the rod would exert upon the needle when placed at the distance (1) from it, and its effect beyond this approximation should always increase in the same proportion in which the cube of the distance decreases. But this relation between action and distance does not hold good for short distances; this, however, does not prevent the use of the moment of resolution for M reduced to the unit as a measure of the magnetism of the rod.

=

Multiplying equation (1) by 3, and placing fr3 M, we get

Assuming the deflecting bar and the needle to be equally magnetic, let the magnetism in both be so developed that the reduced moment of revo lution Mis equal to the pressure which the weight of a milligramme would produce on a lever-arm of one millimetre, if, instead of the force of gravity, this weight be acted upon by a force under whose influence double the space traversed in the first second is equal to the unit of length, (one millimetre,) then this would be the unit of free magnetism.

With this unit the terrestrial magnetic force is also to be measured, or, in other words, T is to be expressed in terms of this unit. The manner in which the value of T is determined, adopting that just defined as the absolute measure, may be found in Weber's original treatise on this subject, and in an elementary account of it in my Treatise on Physics, (3d edition, 2d vol., p. 48.)

If the value of T is determined according to the absolute measure, then equation (2) gives the reduced moment of revolution of a magnetic bar expressed in the same unit.

But the quantity M has still another meaning than the one already mentioned, namely, CT M is the moment of revolution with which the terrestrial magnetism tends to draw the bar, placed perpendicular to the magnetic meridian, out of this position. (Treatise on Physics, 3d edition, 2d vol., p. 44.) Thus, M denotes the magnitude of this moment of deflection for the case in which T = 1.

By observing how many degrees a magnetic needle is deflected by a bar placed north and south of it in the position Fig. 4, we can, from this observation, compute by means of equation (2) the moment of deflection with which the terrestrial magnetism tends to draw the bar, lying perpendicular to the magnetic meridian, out of that position.

By placing the magnet east or west of the needle, as indicated in Fig. 5, the former, at the same distance, deflects the needle more,

Fig. 5.
N

and so that the tangent of the angle of deflection w is exactly double the tangent of deflection v, which the same magnet, at the same distance, would have produced in the position Fig. 4; hence, under circumstances otherwise the same, we have

By making the experiment, not in the position Fig. 4, but that of Fig. 5, we get

M=

Ttang. w
2

The relation of the circulating current, which traverses the ring of the tangent compass, in the magnetic meridian, to the terrestrial magnetism, as well as to the magnetic needle, may now be compared with the effect of the magnetic bar placed in the position of Fig. 5. If the circulating current of the tangent compass deflects the needle w degrees, we have

denoting by g the force of the current, and by r the radius of the ring: thus we have for the reduced moment G of the deflection of the circular current, which corresponds to the moment of deflection M of a magnetic bar

G=

Tr3 tang. w
2.

= r2 g.

(3.)

This G is the force with which, under the relation stated above, the circular current would be deflected from the plane of the magnetic meridian, if the force of the terrestrial magnetism were

Making r2 = 1, we will have G=g;

1.

hence q is the moment of a circular current which circulates in unit g of surface.

From equation (3) we get for g the value

g =

Tr tang. w
2 π

(4)

thus we obtain a value for the force of the current g, measured by the moment of deflection of a current traversing around the unit of surface, expressed in absolute measure, by substituting for T its absolute value.

$5. Comparison of the different current units.-Theoretically these three units of force are determined with perfect exactness, and if the matter were considered only in a scientific point of view, each of them would seem acceptable, though the preference would be due to Weber's unit.

But the selection must be different when practical wants are also considered.

The galvanic battery enters so multifariously into a process of art, that it is of great importance to have methods by which the constants

of a galvanic arrangement can be determined with accuracy. Unfortunately, such methods hitherto have been but little known, and thus it is that we have descriptions of the useful effect of many different combinations of galvanic apparatus, but none such as give an accurate comparison of different apparatus, and a consequence is that we are frequently deceived in their value.

For determining the constants of a battery, it is essential to understand, in the first place, with reference to the unit of current, whether the observations made for that purpose are comparable with other observations at different places with different instruments. To render such a unit popular, it should be accessible to practical men, who though acquainted with the principles of electricity, are unable to enter into the specialities of the science; hence it is fit to select such a unit only whose definition is easily and generally comprehensible; moreover, the unit should be such, that the determination of the force of the current for obtaining it may be accomplished with the least possible apparatus.

Considered in this light, the unit first brought into use by Jacobi has by far the preference. I will endeavor to justify this opinion.

§ 6. Reduction of Pouillet's unit to chemical measure.—To compare the indications of any compass with Pouillet's unit, we must have a thermo-electrical element exactly equal to that used by him; and for that purpose it is necessary that the entire resistance of the circuit, including the wire of the compass or multiplier, should be equal to the resistance of a copper wire 20 metres long and 1 millimetre thick. But the current which such a thermo-electrical element produces under the indicated conditions is exceedingly feeble, or at least much weaker than the current of hydro-electric batteries, which yield a practical useful effect; and in instruments with which ordinarily the force of the current of hydro-electric batteries is measured, as tangent compasses, sine compasses or Mohr's torsion galvanometer, Pouillet's unit will produce but a very small deflection. This unit produces, for example, in Weber's tangent compass, having a ring 40 centimetres in diameter, a deflection of from 5 to 7 minutes; in Mohr's torsion galvanometer, a deflection of about 1 degree; thus it is requisite to have very small subdivisions of a degree in these instruments with accuracy, for determining this angle of deflection with sufficient exactness to make the angle itself, or its tangent, the unit in measuring strong currents.

