Total year....

24

4

4275

6665

4 5

5...

4 11 ... 15 ... 23 31 1 551 57 248 90 3 35... 11 ....... | ... | ... | ...... 232 46 7 87 16 91 137 80 314 2

* No observation.

2:::

13

33

10 11

:

12 10

34

23

374

3361 σr

84

9786

8

TABLE XXXV.

Abstract of the prevailing Winds in Dukhun during the years 1826, 1827, 1828, 1829, and 1830.

XI. Geometrical Investigations concerning the Phenomena of Terrestrial Magnetism. By THOMAS STEPHENS DAVIES, Esq. F.R.S. Lond. and Ed. F.R.A.S. Royal Military Academy, Woolwich.

Received December 18th, 1834,-Read February 5th and 12th, 1835.

THOUGH the experiments of MICHEL and COULOMB have satisfactorily determined the law according to which magnetic forces vary as the distance of the needle acted on is made to vary, yet, so far as I know, no one has attempted to deduce from that law any method of accounting for the phenomena of terrestrial magnetism. Till that is done, however, we cannot assure ourselves whether the poles (I use the term to designate the centres from which the forces emanate,) be two or more; nor even whether it be necessary to consider them infinite in number and distributed over the whole surface or through the whole mass of the magnet. The agreement of the results as to quantity with the actual phenomena would be decisive in favour of any hypothesis. The necessary consequences have not, however, yet been deduced from any one hypothesis whatever: and even had it been otherwise, there is so much uncertainty attached to magnetical observations, and so many anomalous and unaccountable discrepancies and disturbances continually mingling in the registered results, that it is not possible, in the present state of the numerical data, to bring any hypothesis fairly to the trial, however complete the mathematical development of its consequences may be.

Notwithstanding the great difficulty of conducting a series of observations in a perfectly unexceptionable manner, and the utter impossibility, with our present knowledge, of properly determining the correction to be made at any given place and period with any given instrument, there are yet several features in the phenomena which are of too decided a character to be overlooked in comparing the results of any theory with observation. We may not, indeed, be able to avoid considerable discrepancies in our comparison, but still there should at least be a general tendency towards agreement, and in no case a direct reversal of the phenomena presenting itself as the result of any hypothesis which prefers its claims to our adoption. In respect to terrestrial magnetism, no direct attempt has, however, been made to embody the results of any hypothesis in a series of appropriate formula; and hence the conjectures which have been made respecting such agreements have been made from extremely vague and inconclusive considerations.

The duality of the terrestrial magnetic poles is the oldest hypothesis, and perhaps that whose consequences will be found most easy to examine. The hasty comparisons

that have been made between its supposed results and the observations made on the needle at certain places, and especially respecting the Halleyan lines, and HANSTEEN'S poles of greatest intensity, have caused the hypothesis to be rejected by many persons, who, if they had looked more closely into the question, could not have failed to discover that their conclusions were altogether premature, and probably erroneous. I speak now of the broad features of the phenomena compared with a popular rather than a calculated series of deductions from the hypothesis. Whether, however, when the results come to be more closely tested by an appeal to the numerical values of the quantities in question, the same accordance would be found, is a question altogether different: and it is one which we are not at present in a condition, for want of numerical data, upon which to offer a distinct opinion, much less are we entitled to express a positive decision concerning it.

The present series of papers is chiefly intended to deduce the mathematical consequences of the theory of two poles situated arbitrarily within the earth, and especially to investigate the singular points and lines which result from the intersection of the earth's surface with other surfaces related to the magnetic poles. Amongst these are the magnetic equator, the points at which the needle is vertical, the lines of equal dip, the Halleyan lines, the isodynamic lines, and the Hansteen poles. If it shall appear in the course of these developments that the general features of all these are roughly represented by the hypothesis of two poles, then it will be a strong argument in favour of that simple theory; but should a result, in any one of these cases, follow from that hypothesis which is very decidedly opposed to the corresponding observed phenomena, we shall be compelled, if our observations are authentic and to be depended on as unaffected by an extreme degree of foreign influence, to abandon it altogether.

Our hope of being able to separate the disturbing from the primary forces must depend altogether upon their relative quantities. The success of astronomical research has hinged wholly on the relative smallness of the disturbances in comparison with the primary forces that govern the motions of a planet: and if the same order of magnitude should exist in the magnetic forces in question, the same success will, there is every reason to hope, follow in due time. If not, the research should be placed at once amongst the desperata. However, till some method is discovered of ascertaining whether such is the case or not, we should leave no effort untried either to accomplish the proposed object, or to render manifest its impracticability. With that view the present investigations, which are conducted in a manner altogether untried before by any one, are offered to the attention of geometers, as being calculated, besides exhibiting the general consequences of the dual hypothesis, in some degree to point out where we should look for the influence of foreign forces, and especially showing that in reference to one great circle, the want of symmetry in the results at positions taken symmetrically with respect to it, gives us great cause to suspect the action of such foreign forces.

The nature of this paper does not allow of much numerical experiment upon the observation-data; but still, in illustration of the method of determining the position of the magnetic axis, I have entered into a little. The results are not very favourable to the hypothesis; but when it is considered that the observations were selected almost at hazard, all made with different instruments, by various persons, and in geological regions extremely dissimilar, we could hardly, in the confessedly imperfect state of the art of observation, expect to obtain satisfactory results. Taking all things into account, the results are, unfavourable though they be, as satisfactory as we could expect. However, whatever conclusion may be drawn from them, they furnish at least a pattern for calculating any better and more consistent observations we may at any future time be able to procure; and if any that are beyond question can be obtained, it will enable us to bring the linearity of the magnetic axis to an indisputable test. The duality, should the linearity be established, can be at once put to the test by means of the process in Section III.

of

Now that this method of investigation is proposed, it will doubtless occur to some my readers that a more direct course would have been to assume the undetermined coordinates a, b, c, a, b, c, of the two poles, and express the equation of the sphere in reference to the same axes, and hence the directive effect of the two poles upon a needle placed at a point xyz on the spherical surface. Such a method, they will believe, must also have occurred to me as the most natural; but if they will take the trouble to form the equations of condition that this method will require, they will see the utter impracticability of effecting the reductions under the mere motive of making an experiment upon the results of an hypothesis when no confidence was felt in the numerical data which entered into the formula. The reason for adopting the less direct, but incomparably more simple, preliminary test illustrated by Sections XI. and XII. will then be sufficiently obvious.

I. Given the dip and variation of the magnetic needle and the geographical coordinates of the place of observation, to find the geographical coordinates of the place where the needle, sufficiently prolonged, will intersect the surface of the earth again.

We assume, for reasons too well known to need spe- E cification here, that the orthogonal projection of the dipping-needle upon the horizontal plane gives the position of the horizontal needle; or, which comes to the same, that the dipping-needle and the horizontal needle are in the same vertical plane. This plane cuts the sphere in a great circle, which we shall for the present suppose to coincide with the plane of the paper, and to be represented in the annexed figure by CK B. Let EC be the