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Therefore, conformably to the observations we have previously given, the extra current proceeding from P is sensible in Q, that from Q in P, but the magnetic effect of each is nil in the coil from wbich it emanates.

If the coils P and Q are placed as derivations with respect to one another, the direct extra current of each coil traverses the other in a direction contrary to the current from the pile. In this case experiment shows that the interruptions occasion a diminution of the magnetic moment, and the establishments an augmentation, as might have been foreseen from what precedes.

III. The circuit comprises a coil and a condenser.

This case is realized with a Ruhmkorff coil in the following manner: the condenser C (fig. 4) is fixed in a derivation destitute of resistance, on which the interruptions are practised. The coil B may also be placed in the derivation, and the interruption be produced in a point of the principal circuit (fig. 5).

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In each of these arrangements the condenser forms with the coil a closed circuit apart from the interruption, either directly or by the intermediation of the pile. It would not be the same if the interruption were made in the principal circuit with the arrangement of fig. 4, or in the derivation in the case of fig. 5: these last two arrangements are useless, as experiment has shown ; but if one of the former be employed, the rupture of the circuit is observed to produce a diminution of the magnetic moment of the needles magnetized by being passed to the coil.

The effect obseryed cannot be attributed to the extra currents of the condenser ; for these act in a direction contrary to the result obtained. The condenser employed * behaves, with respect to the production of extra currents, like a coil of negligible power.

* That of a Ruhmkorff coil, of which the explosion-distance is 3 centims. But it must be remarked that the intensity of the extra current of the coil is superior to that of the principal current. The direct extra current strongly charges the condenser, which must consequently discharge itself after the extra current has ceased. The coil thus receives, by reflection, the extra current which it has produced, and is traversed by it in the opposite direction to that of the principal current. The reflex effect is, for a given condenser, the more intense the more powerful the coil employed; and experiment shows that with a very feeble coil the effect of the interruption vanishes.

IV. Magnetization by induced currents.—We have studied only the following case :

When the circuit includes two coils, the slow introduction of a core of soft iron into one of them, or its extraction, is without effect upon the magnetism of a needle placed in the other coil. But if the core be introduced slowly and withdrawn suddenly, the direct induced current augments the magnetic moment of the needle placed in the second coil. Repetition of the same operation causes the moment of the needle to tend towards a limit, which it rapidly approaches. The formula y=A+B(1-e-as),

(2) in which A, B, and a are constants, appears very well adapted to represent the magnetic moment y after x passages. The following Table permits the appreciation of the degree of accnracy of the formula :

TABLE VII.

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The differences between calculation and observation are a little more considerable in this Table than in the preceding ones ; but account must be taken of the difficulty of always withdrawing the soft iron in identically the same manner, in order to produced induced currents of the same intensity. Regard being had to this consideration, the agreement is satisfactory. On the contrary, hyperbolic formulæ are not at all suitable for the representation of experiments of this sort; they correspond to a much slower increase, sensible even after twenty operations of the same kind, while here the augmentation ceases to be appreciable after seven or eight operations. Quetelet represents by a formula of the form y=B(1-0)

(3) the magnetic moment acquired by a steel bar magnetizea by one, two, ...x frictions. This formula represents also, as we have just seen, the increment of the magnetic moment produced in a needle by equal induced currents; it does not suit for that produced by the interruptions of one and the same continuous current, acting on a needle innocent of all anterior magnetization. A complete theory of magnetism should account for this difference.

To avoid the intermissions of the principal current in the preceding experiment, the needle should have been placed in a fixed position in its spiral, and its total magnetic moment measured by the method of deviations. The variations observed are related to the permanent magnetism, which may alone be altered by the passage of the induced currents.

We have seen that the induced currents produced by wresting from contact an electromagnet placed in the circuit produce absolutely similar effects.

V. Effects of piles the current of which is not constant.--If the current of the pile employed is not rigorously constant, the effect of the polarization of the electrodes modifies profoundly the phenomena. The following results were obtained with a pile containing bichromate of potash, prepared several days previously.

