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intersection of the plane BKC with the horizontal plane, and let CB be the line along which the dipping-needle disposes itself.
Join CO, OB, (O being the centre of the circle and, of consequence, of the sphere) : then the arc CKB measures the angle COB, which is twice the angle ECB, or twice the dip of the free needle. This arc, then, is known from observations at several particular places on the earth's surface.
Next, let the spherical triangle A B C denote that whose vertex A is the geographical pole of the earth ; C the place of observation ; and the angle B that determined as above from observations made at C: and let the angle ACB denote the observed variation of the horizontal needle at C. Then we have the sides A C, CB, and included angle ACB, from which to determine the colatitude A B and polar angle B AC.
We have therefore the polar spherical coordinates of the point B, the polar distance A B at once, and the polar angle by adding B AC to the longitude of C with its proper sign.
I shall designate the coordinates of C and B by a, b, and a., B, respectively, as is done in my paper on Spherical Geometry in the twelfth volume of the Edinburgh Transactions; a denoting the polar distance and ß the longitude of the point.
II.-Given the dip, variation, and geographical coordinates of the place of observation,
to express the equations of the line in which the dipping-needle disposes itself. Let a, b, c, and a, b,c,, denote the coordinates of two points in space: then the
equations of the straight line through them are
But in the present case the points a, b, c, and a, b, c, are on the surface of the sphere; and if we consider the axis of the sphere to be the axis of z, the intersection of the meridian with the equator to be the axis of y, and that of the meridian at right angles to it with the equator to be the axis of x, then a, , and «, B, being, as before
, stated, the coordinates of the extremities of the chord in which the dipping-needle disposes itself, we shall have, for determining the equations of the needle, the following values of the constants : = PCOS a,
C,,=r COS CI
a,=r sin a,, sin B Hence the equations of the needle take the following forms :
III.-Let M, N, P be the centres of three dipping-needles at known positions on the surface of the earth, and denote the poles by T and U. Then the needles will arrange themselves so as that each shall be in a plane passing through TU; and hence each needle prolonged will cut the magnetic axis T U in some point, as A, B, C, respectively.
Take any point, 0, in TU, and refer all the points to this origin; denote the several distances O U, OT, OC, OB, O A, by u, t, c, b, a respectively; the angles MAO, NB 0, and PCO, by A, B, C; and the distances MA, NB, PC, by f, g, h. Then we have
MT? = (a – t)2 – 2 f (a – t) cos A + f2 )
) Again, by the properties of a needle subjected to the action of the magnetic poles T and U, we have (those needles being small in comparison with its distance from those poles,) the following proportions :
TA : AU :: TM3 : MU37
(4.) TC: CU :: T P3 : PU3. Inserting in (4.) the values of the lines TA, &c. given in (3.), we get the three equations
S(a – t)2 – 2 f(a – t) cos A +f?
(5.) (6 – t)? – 2 g (6 – t) cos B + go 7
b-t b u
IV.-Given the equations of the dipping-needle at three given places on the surface
of the earth to find the magnetic poles. Let M, N, P be the places on the surface of the earth, and denote the needles as follows, viz. MA by x = a' z ta' and y=b' x + B'
(8.) NB by x = a" z + a" and y=b" % + ß" .
(9.) PC by x= a" % + '" and y= b'"% + B'",
(10.) and denote the magnetic axis itself (TU) by x =āztā and y=5z + B.
(11.) Then, since the line (11.) intersects the lines (8.), (9.), (10.), we have the three equations of condition
la' ) (b' – 7) = (B' – B) (d' – ā)
(13.) (ce" – a) (b" — 5) = (f" – ) (a" – a). "
(14.) Taking now as the unknown coordinates of the magnetic poles, T and U, the symbols a, b, c, and a, b,c, we have a b, a B, given functions of a, b, c, and a, ,
b, c, hence we have in the equations just given three equations for the discovery of these six quantities which determine the poles. The three other requisite equations are thus derived :
By means of (11.) combined separately and successively with (8.), (9.), (10.), we can find the coordinates of the points A, B, C in terms of a, b, c, and a, b, c,, and given
CI quantities a' a' b' B', &c.; and the coordinates of T and U are themselves a, b, c, and a, b, c, We hence have the several quantities ą – t, b -- t, c – t, a – u, b – u, and
t c-u in terms of a, b, c, and a, b, c,. Also the distances MA, NB, PC, that is, f, g, h,
, are also given in terms of the coordinates of M and A, N and B, P and C respectively, and hence in terms of a, b, c, and a, b, c, and given quantities. And lastly, the equations of the lines TU and MA, NB, PC being given in terms of a, b, c, and
b a, b,c, and known quantities, the cosine of the inclinations of T U to each of them, that is, of the angles A, B, C, are given so as to involve no quantities but known ones and the coordinates of the magnetic poles. It hence follows, that all the terms which enter into the composition of the equations (5.), (6.), (7.), are functions of the coordinates of the poles and of given quantities. The three remaining requisite equations for the actual determination of the magnetic poles are furnished, then, by those equations marked (5.), (6.), and (7.); the whole six equations which we have laid down being each, obviously, independent of the others under every combination.
