 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, $S$, surrounding the earth, and call $S'$ all the space included by $S$, and $S''$ all the space external to $S$. Whatever may be configuration of the galvanic currents, we can always substitute for them a fictitious distribution of the magnetic fluids in the space $S$, the magnetic action of which, in all other spaces $S'$ and $S''$, 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 $S$, a surface bounded by it and situated wholly inside $S$.

If we designate by $-v$ the aggregate of all the quotients produced by dividing all the elements of the imaginary magnetic fluid by the distance of an indeterminate point, $O$ in $S'$ or $S''$; 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 $v$, (just like those of $V$ in our theory) express the components of the magnetic force which the galvanic currents produce at $O$.

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, $F$ and $F'$, each bounded by the same linear current $G$,—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 $O$ be situated without this space, we obtain for that constant portion of $v$ which belongs to $G$, one and the same value, whether we assign the magnetic fluids to $F$ or $F'$; and this value is equal to the product of the intensity of the galvanic current $G$ (measured by an appropriate unity) multiplied by the solid angle, the summit of which is at $O$, and which is included by straight lines, drawn from $O$ to the points of $G$; or, which is the same thing, multiplied by that portion of the spherical surface described with radius $1$ round $O$, which is the common projection of both $F$ and $F'$.

If, on the other hand, $O$ be situated inside the space included by $F$ and $F'$, it is true that the two values of the part of $v$ in question will not be the same, whether we assign the magnetic fluids to $F$ or to $F'$, 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 $F$ and $F'$, 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 $v$ in question is equal to the product of the intensity of the current multiplied by the whole spherical surface, or by $4\pi$.

Hence it may easily be deduced, that if $O$ is situated in $S''$, the value of $v$ remains independent of the choice of the connecting surface; that if, on the other hand, $O$ is situated in $S'$, the absolute value of $v$ does indeed depend on that choice, but the differential of $v$ 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 $O$ outwards, has the same import in regard to attracting or repelling forces, as the parts of the mass divided by the distance from $O$,—still requires in its generality many fuller explanations, which must be reserved for a detailed treatment of this subject.

39.

The value of $v$, which in general is a function of $r$, $u$, and $\lambda$, passes on the surface of the earth into a function of $u$ and $\lambda$, and

$-{\frac {d\,v}{R\,d\,u}},\quad -{\frac {d\,v}{R\sin u\,d\,\lambda }}$

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, $v$ must first be considered as a function of both $r$, $u$, and $\lambda$; it must be differentiated according to $r$, and then $r=R$ must be substituted.

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

${\frac {V}{R}}=p^{0}+{\frac {r}{R}}\centerdot p'+{\frac {r^{2}}{R^{2}}}\centerdot p''+{\frac {r^{3}}{R^{3}}}\centerdot p''',\;\mathrm {\&c.}$

$p^{0}$ is a constant magnitude, namely, the value of ${\frac {V}{R}}$ at the centre of the earth; $p'$, $p''$, $p'''$, &c., on the other hand, are functions of $u$, and $\lambda$, which satisfy the same partial differential equations as $p'$, $p''$, $p'''$, above.

Hence it follows, in a similar manner to Art. 20, that the knowledge of the value of $v$ at every point of the earth's surface is sufficient to enable us to deduce therefrom the general expresssion for the space $S'$; 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 $p'$, $p''$, $p'''$, &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

$=2p'+3p''+4'''+{},\;\mathrm {\&c.}$

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

$=-p'-2p''-3p'''-{},\;\mathrm {\&c.}$

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 $V$, $P^{0}$, $P'$, $P''$, &c., $v$, $p^{0}$, $p'$, $p''$, applying the former to the causes acting from within, and the latter to the causes acting from without; and if we further put $V+v=W$, $P^{0}+p^{0}=\Pi ^{0}$, $P'+p'=\Pi '$, $P''+p''=\Pi ''$, &c., then on the surface of the earth,

${\frac {W}{R}}=\Pi ^{0}+\Pi '+\Pi '',\;\mathrm {\&c.}$

where $\Pi ^{(n)}$ satisfies the same partial differential equation as $P^{(n)}$ (Art. 18.); and the two components of the horizontal magnetic force there existing are expressed by

$-{\frac {d\,W}{R\,d\,u}},\;-{\frac {d\,W}{R\sin u\,d\,\lambda }}.$

The propositions mentioned in Articles 15. and 16. retain therefore their validity in this case, and we can determine the magnitudes $\Pi '$, $\Pi ''$, $\Pi '''$, &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

