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(c" -a) (6" – ) (B" — B) (a" – a)
(31.) !(cz" – a) (" – ) = ($'"' – B) (a" – a) " '
(32.) (o" – a) (6" - 5) = (B" — B) (a" – ā). .
(33.) This reduction is easily effected by subtracting the first from each of the others, in which case we obtain equations of the first degree, giving each of the other three the quantities a b a B in terms of the fourth, as of ã. These substituted in any one
b of the four equations give a quadratic equation involving a; and hence we obtain two values of ū, and hence again of 5, ofã, and of B. We should then obtain, by a simple and direct process, the equations of the magnetic axis.
The next inquiry is into the signification of this double result. Are there two magnetic axes which fulfill the condition? If so, are they both occupied by magnets? Or if not, why is one to be selected in preference to the other? Can they both belong to every quadruple combination of the magnetic needle ?
The last question may be answered at once. If they both belonged to all the combinations of the needles, then they must form two of the directrices of a rule surface, to which the needles themselves were always tangents. The third directrix not being yet fixed, there is no inconsistency in the conclusion thus derived; for the needles are at liberty to rest upon any magnetic surface, whatever be the number or intensity of the poles, or whatever be the parameter which determines the particular stratum of surface which corresponds to the place of observation on the sphere. There is hence nothing to prevent their belonging to every position of the magnetic needle, so far as we at present can discover from the conditions in their arbitrary form. How far this is consistent with the particular data is another question, and will be presently discussed.
There is no necessity that they should be both occupied by magnets; and it is at once giving up the duality of the poles, and even their being situated in a right line, to make such an hypothesis. They are both, it is true, solutions of the algebraical problem which we have proposed ; but as the algebraical rarely includes all the conditions of the physical problem, it is easy to suppose that one of these solutions may be foreign to the inquiry, without violating our knowledge of the nature of the relations subsisting between the algebraical and physical problem. To prove that it actually is a foreign result must be subsequent to the determination of the particular values of the coefficients of the quantities involved in the inquiry. All we can say at present is, that there are two axes which fulfill the algebraical condition ; but as there is only one which enters into the physical hypotheses, one of these two algebraical axes must be rejected. We cannot, however, ascertain which, except by other conditions than have yet been taken into the formula. For the present, then, we can only compute them both, and take that which best answers to those other physical conditions of which the algebraical problem has taken no account.
VII.—Having obtained the equations of the magnetic axis by the determination of ab a ß, we may put the rectilinearity of the positions of the poles to the test at once. For if we take a fifth needle x = a % + " and y = 6% + Bu
(34.) and combine it with (29.), viz.
x =ā z + ū and y = ū x + B, so as to ascertain whether they intersect or not, by means of the equation of condition
(QS – a) (Gv – 7) = (BY – B) (a' – a)
(35.) The approximation to a fulfilment of this condition for a'a', bu Bo, derived from all the other observations upon which reliance can be placed, and to the extent of the probable accuracy of those observations, will establish this hypothesis to the same extent.
In order to estimate the real amount of the error in the application of this equation of condition, we must recollect that the formula itself is derived from that which gives the shortest distance between two given lines, viz.
a)(61 - 7)
b) – (Bv
(BY – B) (a' - a)
(36.) var – a) + (0" —B)+ (a bv – av 6)2 Hence, in order to estimate the number of miles which the needle would be from fulfilling the condition, we must calculate the denominator of the fraction (36.), and divide the result of (35.) by it. By then calculating the angle which D would subtend at the place of observation, we shall be in some degree prepared to judge whether such an error might possibly have arisen from unskilful observation, the imperfect structure of the instrument, from any probable geological or meteorological causes, or from any temporary local disturbance. Or, conversely, were this line satisfactorily determined by a great number of tests, and the instrument and observer well prepared for the task, then we should be able in some degree to estimate the amount of the disturbing forces that climate, geological structure, and local attraction do actually exert at that place, and perhaps in some cases also to form a probable conjecture respecting the separate contribution of each of these causes to the total amount of the disturbance.
VIII.-Let us now suppose the magnetic axis satisfactorily determined and tested ; and proceed to inquire whether the poles be two or more, and whether equal or unequal in the intensity of their action : and in the first place we shall suppose the intensities equal.
Recurring to the figure in (III.) and the conditions that are tabulated, as (5.), (6.), (7.), we have the quantities designated as a, b, c, A, B, C, f, g, h, and the corresponding ones for any number of points M, N, P, Q,, &c. from actual observation and the calculated equation of the magnetic axis. From any two of these, as (5.), (6.), we then shall be able to compute t and u. The remaining equations, whatever be their number, will serve as tests of the truth of the dual hypothesis with the poles of equal intensity.
It will be the most convenient method of taking the point 0, to assume for it the intersection of the magnetic axis by the perpendicular from the centre of the sphere. Having, then, the distances of the poles from this point, and the equations of the line in which they lie, we can easily determine their coordinates, the great object of our inquiry.
