Scientific Memoirs/1/On the Laws according to which the Magnet acts upon a Spiral, when it is suddenly approached to or removed from it; and on the most advantageous Mode of constructing Spirals for Magneto-Electrical purposes

by Heinrich Friedrich Emil Lenz

Article XXX.

On the Laws according to which the Magnet acts upon a Spiral when it is suddenly approached to or removed from it; and on the most advantageous mode of constructing Spirals for Magneto-electrical purposes; by E. Lenz.

From the Mémoires de l'Academie Impériale des Sciences de St. Petersbourg, vol. ii., 1833, p. 427. Read on the 7th of November, 1832

^[1].

From the great interest which the late discoveries of Faraday in the field of electro-magnetism must awaken in all the natural philosophers of Europe, it is to be expected that we shall soon receive many and various explanations of the momentary action of an electric current on an electrical conductor; and as it is allowed according to Ampère to reduce the action of a magnet entirely to that of circular electric currents, the same may be expected with respect to the action of the magnet upon such a conductor. Up to the present moment we here in the north are only acquainted with the papers of Becquerel, Ampère, Nobili, and Antinori and Pohl; and as none of these authors have occupied themselves with that branch of the subject to which I have directed my particular attention, I hasten to make known as quickly as possible the following contribution to the science of magneto-electrism.

After having repeated Faraday's chief experiments^[2], I first proposed to myself to find out in what manner the phænomena of the magnetic action on a spiral suddenly approached or removed might be produced in the easiest and most powerful manner. For this purpose I had to determine what influence

1. The number of coils,

2. The breadth of the coils,

3. The thickness of the wire,

4. The substance of the coils,

of the electromotive spirals (i. e. of those which are acted upon by the magnet) had upon the pæhnomenon; and this determination, together with the necessary consequences following from it, are contained in this present memoir.