Since the instruments do not admit of sufficiently accurate reading of such small angles, an indirect method must be introduced. The following, perhaps, is the simplest for this purpose:

Pass the current of the thermo-electrical element, serving as unit, through a multiplier, and observe the deflection produced: suppose it is 16, the entire resistance here is equal to the resistance of a copper wire 20 metres long and 1 millimetre in diameter.

Now pass the current of a hydro-electric element through the same multiplier, but insert, in the form of platinum or German-silver wire, resistance until the deflection is as great as that produced by the thermo-electrical element, or until it amounts, as before, to 16°.

The whole resistance which the hydro-electrical current has now to overcome must be determined and reduced to that of copper wire. Suppose it is equal to the resistance of a wire 1 millimetre in diameter and 22,000 metres long.

By making the entire resistance less by the removal of wire, the current will become stronger in equal measure. Make the resistance, for instance, 200 times less, so that the entire resistance to be overcome by the current of the hydro-electrical element is equal to the resistance of a normal copper wire only 110 metres long; the current will now be 200 times stronger than that which produced a deflection of 16° in the multiplier. This current will produce a considerable deflection in each instrument adapted to measuring stronger currents, as a Weber tangent compass; let it be 19°.

Thus a current which indicates in the tangent compass an angle of deflection of 19°, of which the tangent is 0.344, is 200 times as strong as the unit of the current, thus we have for the tangent of the angle to which the unit corresponds—

0.344

200

= 0.00172.

By this result all the indications of the tangent compass can be easily reduced to Pouillet's unit.

Pouillet used, not a tangent compass, but a compass of sines, in all his researches on this subject.

To decompose one gramme of water in one minute, the current passed through the water must have a force 13,787 of Pouillet's unit. Each gramme of water yields 1862.4 cubic centimetres of detonating gas (at 0° and a pressure of 760 metres); hence to obtain one cubic centimetre of detonating gas per minute, a force of current of 187877.4 Pouillet's unit is necessary.

The above examples will suffice to show that the reduction of the data of a rheometer for stronger currents to Pouillet's unit can be obtained only by a whole series of operations by no means simple. First, the resistance of the thermo-electric element, and of the multiplier, must be determined, and so much. resistance must be added that the sum of the resistances shall have the value given above; then the resistance of the conductor of the hydro-electrical element must be found, and after inserting as much resistance in its circuit, the quantity of this resistance is to be determined; then the entire resistance must be reduced to an aliquot part, and the corresponding deflection of a rheometer used for stronger currents observed, &c. The end here is attained only through a circuitous process, and errors of observation are unavoidable in each operation, which affect the final result; the complexity of the process also has a prejudicial influence on the accuracy of the determination.

The above comparison of Pouillet's unit with the chemical effect produced, gives us the means of easily converting the data of a rheometer into this unit; we have only to pass the current simultaneously through the rheometer and an apparatus for decomposing water, to determine how much detonating gas will be evolved while the rheometer indicates a certain number of degrees. Since each cubic centimetre of detonating gas corresponds to 7.4 of Pouillet's unit, it is

known also how many of Pouillet's unit correspond to the observed deflection of the rheometer. Pouillet's unit has been used here only nominally; the deflection of the rheometer alone has, in fact, been compared with the chemical effect, and there is no reason why this comparison should not be adhered to.

87. Reduction of Weber's unit to the chemical measure.-The definition of Weber's absolute measure of the force of a current is by no means so simple as to encourage the hope of making this unit easily very generally comprehended. This inconvenience, however, might be disregarded, if the determination of the force of the current were easily derived from this absolute measure.

If a Weber's tangent compass (which should not be less than 40 centimetres in diameter) be used in getting the angle of deflection which a current produces, it is made to appear stronger in absolute measure, as expressed by the formula,

According to this formula the value of the force of the current is very easily obtained, if the correct value of T be ascertained; that is, if at the place of observation the horizontal part of the intensity of the earth's magnetism, expressed in absolute measure, be known.

The determination of T (Müller's Lehrbuch der Physik, 3d Aufl. 2 Bd.) has for special physicists no great difficulty, but for many artisans who wish to determine the power of their batteries it is too complicated; at least it is more difficult than the comparison of the data of a rheometer as made by Jacobi, with the chemical effect of the current. It would not be necessary to determine the value of T by experiment at the place of observation; it might be derived from the magnetic chart of Gauss, if it were certain that at the place of observation the effect of the horizontal magnetism of the earth was not modified by iron deposited in that locality, which would produce a considerable deviation from T. For instance, we have from Gauss' chart, as well as from direct observation made in the open air, that for Marburg T=1.88, while Kasselman found the value of T, in the locality in which he instituted the experiments for comparing the force of the currents of different galvanic batteries, equal to 1.83, (Über die galvanische Kohlenzink Kette von Kasselman: Marburg, 1844, p. 75); hence it is unavoidably necessary to determine the value of T in the locality in which the experiments on the strength of currents are instituted.

Weber's unit decomposes 0.000009376 grammes of water in one second; in one minute 0.00056256 grammes; or, what is the same, it yields 1.0477 cubic centimetres of detonating gas per minute.

To determine the force of a current according to this measure, a tangent compass of Weber is needed, whose ring should not be less than 40 centimetres in diameter, while rheometers of different kinds can be used if the unit of the current yielding one cubic centimetre per minute of detonating gas be adopted.