for the action of each wire, results which perfectly agree with each other, after allowing for the effect of the original directive force, independent of the wires, which was suffered to remain ; and this agreement takes place also, whatever be the intensity or the direction of this force, provided that it is not so feeble as to allow the needle to be inverted by the action of the wires, when the electric current is transmitted in a direction causing an opposite force. For, unless the exterior magnet employed to weaken the directive terrestrial force, be removed from the needle to a great distance, compared with the dimensions of the latter, such an inversion will always produce some slight alteration in the action exerted upon the needle, and thus modify in some degree results so delicate in themselves, and which cannot be too carefully guarded. I was led to perceive by degrees, the absolute necessity of all the precautions here recommended. Some of my first observations taught me that the action of the oblique wire diminished as the angle comprehended between its two branches diminished, and apparently according to the same ratio ; this law would in fact hold true in the extreme case of the phenomenon; for the action ought evidently to be nothing when the angle is nothing, the two halves of the wire being then applied the one to the other, and traversed by the voltaic current in opposite directions; and this same action must be equal to that of a straight wire when the inclination of each branch to a horizontal line is 90°, since they then form together the same vertical straight line. But, on account of the inaccuracy of the experiments, other laws might also have been admitted, and we might, for example, have substituted instead of the inclination i to the horizontal line, the tangent of half this inclination, that is, tangi; so that calling F the observed action of the vertical wire upon the needle at a certain distance, F tang i would have been the action of an oblique wire traversed by the same voltaic current; whereas, according to the preceding supposition, it would have Fi been expressed by a value which can differ from the pre 90°' ceding only in hundredth parts. In order to decide this question, it is only necessary to repeat the experiment under a single angle, with extreme rigor. For this purpose, I chose the case where i 45°, which renders the action of the oblique wire equal to F tang 22° 30', or F 0,414214. As the coefficient of F differs then very little from, I doubled the oblique wire avoiding all contact between its parts, in order that by passing twice before the needle, it might exert upon it an action which should be double, and consequently more easily observed. I also took care to vary the communications of the two wires with the poles of the voltaic apparatus, so that these two poles should communicate alternately with the bottom and top of each wire, which I have expressed by the letters ZB, CH, and CB, ZH. Since in these two modes of communication, the action of the wires upon the needle was inverted, it is evident that if, in the one case, it augmented the original directive force, it must have diminished it in the other. Let us, therefore, suppose N to be the number of seconds employed by the needle in performing a constant number of oscillations under the influence of the original force alone, and that N' is the corresponding num ber when the action of the wire is additive; it is easy to see that the proper action of the wire must have for its value K being a constant quantity depending on the dimensions of the needle, on the other hand, it becomes when the action is subtractive; which enables us to estimate it in either case. It is in this way that the following tables were formed from the experiments made with the two small rectangular needles, the dimensions of which have been given. Length of the Time of 40 oscillations Time of 40 oscillations Ratio of the actions of N. The value of 2 F tangi, would give for this ratio 0,828427. The difference is insensible in these experiments. But it is still more diminished when we consider that the two branches of the oblique wire being removed each 3 millimetres, or 0,12 English in. one to the right and the other left of the vertical wire, their distance from the centre of the needle was not d, as in the case of this wire, Alg. 102. but (d2 +9)13 or d (1+), which may be reduced to (d+2), 9 on account of the smallness of the fraction Thus, in order to d2 reduce the corresponding observations of the two wires to the same distance, it is necessary to multiply the direct ratio by 9 the inverse ratio of the distances 1+ Now, the value of 2 d2 d was,in the first experiment, 28,5 millimetres or 1,11 English in.; in the second, 33 millimetres, or 1,3 English in.; whence 9 it follows, that the fraction is in the one case, 2 d2 in the other,, by which it is necessary to augment the two means directly found. This correction changes the first to 0,845451, the second to 0,809641; and the mean 0,827545 then coincides almost exactly with 0,828427, the value of F tangi. There can be no doubt, therefore, that this expression represents generally the total action of an oblique wire bent into two branches forming with each other the angle i. Now, if we consider a small infinitely thin lamina of such a wire, situated in μ, and if μm or R be its distance from a particle m of magnet- Fig. 136. ism, whether boreal or austral, we have deduced from our first experiments, that the action of this lamina upon the particle is inversely as the square of the distance μm, multiplied by an unknown function of the angle mu M, which we shall here denote by w. It only remains, therefore, to determine what form it is necessary to give to this function in order that the whole sum of the actions, thus exerted upon m by all the lamina of the wire, perpendicularly to the plane CMZ, may form a resultant proportional to tang. We satisfy this condition by taking sin w for the function sought, which renders the elementary action of any lamina whatever proportional to ; and uniting with this expression, which is founded upon experiment, the knowledge of the absolute direction of the force which is perpendicular to the plane drawn through each distance and through the direction of each longitudinal element of the wire under consideration, we may assign by calculation the total resultant of the action exerted by a wire, or by any portion of a wire, whether straight or curved, limited or indefinite. R sin w R2 269. Let us suppose, for example, that a very long wire is taken, and coiled into a small space, as represented in figure 148, after being first covered with silk thread, or any other substance impermeable to a feeble electricity, for the purpose of preventing the parts from touching each other. If we suspend a magnetic needle AB in the space surrounded by the wire, and make the extremities Z, C, of the wire communicate with the poles of the voltaic apparatus, it is manifest that all the infinitely small elements of each circuit of the wire will exert upon pole A parallel actions, directed the same way; and that their respective forces will vary according to their distance, and also according to the value of o for each element. Thus, in virtue of their parallelism, all these unequal actions will form a single resultant, which will tend to drive the magnetism of A, either to the one side or the other of the plane of the figure, but always in a direction perpendicular to the revolutions of the wire. Since the the same reasoning applies equally to the pole B, this pole will be acted upon by a force similar and equal to the former, if the position of the needle within the circuit of the wire is symmetrical; but the absolute direction of this second force will be op posite to that of the first, on account of the opposite nature of the magnetisms in A and B. These two forces being thus applied, will tend therefore to turn the needle in the same direction; and their effort will be the more powerful, according as the circuits of the wire are more numerous, their distance from the needle less, and the absolute intensity of the electric current greater. We may, therefore, increase at pleasure the action of a given electric current, by means of this ingenious arrangement, and multiply it in an almost indefinite ratio. M. Schweiger, who invented this apparatus in the beginning of September, 1820, soon after the discovery of M. Oersted was known, gave it the appropriate name of electro-magnetic multiplier, which is now generally adopted. In order to give this instrument all the sensibility of which it is susceptible, and render it convenient at the same time in its application, it is necessary first to place the needle in a small paper case, suspended by simple thread, of the silk-worm, the circuits of the multiplier being directed acFig. 149, cording to the natural magnetic meridian. We then place under the needle a horizontal divided circle in order to measure its deviations; which may be done by fixing upon the needle, perpendicular to its length, a small pasteboard index, the point of which being thus directed without the circuits of the wire, traverses a visible part of the circular division. The whole is covered with a glass case to preserve the needle from the agitations of the air; and from beneath this case proceed the two extremities of the uniting wire, which are inserted in small glass vessels filled with mercury, and placed upon the table where the experiments are to be performed. If we wish to transmit through the multiplier the current developed by any electromagnetic apparatus, it is sufficient to establish the communication between the mercury of the small vessels and the poles of this apparatus. When the multiplier is prepared in the way we have explained, with a suitable number of circuits, its sensibility is so great that it is sufficient to touch the two extremities of the wire with the two opposite faces of a single copper and zinc pair slightly moistened, to cause immediately a great deviation in the |