The following was the apparatus I employed for my experiments. A multiplier (with a very sensible double needle of Nobili) of seventy-four coils of copper wire of 0·025 of an English inch in thickness^[3] was placed in connection by means of conducting wires with the electro-motive spirals, so that the horseshoe magnet which acted on the spirals was at a distance of nineteen feet from the multiplier, and had no immediate influence on its needles. I had assured myself of this by previous experiments. The horseshoe magnet consisted of five single bent steel bars, firmly connected with one another by screws; the middle one protruded at the ends about 0·7 of an inch; it might together with the armature weigh somewhat more than twenty-two pounds. The length of the bars was twenty-three inches, the breadth 0·8, and the thickness 0·22; the middle one projecting beyond the others was 0·4 in thickness; the distance of the arms was 1·64 inch. In order to be able to approach and remove the spirals, and at the same time to read off the deviation of the needle without any aid, I constructed my apparatus in the following way:—I did not cover the multiplier with its bell glass, but with a glass cylinder open at both ends, and closed these by means of a plate of mirror glass; I then placed over it a good mirror under an inclination of 45°, and from a point near the magnet I observed by means of a good Munich telescope the reflected image of the scale of the multiplier. The reading off was thus performed very precisely, and was more certain than with the naked eye close to the scale, because at this distance and with a fixed position of the eye the parallel axis of the index which stands at some distance from the graduated circle may be considered as evanescent. The method of exciting the electric current M'as the same as that given by Nobili: I wound the electromotive wire about a soft iron cylinder, which served as an armature and was filed smooth at those places where it was laid on the magnet, and laid it then on the magnet, or removed it suddenly from it, by which the magnetism arising at the moment, or vanishing again in the armature, thus produced the momentancous electric current. But as the removal of the armature can be performed in a more certain, prompt, and uniform manner than the placing of it on, I have in all my following experiments only given the results which were caused by the taking off of the armature, or the sudden removal of the magnetism in the iron. I must here at the same time remark that in my experiments it made no difference whether the magnetism of the iron disappeared really and entirely all at once, or there still remained a part, provided only the remaining quantity of magnetism was the same after each removal. I frequently convinced myself of this by the identity of the results in several repetitions of the experiment. This also showed me that the electromotive power of the magnet, at least after having already undergone several removals, did not become weaker; proofs of this will also be furnished in some experiments hereafter to be mentioned. In the above-described arrangement of the apparatus, I could now with the right hand perform the removal of the armature from the magnet, which was fixed to a table, while at the same time my eye observed in the telescope the consequent deviation of the index of the multiplier. This index was a thin lath, which was fixed by means of some wax to the wire which served as a common axis for the two needles of the multiplier, and formed a diameter of the graduated circle. Being thus able to observe the deviation for every result which was to be deduced therefrom, first on the one and then on the other end of the index, I freed this result from the influence of the eccentricity of the axis of the needles, and turning first the end A and then the end B of the spirals towards the north arm of the magnet, and allowing the needles of the multiplier to deviate first on the one side then on the other, I made the result independent of a second error which arises if the cocoon threads to which the needles of the multiplier are suspended possess a rotatory motion. Further, I carefully avoided every disturbance of the multiplier during a series of combined experiments, because it is impossible that every coil of the multiplier could act in the same manner as another (this would presuppose that they were all in the same plane, and parallel to one another), and because even if this might be presupposed, the action would still vary according as the needle when stationary might be exactly parallel to the coils, or form a greater or less angle. The positions of the needles when at rest seldom differ more than 0°·3 from one another. According to the above statement, a complete experiment always demanded four observations, namely, two (at both ends of the hand) for the position where the end A of the spiral was turned to the north pole, and two where B was directed to the north pole. Besides this I have repeated almost every experiment twice over in order to convince myself that no accidental fault had crept into the reading off; if the two observations differed much from one another, I again repeated each of them. The first preparatory experiments were made on the influence of the combinations of the conducting wires with the electromotive spirals and with the wires of the multiplier, in order to see whether I should content myself with winding the ends of the wires, which had been freed from their silk and were clean, very closely round one another, or should be obliged to produce a closer connection, for instance by immersing them in quicksilver. I proceeded on the supposition that if the connection effected by winding them many times closely round one another was not sufficient, an increase of convolutions would necessarily increase the force of the electrical current, I therefore made the following experiment. I wound round the armature ten convolutions of copper wire bespun with silk, and the conducting wires were connected with the ends of this spiral only by a single twist of the wires; the result of the four readings off amounted to 36°·8; upon this the same connection was made by twisting the ends of the wires ten times round one another as tightly as possible; the deviation amounted again to 36°·8; I finally pressed the last connection as tightly as possible together with a pair of pinchers, so that they were very much flattened; the deviation was 36°75. We may therefore consider the connection made by tightly twisting the wires ten times round one another as quite sufficient, and this was therefore made use of in all the subsequent experiments. The places where the connection was made were then wound round with silk stuff in order to secure them from reciprocal contact.

The second preparatory experiment I made in order to see whether, when I advanced the electromotive spiral on the armature more to the north limb or to the south limb of the magnet, it had any influence on the electric current. For this purpose I obtained with two convolutions the following results:

The convolutions advanced till in contact with the north limb of the magnet, gave a deviation	=	5°·55
The convolutions advanced until in contact with the south limb of the magnet, gave a deviation	=	5°·55
The convolutions advanced to the middle of both limbs gave a deviation	=	5°·60

therefore this influence also of the different positions of the spirals on the armature is imperceptible: from this time I always placed them so that the spirals occupied the middle of the armature.

I thirdly determined, before I proceeded to the proposed experiments, the thickness of the copper wires employed; I weighed two feet of each having wound off the silk, by which I obtained the proportions of their diameter on which it principally depended; but in order to obtain also their absolute thickness, I measured the thickest by means of a micrometrical contrivance: I obtained the following results, in which I have designated the wires, beginning with the thinnest, Nos. 1, 2, 3, and 4.

					grains			inch.
2	feet of wire	No. 1	weighed	=	23·3;	absolute thickness	=	0·023
2	————	No. 2	———	=	27·4;	————————	=	0·025
	(wire of the multiplier)
2	————	No. 3	weighed	=	83·9;	absolute thickness	=	0·044
2	————	No. 4	———	=	166·1;	————————	=	0·061

All the four kinds of wires were well covered with silk, so that no metal could be perceived except at the ends which served for connecting them.

I now proceed to the experiments themselves.