It can be no objection to this process that it necessarily requires the solution of equations of a high order, since it is only the solution of it in the case of given numeral coefficients, and not with literal coefficients. The method of effecting these solutions with rapidity and precision is now well known, and need not here be dwelt upon. We shall have occasion hereafter to employ them in the numerical solution of this special problem*
IX.-If we suppose the intensities unequal, we can assume their ratios to be that of F, to F,, or R. The relation upon which (5.), (6.), (7.),... are founded no longer holds good in this case. Nevertheless, by reference to (XIV.) we see that the difference is only in the numeral coefficients of the equations, and not in the form or the number of terms, or in any circumstance that alters in the slightest degree the labour or the difficulty of the actual solution. We have however, in consequence of the new quantity R which is thus introduced, to employ one equation more derived from observation t, and one only. Hence still in this case, too, four observations are sufficient, not only for the determination of the actual position of the poles, but also to furnish a test of the accuracy of the hypothesis. As a method, then, this also is complete, and the problem is fully brought within the reach of known and familiar operations.
X.-As a specimen of the method of computing the equations of the magnetic needle, I have given calculations for Chamisso Island, Valparaiso, Paramatta, Port Bowen, Paris, and Boat Island; and that the whole process may be distinctly seen, I have also given the equations of the magnetic axis itself as deduced from the equations of the first four needles, and a comparison of the result with the Paris needle. That result is not very favourable to the theory, provided the observations themselves are considered trustworthy. But since those philosophers who have had most experience in the use of magnetical instruments, and especially of the dipping-needle, are most strongly convinced that there are errors attached to all our present instruments and modes of observation whose amount vitiates any result obtained by them, I have not thought it necessary to add any further discussion of the question in the present stage of my investigations, in as much as till the results can be ensured as unaffected by extraneous sources of error, all methods must alike be useless, since they are alike dependent upon data that are at least unsatisfactory if not erroneous. There is some reason, however, to hope, since the attention of the scientific world is now so intensely turned to researches of this nature, that there will at length be discovered some methods of observing which shall be free from this class of errors. However, till this is done, it would be useless to attempt the discussion of the present or any other method of mathematical investigation into its numerical details: and the utmost we can now perform is to lay down methods of investigation by which, when satisfactory experimental data are obtained, the question may be brought to a decisive test at once.
* I refer, of course, to Mr. HORNER's method, published in the Philosophical Transactions for 1819, and in the fifth volume of Professor LEYBOURN's Mathematical Repository. It is unnecessary to add, that all the effective methods of solution of algebraical equations that have since appeared have been but imitations of Mr. Horner's, however much the notation and form of the reasoning employed in them may differ from his. + Or if we seek to determine the actual value of F, and F,, we shall want all the four complete observations.
I should also state here, that in consequence of the great labour attending the calculations of the axes, I have been led to examine the method of construction by descriptive geometry, (especially on account of the facility and the considerable degree of certainty which may be attached to its solutions,) of the problem of describing a
a line which shall rest upon four given lines. In any case where the data are so uncertain as the present, such a method is sufficiently accurate, since in very few cases will the errors of construction be probably near so great as the errors in the data themselves. The geometrical problem itself is not in practice so simple as could be desired, at least by any method yet made public, but still it offers far greater facilities than the algebraical one. It has, moreover, one important advantage which the algebraical has not, viz. the ready and visual exhibition of those cases which are unfit for this method, or those in which a small error in the data will greatly increase in the result. It should hence be always used before the algebraical. No doubt the algebraical method may be rendered subservient to the same purpose, but then the method is intolerably operose, and hence, practically, almost useless.
By employing such constructions, I find that there is a greater degree of approximation in the few magnetic axes which I have determined by them from existing data than appears compatible with any other theory of the constitution of the terrestrial magnet than that which considers the magnetic force situated in two isolated centres or poles. By this I would not be understood to say that the approximation is close, but simply that, in comparison with all the positions which lines may take, there seems to be one region of space, in reference to the coordinate planes or planes of projection, into which they dispose themselves, but dispose themselves very irregularly in it.
As I expect to be favoured, by the kindness of my distinguished colleague Mr. CHRISTIE, with the results of the observations of the late lamented Captain Foster, I shall probably resume this branch of the subject at an early period; and hence any further details respecting these constructions which may appear necessary will be with more propriety included in a future than in the present paper.
The equations of the magnetic needle for different places, in reference to rectangular
coordinates, when the geographical coordinates, dip, and variation are given. Let A O A' be the meridian of Greenwich, E O Q the equator, C the position of a place where a magnetic observation is made, A CB the variation, and B C equal to twice the dip of the needle. Then B is the point on the surface of the earth towards which the dippingneedle is directed, and that in which the straight line which coincides with the needle intersects the earth a second time.
Estimating (as is done in my paper on Spherical Loci before referred to) the positions of places on the surface of the earth by means of the polar angle CA O and radius-vector C A, we have the coordinates of C directly from observation; and by means of the triangle A C B, whose sides B C, C A, and included angle A C B are given, we can compute the coordinates of B. Denote the polar distance and polar angle of C by a, Bi, and those of B by, B,
In the next place, by the employment of equations (1.), (2.) of II., we may obtain the equations of the needle, referred to three rectangular axes, the coordinate planes of which are the meridian of Greenwich, the meridian of + 90°, and the equator. The results for the six different places before mentioned are given in the last column of the following Table. The construction of the Table itself is indicated at the head of each column, in a way that renders further explanation unnecessary.
Chamisso Island. 1827. Lat. 66° 12' O" N. Dip. 77° 39' ON. «, =
23° 48' 02, Captain Beechey.
Long.161 46 0 W. Var. 32 0 30 W. B, = – 161 46 0 Bar Voyage, App. p. 752.