Let M, Fig. 4, be a magnet, of which F is the focal point, and C a compass at the distance of at least four focal lengths. Let'ns be the position of the needle under the influence of terrestrial magnetism only, and n' s' the position which it assumes under the action of the magnet. Then the action of the nearest pole F is represented by the sum of the squares of the distances Fs and Fn' inversely, which does not, in this instance, materially differ from the sum of the squares of the distances Fs and Fn inversely. But in the case represented in Fig. 5. the result is far otherwise. Here the bar M, placed at the distance of one focal length, occasions such a great deviation of the needle that a very considerable increase of action on the pole s' is gained, beyond the diminution of action sustained by the pole n',—the increase of force being in the proportion in which the sum of the reciprocals of the squares of the distances Fs' and F n', expressed fractionally, exceeds the sum of the reciprocals of the squares of the mean distances Fs and F n *. *The sum of these direct squares must in all cases be equal. For F s n being a triangle which is bisected in the point C by a line FC drawn to the apex, the sum of the squares of F s and Fn, is equal to twice the squares of FC and Cs. In the triangle Fs'n', for the same reason, the sum of the squares Fs and Fn', is equal to twice the squares of FC and Cs'. But the lines sn and's'n', representing the same needle revolving on a centre, are equal, and But the nature and extent of this disturbing influence will be more evident, if we work out the case represented in Fig. 5, exhibiting the effect of the first distance of a 12-inch magnet (C) of 10 inches focal length, the powers of which, to the extent of six lengths, are exhibited in the following Table. and FC common to both; hence twice the squares of each must be equal. Therefore the sum of the squares of F s, F n, and the sum of the squares of Fs, Fn' being each equal to the same thing, must be equal to one another. Though, however, the sum of these several squares are equal—not so their reciprocals, as is clearly shown by working out the case referred to in Fig. 5. Or, take a more simple case: In like manner, Fs + Fn' = 12 + 72 =50. Thus, as before stated, the sum of the squares of F's of the squares of Fs' Fn' are equal, each amounting to 50. 2 Fs ciprocals. The reciprocals of Fa2 and Fn' (= 2 F n', and the sum = 25 and 25, or 2 2 25 and their sum being But the reciprocals of F s' and Fn' (= 1 and 49) 25 25' two sets of squares are equal, their reciprocals are found to be in the relation No. 1. repeated with a small compass, with a needle of 15% inches. 1 0 9 n = 55 0 55 45 146870 14687 Now, the position represented in Fig. 5. is that of No. 1, in the first line of this table, in which the magnet was placed at the distance of one focal length from the compass-not from the centre, but, in this instance, measured from the focal poles of the needle n, s. Therefore the distances Fs and F'n are each = 1; and F's and F'n the distances of the remote or counteracting pole are each 2. But owing to the considerable length of the compass needle, and the great deviation which occurred on this occasion (namely 59° 23′), the actual distances of the two poles of the needle from the nearer focus F were and and from the re 2 8. 109 mote focus were and 2 Let us now see what relation of 10° forces these distances afford in comparison with those belonging to the distance 1 and 2. First, As to influence of the magnet on the compass, whilst the needle is in the meridional position s n. The distances Fs and Fn being each = 1, their combined influence will be inversely as 12+12 or, which represents the 2 whole force of the nearer focus of the bar which would act on the needle in the position n s. = Again, the distances of the remote focus F's and F'n being each 2, both their influences will be inversely as 22, that is 4 or; the reciprocal of which,, represents the respective forces. And+represents the counteracting force operating on the needle in the position n s, at the distance 2. Hence -represents the resultant influence of both foci, or of the whole bar in the given position. 2 Secondly, As to the actual influence exerted by the magnet in the deflected position of the needle s'n'. The distances F & and Fn' being and and, their squares, representing the inverse power of their action, are and 36 25° 8 12 10 = 4 16 25 25 16" Hence the reciprocal representing the attractive force is and the reciprocal representing the repulsive force is 25 Then 36 represents the whole in fluence of the nearer focus in the actual position, s'n', assumed by the needle *. Again, the distance F's', in the case before us, was found to be 1%, and that of F'n' = 2%, or 18 and and, the 10 the whole influence of the remote focus, or counteracting forces, in the actual position, s' n', as sultant influence of both foci in the deviated or actual position assumed by the needle, whilst the excess of this above the assumed force in the position n s = indicates the quantity of power gained by the magnet in consequence of the length of the compass needle. * It is here assumed that the attractive and repulsive forces are parallel to each other, which is not the case; hence the results obtained, though sufficiently near for our present object, can only be considered as approximations. 2117016 2458125 1411344 to 2117016 or 1411344 Reducing, now, the fraction to the same denominator as the above, we have as the resultant influence of the whole bar in the assumed position n, s. Therefore the entire or resultant force acting upon the needle, in the deflected position n', s, is to the force in the assumed position n, s, as asto to nearly. If, then, we apply this proportion to the observed deviation of No. 1. (series in page 108), 59° 23′, the tangent of which is 168979, we have ::: 168979: 144839, = 55° 23′, which, it is satisfactory to find, corresponds very nearly with the deviation observed when a very small compass was substituted for the large one; in that case, the angle formed by the needle, as near as could be observed, being 55° 45'. 6 4 Still, however, the deviation thus reduced is found to be considerably greater than that given by a mean proportional, namely, 54. The cause of this difference is probably to be found in the peculiar direction, Fs, of the strongest force, which evidently is not strictly tangential to the meridional position of the needle; but must operate more favourably for overcoming the directive force of the earth, than if, acting in the direction &M, it were precisely at right angles to the terrestrial magnetism. Since now the calculated deviations of the three feet magnet (Table, p. 24), as obtained from the mean ratio 121600, are all, except the first, within the limits of the possible error of observation; and since the ratios obtained from experiments with the twelve-inch magnet (Table at p. 108) are all, with the exception of the first, uniform within the probable limits of error, —whilst the discrepancy at the first focal length has been sufficiently, I trust, accounted for,-the position of the foci in both these magnets may be considered as rightly determined *. For all practical purposes, therefore, connected with the proposed * Though I have hitherto spoken of a fixed and determinate focal position representing the whole of the magnetic forces of either half a regularly magnetized bar, yet I am aware that that very focal position will be liable to a small variation at very short distances, in such cases being nearer the extremity than the calculated position. Nevertheless, at distances beyond the length of the magnet, no alteration in the position of the foci, I apprehend, will be discernible in practice.. |