$Q^{0}+Q'+Q''+Q'''+,\;\mathrm {\&c.}$

so that $Q^{(n)}$ satisfies the above-mentioned partial differences, then

${\begin{aligned}&Q^{0}=P^{0}\\&Q=2P-p'\\&Q''=3P''-2p''\\&Q'''=4P'''-3p''',\;\mathrm {\&c.;} \end{aligned}}$

and, consequently,

${\begin{aligned}&3P'=\Pi '+Q',&3p'=2\Pi '-Q'\\&5P''=\Pi ''+Q'',&5p''=3\Pi ''-Q''\\&7P'''=\Pi '''+Q''',&7p'''=4\Pi '''-Q'''.\end{aligned}}$

Thus, by the combination of the horizontal force with the vertical, we obtain the means of dividing $W$ into its constituent parts $V$ and $v$, and thus of learning whether a sensible value must be assigned to the latter. Only the constant part of $v$, namely, $p^{0}$, 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 $\Delta X$, $\Delta Y$, $\Delta Z$, it must first be examined whether the alterations of the two horizontal components comport themselves in correspondence with our theory, according to which $\Delta X$ and $\sin u.\Delta Y$, must be values of the partial differential quotients of a function of $u$ and $\lambda$ 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.
$\phi$	$X$	$Z$	$\phi$	$X$	$Z$
	$a^{\circ }$	$c^{\circ }$		$a^{\circ }$	$c^{\circ }$
°			°
+ 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.
$\phi$	$X$	$Z$	$\phi$	$X$	$Z$
	$a^{\circ }$	$c^{\circ }$		$a^{\circ }$	$c^{\circ }$
°			°
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.
$\phi$	$X$		$Y$		$Z$
	$A^{1}$	$\log a^{1}$	$B^{1}$	$\log b^{1}$	$C^{1}$	$\log c^{1}$
°	° ′		° ′		° ′
+ 90	292 9	2·07430	22 29	2·07430	172 29	− $\infty$
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.
$\phi$	$X$		$Y$		$Z$
	$A^{1}$	$\log a^{1}$	$B^{1}$	$\log b^{1}$	$C^{1}$	$\log c^{1}$
°	° ′		° ′		° ′
+ 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.
$\phi$	$X$		$Y$		$Z$
	$A^{1}$	$\log a^{1}$	$B^{1}$	$\log b^{1}$	$C^{1}$	$\log c^{1}$
°	° ′		° ′		° ′
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.
$\phi$	$X$		$Y$		$Z$
	$A^{1}$	$\log a^{1}$	$B^{1}$	$\log b^{1}$	$C^{1}$	$\log c^{1}$
°	° ′		° ′		° ′
−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	— $\infty$

TABLE III.
$\scriptstyle {\phi }$	$\scriptstyle {\mathrm {X} }$			$\scriptstyle {\mathrm {Y} }$			$\scriptstyle {\mathrm {Z} }$
	$\scriptstyle {A^{\mathrm {II} }}$		$\scriptstyle {log~a^{\mathrm {II} }}$	$\scriptstyle {B^{\mathrm {II} }}$		$\scriptstyle {log~b^{\mathrm {II} }}$	$\scriptstyle {C^{\mathrm {II} }}$		$\scriptstyle {log~c^{\mathrm {II} }}$
°	°	′		°	′		°	′
+90	347	16	$\scriptstyle {-\infty }$	77	16	$\scriptstyle {-\infty }$	176	59	$\scriptstyle {-\infty }$
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.
$\phi$	$X$		$Y$		$Z$
	$A^{II}$	$\log a^{II}$	$B^{II}$	$\log b^{II}$	$C^{II}$	$\log c^{II}$
°	° ′		° ′		° ′
+ 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.
$\phi$	$X$		$Y$		$Z$
	$A^{II}$	$\log a^{II}$	$B^{II}$	$\log b^{II}$	$C^{II}$	$\log c^{II}$
°	° ′		° ′		° ′
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.
$\phi$	$X$		$Y$		$Z$
	$A^{II}$	$\log a^{II}$	$B^{II}$	$\log b^{II}$	$C^{II}$	$\log c^{II}$
°	° ′		° ′		° ′
– 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 $V$ could have any arbitrarily chosen form, then in particular cases a maximum or a minimum value of $V$ 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 $V$ 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 $P^{*}$ 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 $P^{*}$.
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.