I. On the Influence of the number of Convolutions upon the Electro-motive Power produced in them.

In these experiments I connected the wire No. 3 with the multiplier so that the conducting wire and the electromotive spirals were formed of one and the same piece ; the length of this wire was about fifty feet: here however this is of no consequence, as it remained the same in all the experiments. The experiments themselves are contained in the following table.

Number of the Convolutions	INDIVIDUAL DEVIATIONS.				Mean deviation, or $\alpha$.	$\alpha$ in minutes		${\frac {1}{2}}\,\alpha$.
	Side A of the Spiral to the north pole.		Side B of the Spiral to the north pole.
	End a of Index.	End b of Index.	End a of Index.	End b of Index.
	°	°	°	°	°
2	5·7	5·8	5·3	5·8	5·65	5°	39'	2°	49'
4	12·1	12·9	11·1	12·0	12·00	12	00	6	00
8	25·7	25·8	22·9	25·2	24·90	24	54	12	27
9	29·5	30·1	26·2	28·5	28·32	28	19	14	15
10	32·5	33·3	29·4	32·0	31·80	31	48	15	54
12	39·8	40·9	35·8	38·6	38·77	38	46	19	23
14	47·4	48·8	40·8	45·9	45·43	45	43	22	51
16	55·7	56·8	47·6	52·3	53·10	53	6	26	33
18	63·1	64·4	54·1	57·8	59·80	59	48	29	54
20	71·0	71·8	62·8	66·6	68·05	68	3	34	1

From this series of experiments we must now deduce the electromotive power of the spirals for each number of convolutions, for which purpose the following considerations will be of service.

The action of the electric current in the wire of the multiplier upon the magnet needle, is a momentary one, since the current itself exists only for a moment; we may therefore consider this action as an impulse given to the needle, and shall be able to measure its force by the velocity which it imparts. But the velocity of the needle at its exit is evidently as great as that which it acquires when it springs back to the point of exit; it may therefore be expressed ($f$ being constant) by

$A=f{\sqrt {(\sin .{\mbox{vers.}}\,\alpha )}}$

where $A$ represents the sought for velocity of the exit; or according to what has been above stated, the magnitude of the current in the wire of the multiplier, and $\alpha$ the angle of deviation of the needle produced by this force. This expression changes however by the substitution of $2\sin .^{2}{\frac {1}{2}}\alpha$, instead of $\sin .{\mbox{vers.}}\alpha$ into the following

$A=p\cdot \sin .{\frac {1}{2}}\alpha$

if we put $p=f{\sqrt {2}}$.

In order now to find the resistance which the electric current suffers in its passage through the different wires, I first reduce their lengths all to one diagonal, and indeed to that of the wire of the multiplier, on the principle that two wires of the same metal offer then the same resistance to conduction when their lengths are in the same proportion as their diagonals (See Ohm's Galvanic Chain). In this case therefore the reduced lengths of the wires express their resistance to conduction: to have therefore a general idea of the problem, I suppose the multiplier, the conducting wires, and the electromotive spirals (with their free ends) to have the three reduced lengths, $L,\,l,$ and $\lambda$, and the electromotive power produced in the spirals to be represented by $x$, then ${\frac {x}{L+l+\lambda }}$ will be in effect the current which takes place, and we therefore have

${\frac {x}{L+l+\lambda }}=p\sin .{\frac {1}{2}}\alpha$

$x=(L+l+\lambda )\cdot p\cdot \sin .{\frac {1}{2}}\alpha$

(A.)

If we now consider the electromotive power in a convolution of the wire as unity, representing the unknown deviation produced by a convolution by $\xi$, and its reduced length by ($\lambda$); then granting the probable hypothesis, that at one and the same distance of the convolutions the electromotive force is directly as the number of convolutions, the following relation will take place for the number $n$, and for the reduced lengths $\lambda _{n}$ belonging to it (this is not necessarily $n\,\lambda$, because the free ends of the spirals need not increase in the same ratio for every number of convolutions)

${\frac {1}{n}}={\frac {(L+l+(\lambda ))p\cdot \sin .{\frac {1}{2}}\xi }{(L+l+\lambda _{n})p\cdot \sin .{\frac {1}{2}}\alpha }}$

therefore

$\sin .{\frac {1}{2}}\alpha =n\cdot {\frac {L+l+(\lambda )}{L+l+\lambda _{n}}}\cdot \sin .{\frac {1}{2}}\xi$

(B.)