V.—The preceding processes show that the determination of the magnetic poles, their duality being admitted, as well as the equality of their intensity, can be effected from three observations of the magnetic needle as to dip and azimuth ; and hence
that the problem is now reduced to a purely arithmetical state, requiring only the application of known processes, and perfectly capable of execution, for the actual assignment of the positions of the poles themselves. A very slight attention to the processes themselves must, however, convince us that the operations will be very laborious; but at the same time, the symmetrical forms in which the two triads of equations are presented, might induce us to hope that a greater degree of simplification would result in the final formulæ than our passage through so many operations could at first sight lead us to expect. Such, indeed, proves to be the case; and the results are not altogether destitute of elegance, as well as simplification. Fortunately, however, there is no necessity to even attempt the solution under the present aspect of the problem ; and having learnt from it, in its present state, the number of observations necessary for the determination of the poles, we shall exchange it for another, which is in some degree different as to general plan, and considerably more simple in its requisite calculations.
VI.-A necessary consequence of the hypothesis upon which we are proceeding, is,—that all the needles must intersect the magnetic axis. If, then, we assume the coordinates of the two poles a, b, c, and a, b, c, we have the equation of the magnetic axis as before,
2,1 - a a, 9 - a, cu
Z c , -c,
C C And if we take the equations of four magnetic needles, as x = a' % + ' and y=b'% + !
า x = a" x + a" and y = 65% + B"
(16—19.) x = a'" % + d'" and y = 6% + B" x = a" +a" and y = 5" x +B", %
% , the intersections of (15.) with (16—19.) give four equations, of condition similar to those of (12.), (13.), (14.), from which to determine a, b, c, and a, b, Cr, viz.
' – =
CThese four equations will determine four of the coordinates, as a, b, and a, bus leaving the two others indeterminate. But still the law of force furnishes fonı other equations from which to determine the two quantities c, and c,; that is, a redundancy of equations, from which redundancy the remaining number may be taken as checks of accurate calculation if the principle be admitted, or as tests of the truth of the principle when we are assured of the accuracy of calculation and of observation.
By this method a greater uniformity of process, and a perfect symmetry in respect to the quantities involved, are obtained; but still the process is very laborious, and it
is probable that the resulting equation will be of a higher degree than really belongs to the problem in its direct form. If so, it will contain foreign factors, which it may be difficult to detect and peculiarize, so as to separate them from the
proper solutions of the problem. The method, besides, is not essentially different from the last.
Another difficulty also presents itself here; nor is it the only one. The mere intersection of the magnetic axis with the magnetic needle is not a test of the duality in point of number, nor of the equality of intensity in point of force, nor is it confined to any law of variation of force whatever ; and hence the mere intersection is not of itself sufficient for the determination of the question respecting the duality or the relative intensity of the poles. Still it is one of the necessary conditions, though not the only one, by which the hypothesis is to be tested; since the poles, being of any number, and of any intensities whatever, if situated in the same straight line, will cause the needle to intersect that line, and hence render that phenomenon incapable of determining the number, intensity, or position of the poles ; yet wherever this intersection is not fulfilled the duality of the poles cannot be admitted, nor yet the position of the poles, however many they may be, be in one straight line. The determination of a, b, c, a, b, c, from the equations (20—24.) cannot then be effected completely.
We shall hence proceed in the following manner. A straight line, which constantly touches three given straight lines, but undergoing all the changes compatible with that triple contact, describes the hyperboloid of one sheet. This surface being of the second order, will be cut by a fourth given line in two points ; and hence there are two positions which a line resting upon four other lines can take. If, then, we imagine these four lines to be four different positions of the magnetic needle, and the line which rests upon them to be the magnetic axis, we shall perceive at once that in case of any number of poles of any variety of intensity, and acting under any law of variation of force depending upon distance, the magnetic axis can be determined in position from four observations of the magnetic needle; and, therefore, of course, in the case which we are examining, where the poles are two, the intensities equal, and the law of force that determined by MICHEL and CouLOMB.
Let us take as the equations of the magnetic axis and four of the needles, respectively, the following:
= % + andy=[ %+B x = a' % + &' and y = b' % + B' x = a % + a" and y = b 2 + B"
(25-29.) x = a % ' z + a" and y = kx + 3""
% r=a"% + all and y=
and y = 6" % + B". Then the condition, that the first of these intersects each of the others simultaneously, gives the four equations,