and PD, destroy each other. The same may be shown with regard to the actions upon the south end of the needle. It follows, therefore, that the needle will place itself at right angles to CPD, the isogeothermal line passing through P the station of the needle. This is a consequence from our theory which, like the formulæ soon to be investigated, is to be tested by making comparisons with observations.

Let us now deduce from the general principles which have been laid down, the horizontal and vertical components of the directive force of the needle.

Let ApB, fig. 3, represent a great circle of the earth, answering to mPs, or nPr in fig. 2, Cp its radius, P the north end of a magnetic needle, and mn two particles of the earth situated at equal distances to the north and south of P. The action of m situated to the south

Fig. 3.

of P, will be in the direction Pa perpendicular to mP, and that of n will have the direction Pb perpendicular to nP. The force Pa may be decomposed into two forces having the directions PC and PH; and the force Pb may be decomposed into two having the directions PD and PH. The sum of the two horizontal components will be the effective horizontal force due to the actions of m and n, and the difference of the two vertical components will be the effective vertical force due to the action of the same particles. Since the temperature of m is higher than that of n, the component directed from P to C is greater than that directed from P to D, and hence the north end of the needle will be urged downward. The horizontal force will solicit the north end of the needle toward the north. The actions upon the south end of the needle will be just the reverse. Now if we suppose the same process of decomposition to be gone through with for each pair of particles situated on AB at equal distances from P, up to a certain distance at which the molecular actions become insensible, by taking the sum of the individual forces along PC and PH, we shall have the entire effects of the arc AB in these two directions. In the same manner we may obtain the effects of any arc below AB and situated in the same plane; and thus the entire effect of all the matter situated in this plane which exerts any action upon the needle. Since the curvature of the arc AB is very slight, and P is very near to it, it is only the particles situated quite near top that will have any material action in the horizontal direction. For arcs below the earth's surface the portion that furnishes the horizontal force will be greater as the depth increases, but will still, doubtless, be small in comparison with the more distant

parts which act nearly in the vertical direction upon the needle. If, as we have supposed, the principle of magnetism be analogous in its nature to light and heat, then it must be more or less absorbed in its passage from the lower arcs to the surface; and there may be a gradual decrease in the extent of the arc which exerts a sensible action upon the needle, as the depth of the arc increases, until at the lower surface of the stratum of sensible action it becomes reduced to zero.

are of a great circle crossing this line perpendicularly and passing through P the station of the needle. The magnetic intensity of the particles of AB is every where the same. Take any particle m and designate the distance Pm, in a right line, by r. Either end of a needle at P will be solicited by a force perpendicular to Pm, and in the vertical plane through Pm. This force will be, for different isogeothermal lines, directly proportional to the magnetic intensity of m, and therefore to its mean annual temperature (t); and will, for the same isogeothermal line, vary from one particle to another with the distance r. Its expression will therefore be of the form At. q*(r); A being an indeterminate constant. Now, let mp, fig. 5, represent the great circle immediately below mP

in fig. 4, and lying either on the earth's surface or beneath it. We shall have force Pa(due to m)=At. q(r). The

Fig. 5.

D

a

R

component of Pa in the direction of the radius or vertical PC will be equal to Pa.cos aPC=Pa. sin mPR=Pa. q'(r, h, R); R being the radius of the circle, which may be taken equal to the radius of the earth, and h the height Pp of the needle above the circle. We have therefore for the action of m in the direction of the vertical PC, the expression At. q(r). q'(r, h); and when the height his regarded as constant, we have At. q(r). q'(r), or At.f(r). To obtain the entire effect in the vertical direction of all the particles in the line GB, fig. 4, let GB be denoted by k, any portion Gm of it by x, and PG by . The action of an elel. mentary portion of GB will have for its expression At.f(r)dx

The letters q, f, F, with and without accents, are used in these investigations to designate different functions, and are therefore to be read "a function of."

=Atf(√l2+x3)dx. Integrating this between the limits 0 and k, we have, vertical action of GB=At. F(l, k). Whence, vertical action of AB=2At. F(l, k). If k may be taken sensibly the same for different isogeothermal lines, this expression will become 2At. F(l). It is to be supposed, however, that the last particle of GB, which has a sensible action upon the needle at P, is at the same distance from this point whatever may be the distance of AB from it. The value of k will therefore be less, in proportion as the distance l is greater. Supposing the most remote particle to be at B, and denoting its distance PB by d, k will be equal to √d2 — l3, and the above expression will become 2At. F (1,√da —l1), or 2At. F'(l). It follows therefore that the entire action of any isogeothermal line AB in the vertical direction upon a needle at P, may be reduced to a single force, proportional to the temperature, and varying from one isogeothermal line to another, with the distance PG of this line from the station of the needle. The entire effect of any single lamina of matter will therefore be the same as if the action was confined to the particles lying in the arc GPH; the effective force of each particle being proportional to its temperature, and also a certain function of its distance from the needle.

