Let the scheme in fig. 6 represent the state of things in a circuit where the discharged point a is contiguous to the zinc pole. The reduced length, ab, and the electromotive force, bc, being given, let d be any point whose tension, de, we wish to ascertain. Let bc=a, de=u, ab=l, ad=λ; then by similar triangles,

or, expressed in words, if the reduced length of the circuit between the discharged point and the point whose tension is sought be divided by the reduced length of the entire circuit, the quotient, multiplied by the electromotive force, gives the tension at the required point.

In submitting this formula to an experimental test, M. Kohlrausch made use of the wooden trough before alluded to. The copper and zinc plates were united, as in one of the experiments already described, by a long fine wire, bent from side to side of a wooden frame in a zigzag manner. The tensions of the points described below were determined by direct experiment. The electromotive force was also determined, the reduced length of the circuit was found by measuring the resistances of its various parts, and from these two, the electromotive force and the reduced length, the tensions due to the same points were calculated by the foregoing formula.

Points examined.

a. The second lower angle of the zigzag. b. The fourth lower angle of the zigzag.

c. The sixth lower angle of the zigzag.

d. The point where the zigzag joined the copper.

e. The solution of sulphate of copper 2.02 inches from the plate of copper.

f. The solution of sulphate of copper 4.02 inches from the plate of copper.

g. The solution of the sulphate of copper 6 inches from the plate of copper.

h. The solution of sulphate of copper 8 inches from the plate of copper.

In the following table the results obtained by calculation are

compared with those obtained by direct experiment; the quantity is the same as that contained in the formula.

The truth of Ohm's formula, which he derived from considerations purely theoretical, appears to be placed beyond the pale of doubt by these results. Hitherto the celebrated law which usually bears his name has rested upon hypothesis merely; and to the extraordinary patience and refined experimental skill of Kohlrausch is due the credit of giving to this conjectural foundation the stability of fact.

It may be stated, in addition, that the same physicist has also examined the thermo-circuit, and has not only demonstrated the existence of electric tension at its poles, but also proved that the electricity obeys the same law of distribution as that true for the voltaic circuit.

ON ELECTRO-MAGNETIC ATTRACTIONS.*

§ 1.

THE subject of the present memoir is embraced by the following four propositions:

I. To determine the relation between the strength of an electromagnet and the mutual attraction of the magnet and a mass of soft iron, when both are in contact.

II. To determine the same relation when the magnet and the iron are separated from each other by a fixed distance.

III. A constant force being applied to the iron in a direction opposed to the pull of the magnet, to determine the conditions of equilibrium between this force and magnetism when the distance between the magnet and the iron varies.

IV. To determine the relation between force and distance, that is to say, the law according to which the magnetic attraction decreases when the distance is increased.

The first of these propositions has engaged the attention of physicists from time to time during a considerable number of years. Experiments have been made and facts multiplied, which, however, are so disunited and contradictory as to render any attempt to reduce them to a common law altogether hopeless.

The most important experiments which have been made in connection with this subject are those of Lenz and Jacobi; who established, that the attraction between two electro-magnets, or an electro-magnet and a mass of soft iron, is proportional to the square of the magnetising current. Here, however, an interval of about th of an inch separated the attracted mass from the magnet,-a condition the importance of which appears to have been overlooked by the discoverers of the law. It has been generally assumed, that if the defects incidental to

* Phil. Mag. April, 1851.

Y

the modes of experiment hitherto adopted could be avoided, the same law would pronounce itself in the case of contact. Were this the case, our two first propositions would be identical, the solution of the one would necessarily imply the solution of the other; it will be shown, however, that the laws of action in both cases are entirely different.

Two principal causes have been assigned as giving birth to the discrepancies alluded to-the incompleteness of contact, and the peculiar shape of the mass of iron attracted; the question naturally occurs, cannot these causes of divergence be removed?

1. To annul, as far as possible, the disturbances arising from mere form, a number of regularly-shaped pieces of good soft iron were procured; they included cubes, cylinders, and spheres of various diameters. It is easy to see the practical difficulty of experimenting with cubes and cylinders; indeed, to render such experiments pure, conditions are required which it is almost impossible to fulfil. Conceive the cube suspended by a wire attached to the middle of one of its faces, and laid with its opposite face flat upon the polished end of the magnet; let the wire ascend vertically, pass over a pulley, and be attached to a scale-pan at the other side; on this scale-pan let weights be laid until the cube is separated from the magnet; the weight which effects the separation expresses the sustaining power. That the experiment, however, shall be faultless, it is necessary that all parts of the surface of the cube should give way at the same moment, otherwise the cube will hold on by its edges and corners, and thus totally vitiate the experiment. To effect this, it would be necessary, first, that the production of the wire should go exactly through the centre of gravity of the cube; and secondly, that all portions of the surface should be equally in contact, or that any deviation from the one condition should be compensated by a deviation from the other. The difficulty of complying with these requirements has compelled me to abandon both cubes and cylinders, and to resort to a body with which the smallness of the surface of contact reduces the irregularity to a minimum; a body, moreover, which is able to accommodate itself to the slight divergences of the wire. That body is the sphere.

2. The magnet used was that formerly applied by my friend Professor Knoblauch and myself in an investigation On the

Magneto-optic Properties of Crystals.'* As then used, it consisted of two soft iron cylinders set upright in a glass case, and united below by a cross-piece of iron. Round the cylinders were coiled 360 feet of copper wire, weighing nine pounds, and upon the top of the cylinders two finely polished parallelopipeds of iron were laid, between which the crystal was suspended. In the present case the cross-piece was removed, and the two cylinders were tightly screwed together-an arrangement provided for in their construction-and thus converted into a single magnet, 9 inches long, 1·3 inch thick, surrounded by a helix containing 360 feet of copper wire. The magnet was made fast in a block of wood, and set vertically upright under one end of the beam of a fine balance; from this end the ball of iron was suspended by a copper wire; the length of the latter was so arranged, that, when stretched full, the ball resting on the end of the magnet, a whalebone index pointing to the zero of a graduated arc showed that the balance-beam was horizontal. From the other end of the beam a scale-pan was suspended which held the weights. The weight of the ball and its attached wire was, in the first place, exactly balanced by a counter-weight; so that when the magnetism was excited, the attractive force measured by the other weights was a purely magnetic force.

After a few experiments a slight modification of the above arrangement was found necessary. The end of the iron core on which the ball rested had a little cavity in its centre, which resulted from its having been turned in a lathe. It being absolutely necessary that the ball should rest exactly upon the centre, a parallelopiped of soft iron was placed upon the end of the magnet. The two diagonals were drawn upon one of its polished horizontal faces, and the iron sphere always rested upon the point of intersection.

3. During the investigation, the battery, the magnet, and the instrument used to measure the intensity of the current (Weber's tangent galvanometer), were in three different rooms. From the poles of the battery two long bands of sheet copper ran side by side, and passed thus under the door into the room which contained the magnet. One of them was carried to the magnet,

* Phil. Mag. March, 1850, and July, 1850.