If the circuit comprises, besides the pile, only one coil, into which a needle magnetized by a great number of passages is introduced, the establishment of the current augments the magnetic moment by a quantity more or less considerable, often enormous. The current therefore possesses a much greater intensity at the moment of its closing than it retains a moment afterwards. When the resistance of the spiral is augmented, the polarization is less strong, and consequently the proper effect of the establishment of the current tends to disappear.

With respect to the interruptions, they bave no very marked effect upon the needles, at least while the resistance of the coil is not very strong; but in the latter case, if we introduce into the coil a strongly magnetized needle with its south pole to the left of the principal current, we always obtain by the interrup

Phil. Mag. S. 4. Vol. 49. No. 323. Feb. 1875. H

tion a diminution of the magnetic moment of the needle. I think that this strange effect must be explained thus :--The direct extra current of the coil, without magnetic effect in its interior, momentarily increases the polarization of the pile, whence a current of very perceptible depolarization in consequence of the interruption; this current partially demagnetizes the needle. We have here, therefore, to do with a reflex effect, analogous to that of the condenser of the Ruhmkorff coil, although incomparably more feeble.

If the circuit contains two coils, P and Q, one very powerful, the other very feeble, the direct extra current of P and the reflex action of the pile succeed each other in Q, and produce a very singular effect. Interruption of the circuit augments greatly the magnetic moment of a needle magnetized in Q by a great number of passages; but the same operation diminishes the magnetic moment of a strongly magnetized needle placed in Q with its south pole to the left of the principal current. This effect is much more pronounced when P contains a core of soft iron. Example :

Moment.
Needle magnetized in Q by passages .

1.16
Interruptions

11.10 Needle strongly magnetized

36•28 Interruptions

35.40 Thus one and the same complex instantaneous current magnetizes needles wbich were not magnetized, and partially demagnetizes needles which were strongly magnetized. The same thing can be easily reproduced by the alternation of a continuous direct current and a weaker inverse current.

We thus see that, in certain cases, magnetized and nonmagnetized steel needles permit us to analyze a complex instantaneous current, and recognize in it a change of sign.

[To be continued.]

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XII. On the Mathematical Theory of Mr. Baillie Hamilton's

String-Organ. By R. H. M. BOSANQUET, Fellow of St John's
College, Oxford*.
THE investigation which follows was completed more than a

year ago (1873). The endeavour was then made to test the theory by experiment; but difficulties were encountered which, notwithstanding the assistance kindly afforded by Mr. Hamilton, were not got over. The subject has now been brought repeatedly before the public by Mr. Hamilton, who has read

* Communicated by the Author.

THE

papers before the Physical Society, the Ashmolean Society of Oxford, and the Musical Association; and in view of the interest generally expressed about it, I have thought that an account of the theory, as far as it goes, may be acceptable to the readers of the Philosophical Magazine. The two sets of observations given towards the close of these remarks were made by me at Mr. Hamilton's laboratory in November 1874.

It is not my purpose to describe the instrument here; it will be sufficient to point out that it produces continuous tones by means of combinations of strings and harmonium-reeds. There is a separate string to each note, and to some point on each string the extremity of a harmonium-reed is attached. The reed is then set in vibration by wind; and the problem is to determine the forms of vibration of the combination.

The method of the following investigation is substantially that employed at p. 139 of Donkin's 'Acoustics,' with the extensions necessary for the purposes of the problem.

The instrument may be regarded as a string loaded at the point of attachment to the reed, subject also at the same point to a force tending towards the position of rest, and varying directly as the displacement of the point from that position ; this force represents the elasticity of the reed. Let u be the load at the point of attachment,

T the tension of the string,
p mass of unit of length of string,
a elastic force of reed per unit of displacement,
y the displacement,
i the length of the string,
b distance of point of attachment from one end,
x distance of any point from the same end,

T
a?=

р Then the equation of motion of the point of attachment is

dy

=TAdt2 - dx

-ay; and for the rest of the string dạy

(2) di2 dx? Assume (see Donkin's 'Acoustics,' pp. 119, 139)

y=sin m(l-b) sin mx(A cos amt+B sin amt) (3) from x=0 to x=b, and

y=sin mb sin m(1-2)(A cos amt + B sin amt) (1) from x=b to x=l.

.

dy

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