In the experiments just mentioned $l+\lambda$ continued of the same magnitude for every number of convolutions, as the conducting and spiral wire consisted of one piece, besides $L$ remains the same, we therefore have $L+l+(\lambda )=L+l+\lambda _{n}$ and the equation B becomes changed into the following:

$\sin .{\frac {1}{2}}\alpha =n\sin .{\frac {1}{2}}\xi$

(C.)

If we now put instead of ${\frac {1}{2}}\alpha$ the values contained in the last column of our table of experiments, we obtain eleven equations, from which after the method of the least square, we shall be able to determine $\xi$, and if we bring this value of $\xi$ into the equation (C), we shall find the deviations $\alpha$ belonging to the number $n$ of convolutions, and the differences between this and the observed values will show whether the assumed hypothesis of the proportionality of the number of convolutions and of the electromotive power is confirmed in reality by the observation.—

The known formula for $\sin .{\frac {1}{2}}\xi$ is after the method of the least squares:

$\sin .{\frac {1}{2}}={\frac {\Sigma (n\cdot \sin .{\frac {1}{2}}\alpha )}{\Sigma (n^{2})}}$

and after having performed the calculation, we have from the foregoing table

$\xi =30^{\circ }\,9'\,{\mbox{or}}\,\log .\sin .{\frac {1}{2}}\xi =8\cdot 43989$

.

This value of $\xi$ gives for $\alpha$ the following values:

$\alpha$		DIFFERENCES.		$\alpha$		DIFFERENCES.
Calculated.	Observed.	In Degrees and Minutes.	In Degs.	Calculated.	Observed.	In Degrees and Minutes.	In Degs.
6° 18′	5° 39′	+ 0° 39′	+ 0°·6	45° 22′	45° 26′	− 0° 4′	− 0°·1
12 38	12 00	+ 0 38	+ 0·6	48 48	48 32	+ 0 16	+ 0·3
25 36	24 54	+ 0 32	+ 0·5	52 16	53 6	− 0 50	− 0·8
28 42	28 19	+ 0 23	+ 0·4	59 26	59 48	− 0 22	− 0·4
31 58	31 48	+ 0 10	+ 0·2	66 50	68 1	− 1 11	− 1·2
38 36	38 46	− 0 10	+ 0·2

the coincidence of the calculated with the observed deviations, confirming our presupposition that the electromotive power increases as the number of convolutions.

A second series of experiments on the same subject were made with the same wire, No. 3, except that the length of the wire through which the current had to pass, was no longer the same in each number of convolutions; we must therefore return to our general formula (B.). It was

$\sin .{\frac {1}{2}}\alpha =n\cdot {\frac {L+l+(\lambda )}{L+l+\lambda _{n}}}\cdot \sin .{\frac {1}{2}}\xi$.

The wire of the multiplier and of the conductors always remained the same, and was reduced to the diameter of the wire of the multiplier

$L+l=673\cdot 25\,{\mbox{inches}}$.

The lengths $\lambda ,\,\lambda _{i},\,\lambda _{ii}$ &c., were however changeable; I have therefore added these values, reduced also to the wire of the multiplier in the following table of the experiments.

$L+l+(\lambda )\,{\mbox{is}}=681\cdot 45$

Number of the Convolutions.	DEVIATIONS.				Individual Means.	Complete Means or $\alpha$.	$\lambda$	$L+l+\lambda$
	Side A of Spiral to the north pole.		Side B of Spiral to the north pole.
	End a of Index.	End b of Index.	End a of Index.	End b of Index.
5	18.5	18.5	19.8	20.5	19.33	19.40	17	690.25
	18.6	18.8	20.2	20.3	19.47
10	37.3	37.6	39.6	39.3	38.45	38.41	28	701.25
	37.3	37.5	39.4	39.3	38.37
15	57.8	58.7	58.6	58.2	58.32	58.15	39	712.25
	57.4	58.2	58.6	57.6	57.95
20	81.4	82.3	80.7	79.8	81.05	80.91	50	723.25
	81.3	82.3	79.7	79.8	80.77
25	110.0	112.7	103.1	101.9	106.67	106.67	61	734.25
	110.0	112.8	103.7	102.2	106.67