a sensible action upon the needle, u the difference between the mean temperature at p and at any point m, y the arc pm, a the arc pA, and r the distance Pm. pm or y may be regarded as depending for its value upon Pm, Pp, and Cp; of which Pp and Cp are constant for the same arc. Thus for any one arc, (representing, according to what has been shown, a single lamina,) y=4(r). If we regard the variation of temperature as uniform for the extent of the arc AB

y

u : t − T: :y : a. .u=(t − T) 2 = (t − T) (r).

a

a

Whence, putting v vertical force due to an element dy at m, and taking the expression for the action of an isogeothermal line, and incorporating the 2 with the constant A,

de = A (T+ (e− T) "(r) F'(r)dy = A (T+(t − T)"{r})}F'(r)dq{r}

dv

a

a

=AT. F'(r)dq(r)+A(t− T)"(7)F«(r)dx(r).

Integrating, v=ATƒF^{r}dq(r)+A(t−T) SHF'(r)dq{r}.

a

Integrating between the limits Pp and PA, to obtain the force due to the arc pA, the two integrals will become two functions of Pp and PA. Now, for any supposed value of Pp, PA will be the same at every different place on the earth, and therefore the values of these integrals will be every where the same. If we denote them by M and N, we have

v=ATM+A(t − T)N=AM.T+AN(t−T).

By the same process we obtain for the vertical force due to the arc pB v'AM. T+AN(t-T). Hence the expression for the effect of the whole arc, AB, is

v-v'=AN(t-t') = c(t-t').

(1.)

If we consider the action of a second lamina, the value of c may be different, but t-t' will remain very nearly the same, except at considerable depths where the rate of variation of the temperature may be different, or the arc AB may be diminished by the absorption of the ethereal waves in their passage to the surface. If we neglect these possible variations of t-t', and add together the actions of the different laminæ, we obtain for the actual vertical force

in which C is the sum of the values of c for the different laminæ. If we take account of the variations of t-t', we shall have the actual force equal to the sum of a series of expressions of the form e(t -t') in which both c and t-t' will be more or less different. It would seem, however, that the changes in the value of t-t', from absorption or other causes, must be very slight. In fact if the absorption be always a certain fractional amount of the intensity, there will be no change of t-t' from this cause. It will only be necessary to regard e as varying. And if the absorption be always the same fractional amount whatever may be the intensity, c and therefore C will have the same value at different places.

The supposition made in the investigation of formula (2), that the variation of temperature is uniform for the extent of the arc AB, is not strictly true. From the equator to the latitude 45°, and even beyond this, the rate of diminution of the temperature for every degree of latitude continually increases. The effect of this will be to make the vertical component somewhat greater, except in the higher latitudes, than formula (2) would SECOND SERIES, Vol. IV, No. 10.—July, 1847.

2

give it, (that is, supposing C to be determined à priori. If C be determined from observations made at the point of maximum variation of temperature, the values of V given by equation (2) will be too small south of this point and too great north of it.)

To obtain a formula for the horizontal component of the directive force, we may proceed in the same manner as for the vertical component, except that we now multiply the force Pa, fig. 5, by the cosine of the angle aPH instead of aPC. We shall therefore have for the entire action of the isogeothermal line AB, fig. 4, the expression A't. F"(l). Hence, that of all the isogeothermal lines, or of the whole acting surface, will be reduced to that of the single arc which crosses these lines at right angles; the magnetic intensity of the different points of this arc being proportional to the temperature, and the effective forces upon the needle varying according to some function of the distance. Now, as in the present enquiry all the active particles lie quite near to P, their temperatures may be considered the same and equal to that of the earth at the station of the needle: or, if there is a sensible variation at the lower layers, the augmentation towards the south will be compensated for by an equal diminution towards the north. Hence, designating the arc pm, fig. 7, by y, and the distance Pm by r, the expression for the horizontal force due to this arc is

Fig. 7.

H

P

A

fdh=fA'T. F"(r)dy=A'T ƒF"(r)dx(r).

Integrating between the limits r=Pp and r=PA, and designating the value of the integral by P, we have

H' A'T. P; 2H'=2A'P. T

and thus finally the total horizontal force

This is the expression for the entire effect of a single lamina. For different laminæ C' may be different; and beyond a certain depth T will increase. If the supposed absorption of the magnetic emanations be a certain constant fractional amount of the magnetic intensity of the molecules, C' will be every where the same. If we take the sum of all the equations (3) answering to the different laminæ, we shall have an equation of the same form for the horizontal component of the directive force, or the horizontal intensity at P. It is only by comparing the results furnished by this equation, with observations, that we can ascertain with certainty whether T is to be taken sensibly different from the mean surface temperature, and whether C' may be regarded as truly constant for all places.