If we now apply the method of the least squares to this table, as we did to the first, we obtain

$\xi =3^{\circ }.97\,{\mbox{and}}\,\log .\sin .{\frac {1}{2}}\xi =8.53944$

and with this value we obtain from formula (B.) the following deviations:

Number of Convolutions.	DEVIATIONS.		Difference.
	Calculated.	Observed.
5	19.53	19.40	+ 0.13
10	39.00	38.41	+ 0.59
15	59.07	58.13	+ 0.94
20	80.67	80.91	− 0.24
25	105.67	106.67	− 1.00

In this place also the calculation coincides well with the observation; as I expected however to attain this coincidence still more completely, if I allowed the length of the conductors to remain the same for all the experiments, I made a second series of experiments similar to the above with another multiplier, where $\lambda ,\,\lambda '$, &c., remained always equal to one another; this series has also been performed with more care than the others above-mentioned, since each of the numbers contained in the following table is the mean deduced from three observations, in which mean however I retained only one decimal place. The columns designated by 1, 2, 3, 4 are intended for the same purpose as the four columns in the former tables.

Number of Convolutions.	Deviations.				Mean Deviations or $\alpha$.
	1	2	3	4
5	8·6	8·7	8·5	8·6	8·63
10	17·5	17·8	17·2	17·1	17·40
15	26·4	27·2	26·6	25·6	26·45
20	35·5	35·3	35·6	34·6	35·25
25	45·2	46·0	45·0	44·2	45·10
30	54·6	56·5	55·0	54·1	55·05

Hence may be calculated by means of the least squares

$\xi =1\cdot 73\;{\mbox{and}}\;\log .\sin .{\frac {1}{2}}=8\cdot 18478$

therefore we have for the calculated values of $\alpha$

Number of Convolutions.	$\alpha$		Difference.
	Calculated.	Observed.
5	8·77	8·60	+ 0·17
10	17·60	17·40	+ 0·20
15	26·53	16·45	+ 0·08
20	35·58	35·25	+ 0·33
25	45·00	45·10	− 0·10
30	54·67	55·05	− 0·38

Here then the coincidence for this kind of experiments is very great, go that we may regard the position as entirely confirmed, namely that

"the electromotive power which the magnet produces in a spiral, with convolutions of equal magnitude and with a wire of equal thickness and like substance, is directly in the same proportion as the number of the convolutions."

Moreover, we must not let it escape our attention, that in all the three series of observations the differences of the calculated and of the observed deviations are in the beginning positive, and then negative; which seems to show that the electromotive power increases in a somewhat quicker proportion than the number of the convolutions; but the differences are so small, and become, when the observations are made with great care (as the third series proves) smaller and smaller, I therefore ascribe this little irregularity to the influence of some peculiar circumstance which up to the present moment I have not succeeded in discovering.

II. On the Influence of the Distance of the Convolutions of Spirals on the production of the Electromotive Power in them.

In these experiments I employed at first the horseshoe magnet, but I soon perceived that from this none but false results could be obtained. By considerably widening the circuit of the spiral, it advanced nearer and nearer to the upper bow of the magnet; so that by removing the armature, not only the sudden disappearance of the magnetism in it, but also the sudden removal which took place at the same time from that upper part of the magnet (the bay of the horseshoe) acts on the spirals, and indeed unequally with unequal diameters of the spiral; the electromotive power becomes thus greater in larger spirals than it would otherwise be. On this account I took two strong rectilinear magnetic systems, each of which consisted of ten single magnet bars; I laid them with their opposite poles against one another so that they lay in a straight direction, and brought the iron cylinder which had served me in the above-mentioned experiments as an armature to the horseshoe magnet, between their poles, while the spirals covered the cylinder; upon this I let the magnet be suddenly drawn by two assistants from each other in opposite directions.

I wound at first ten convolutions of wire No. 2 round the iron cylinder

the diameter of the convolutions ═ 0·73 inch;

upon this I wound ten convolutions of wire No. 2 round a wooden disc,

the diameter of the convolutions ═ 6·57 inches;

the wooden disc was perforated in the centre, into which the iron cylinder was inserted. The observation gave

	Angle of Deviation.				Mean.
	1	2	3	4
Narrow Conv.	24·6	27·1	26·4	26·5	26·15
Wider Convolutions.	22·8	22·7	22·0	22·5	22·50
	23·4	23·5	21·6	23·2	22·92
Narrow Conv.	24·8	27·7	26·3	26·6	26·35

I observed the deviation of the wider spirals between the narrower, in order that the fault which might have originated by diminishing the magnetic power of the magnet systems might be estimated: we therefore have

for the narrower spirals the angle of deviation $\alpha =26\cdot 25$
for the wider spirals the angle of deviation $\alpha '=22\cdot 71$

The length of the wire of the multiplier and of the conducting wires (reduced to the diameter of the first) was as in the former experiments, i. e. they amounted together to $673\cdot 25$, or $L+l=673\cdot 25,\,\lambda$ however is for the narrower convolutions $=28$, and for the wider $\lambda '=203$. By means of formula (A.) we shall therefore obtain for the narrow spirals

$x=(L+l+\lambda )p\cdot \sin .{\frac {1}{2}}\alpha =701.25\cdot p\cdot \sin .(13^{\circ }\,7')$

for the wide spirals

$x'=(L+l+\lambda ')p\cdot \sin .{\frac {1}{2}}\alpha '=876.25\cdot p\cdot \sin .(11^{\circ }\,21')$,

therefore the relation of the electromotive powers, or

${\frac {x'}{x}}={\frac {876.25\cdot \sin .(11^{\circ }\,21')}{701.25\cdot \sin .(13^{\circ }\,7')}}=1.0838$,

}therefore not deviating much from 1, that is, the electromotive power is in both spirals the same.

I endeavoured in a more striking manner to confirm this position by the following experiment: I wound the wire No. 2 in six convolutions round a great wooden wheel of 28 inches in diameter, and placed the wheel on the iron cylinder. After having completed, as in the former cases, the experiment, I wound also six convolutions of the same wire immediately round the same iron cylinder, where also, as above, the convolutions again were 0·73 inch in diameter. The experiment gave

	Angle of deviation.				Mean or $\alpha$	$\lambda$	$L+l+\lambda$
	1	2	3	4
Narrower convolutions.	13·1	15·8	12·8	12·4	13·52	19·2	692·45
Wider convolutions.	7·1	8·7	7·1	8·7	7·90	549·2	1222·75

therefore

${\frac {x'}{x}}={\frac {1222.75\cdot \sin .(3^{\circ }\,52')}{692.45\cdot \sin .(6^{\circ }\,45'.5)}}=1.0107$.

Here the proportion of both electromotive powers approaches still more nearly to unity than in the former case, although the proportion of the diameter of the spirals is $=1:38.3$. We may therefore regard as a tiling proved by experiment, the position, that

"the electromotive power which the magnetism produces in the surrounding spirals is the same for every magnitude of the convolutions."

Since however a spiral wire inclosing the armature presents to the action of the magnetism in the armature a length greater in proportion as its diameter or its distance from the armature is greater, it follows from the law just discovered that the electromotive action of the magnet upon one and the same particle of the wire decreases in the simple ratio of the distance. This is as it were the reversal of the law demonstrated by Biot in the field of electro-magnetism, which, as is known, states that the action of an electric closing wire upon a magnetic needle decreases in the simple ratio of the distance; and it follows from our experiments as from those of Biot, that the action of a particle of the electric currents which encircle the magnet upon every particle of the spiral, is in the inverse ratio of the squares of the distance.

It also immediately follows from the law just demonstrated that the electric current produced in the various wire rings which inclose the armature, by its removal from the magnet, is in the inverse ratio of the diameter of the rings; for the electromotive power is the same in every ring, but the resistance it suffers in being conducted increases as the diameter of the rings; therefore the electric current, the quotient of the electromotive power, by the resistance it suffers, decreases as the diameter of the rings increases.

III. Influence of the Thickness of the Wire of the Electromotive Spirals on the Electromotive Power produced in them.

I have also again made these experiments with the horseshoe magnet, since in this case the convolutions of the wires had always the same magnitude. I here employed ten convolutions, which I formed from the wires No. 1, No. 3, and No. 4, and in which the diagonals were in the same proportion as the numbers 233 : 839 : 1661. The entire length of the convolutions in each sort was 33 inches. The deviations are contained in the following table.

	Angle of deviation.				Mean.
	1	2	3	4
Spirals from No. 1.	39·3	40·4	35·1	37·8	38·15	38·19
	39·3	40·4	35·2	38·8	38·22
Spirals from No. 3.	36·8	39·6	40·2	42·0	39·65	39·60
	36·4	39·4	40·4	42·0	39·55
Spirals from No. 4.	40·5	42·4	37·5	39·3	39·92	39·74
	40·3	40·4	37·5	40·1	39·57
Spirals from No. 1.	38·6	40·6	35·7	37·8	38·17	38·00
	38·7	40·0	35·2	37·4	37·82

If we now combine the observations No. 1, at the beginning and end of the series of experiments, and take their mean, we have the following deviations:

For	No. 1	the deviation	or	$\alpha \;\,{}=38.1$,
—	No. 3	————	or	$\alpha '\,=39.6$,
—	No. 4	————	or	$\alpha ''=39.7$.

From the proportion of the diagonals in which that of the wire of the multiplier is expressed by 274, we find the following reduced lengths (referred to the wire of the multiplier or No. 2) of our three spirals,

$\lambda \,\;=38.81$	therefore	$L+l+\lambda \,\;=712.06$,
$\lambda '\,=10.78$	———	$L+l+\lambda '\,=684.03$,
$\lambda ''=.\,5.44$	———	$L+l+\lambda ''=678.69$,

the equation (A.) gives therefore

$x\,\;=712.06\cdot p\cdot \sin .(19^{\circ }\,\,3'),$

$x'\,=684.04\cdot p\cdot \sin .(19\,\,48\,\,),$

$x''=678.69\cdot p\cdot \sin .(19\,\,51\,\,),$

or, if the two last electromotive powers be compared with the first, the proportions

${\frac {x}{x'}}={\frac {712.06\cdot \sin .(19^{\circ }3')}{684.03\cot \sin .(19^{\circ }48')}}=1.00305\quad {\mbox{and}}$

${\frac {x}{x''}}={\frac {712.06\cdot \sin .(19^{\circ }3')}{678.69\cdot \sin .(19^{\circ }51')}}=1.0085.$

Both propositions differ so little from unity that we are fully warranted in concluding that the electromotive power which the magnet produces in the wire No. 1 is quite as strong as those in the wires Nos. 3 and 4, although the latter possesses a diagonal almost four and seven times greater, and therefore that the electromotive power is independent of the thickness of the wires. A second confirmation of this position is found in the following experiment previously made:

		Angle of Deviation.				Mean.
		1	2	3	4
10 Conv. of wire	No. 3	36·3	37·8	33·5	35·7	35·82
—————	No. 2	36·0	37·0	32·1	34·9	34·0	34·95
		35·4	36·8	32·6	35·0	35·9
—————	No. 3	33·6	35·5	35·7	37·3	35·52

Consequently we have for

No. 2,	$\alpha .=34.95$,	further	$\lambda .=34.00,$	also	$L+l+\lambda .=707.25$
No. 3,	$\alpha '=35.67,$	———	$\lambda '=11.52,$	and	$L+l+\lambda '=684.77,$

consequently

${\frac {x}{x'}}={\frac {707.25\cdot \sin .(17^{\circ }.29')}{684.77\cdot \sin .(17^{\circ }.50')}}=1.013.$

Here also the proportion is so near to unity that we may from this, combined with the above results, regard it as an established truth, that

"the electromotive power produced in the spirals by the magnet remains the same for every thickness of the wires,or is independent of it."

From this law again it immediately follows that in rings of wires of various thickness surrounding the armature of the magnet, the electric current produced by its removal is directly as the diagonals of the wires; for the electromotive power remains the same, but the resistance it experiences in being conducted decreases inversely as the diagonals; consequently the electric currents, or the quotients of the electromotive powers by the diagonals, increase as the diagonals.

IV. On the Influence of the Substance of the Wires on the Electromotive Power produced in the Spirals.

Nobili and Antinori have in their first paper on the electrical phænomena produced by the magnet (Poggendorff's Annalen, 1832, No. 3) already determined the order in which four different metals are adapted to produce the electric current. They arrange them in the following order,—copper, iron, antimony, and bismuth.

It is particularly striking that this order is the same as that which the above metals occupy also in reference to their capacity of conducting electricity; and the idea suddenly struck me whether the electromotive power of the spirals did not remain the same in all metals, and whether the stronger current in the one metal did not arise from its being a better conductor of electricity than the other. With this view, therefore, I examined four metals, namely, copper, iron, platina, and brass, and pursued the following course: In order to avoid entirely the influence of different conduction, I brought at the same time into the metallic conducting circle through which the electric current had to pass, two spirals, equal in all respects excepting that they were of different metals, binding the one end of the first with the one conducting wire, the one end of the second with the other conducting wire, and connected the two ends of the spirals which had remained free with a distinct copper connecting wire. I now brought first the one spiral upon the iron armature of the horseshoe magnet, and proceeded in the same way with it as in the former experiments, and then the other. In this manner the resistance which the electric current suffered in each process was naturally quite the same.—I must also remark that I carefully avoided all thermo-electric disturbing forces, as I surrounded the places of connection of the various wires with several layers of blotting-paper, and after having arranged the apparatus I always waited several hours in order to give the places of connection time to take the temperature of the room.

The experiments themselves are as follows:

		Angle of deviation.				Mean.
		1	2	3	4
		°	°	°	°	°
Copper and iron spirals	Copper spir. on the	17·3	17·4	17·6	17·7	17·500
	armature
	Iron spirals	17·3	17·6	17·5	17·9	17·575
		17·3	17·4	17·6	18·1	17·600
	Copper spirals	17·4	17·4	17·8	18·2	17·700
Copper and platina spirals	Copper spir. on the	15·2	15·4	15·8	15·8	15·500
	armature
	Platina spirals	15·7	15·4	15·9	15·4	15·600
		15·7	15·4	15·8	15·4	15·575
	Copper spirals	15·4	15·8	15·3	15·9	15·600
Copper and brass spirals.	Brass spir. on the	18·4	18·3	18·4	18·2	18·350
	armature
	Copper spirals	18·5	18·2	18·1	18·3	18·275
		18·4	18·2	18·5	18·4	18·375
	Brass spirals	18·4	18·3	18·3	18·3	18·325

If we now combine the single means together and convert the decimals of the degrees into minutes, we obtain from this table the following results:

Copper spirals,	deviation	$\alpha \;\,=17^{\circ }\;36'.0$
Iron spirals,	———	$\alpha '=17\;\;35.2$
Copper spirals,	deviation	$\alpha \;\,=15^{\circ }\;34'.5$
Platina spirals,	———	$\alpha '=15\;\;35.2$
Copper spirals,	deviation	$\alpha \;\,=18^{\circ }\;19'.2$
Brass spirals,	———	$\alpha '=18\;20\cdot 2$

Since in this case the resistances remain the same for every pair of observations, our chief equation (A.) gives, when treated as before, the following proportions of the electromotive powers, if we designate them for copper, iron, platina, and brass with $x,\,x',\,x'',\,x'''$:

${\frac {x}{x'}}={\frac {\sin .(8^{\circ }\;43'.0)}{\sin .(8^{\circ }\;42'.6)}}=1.00033,$

${\frac {x}{x''}}={\frac {\sin .(7^{\circ }\;47'.2)}{\sin .(7^{\circ }\;47'.6)}}=0.99912,$

${\frac {x}{x'''}}={\frac {\sin .(9^{\circ }\;\;9'.6)}{\sin .(9^{\circ }\;10'.2)}}=0.99894.$

These three proportions are all of them so near to unity that there will exist no doubt as to the fact, that wires of copper, iron, platina, and brass suffer one and the same electromotive action; and that I may be allowed to extend the same position by analogy, even to all other metals and substances in general, until direct experiments shall have left the matter beyond all doubt. We shall have therefore the law,

"that the electromotive power which the magnet produces in spirals of wires of different substances, but in every other respect placed in exactly the same circumstances, is completely the same for all these substances"^[4].

Hence again it immediately follows that in two perfectly equal wire rings of different substance, surrounding the magnetic armature, the electric currents which are produced by taking the armature off or placing it on the magnet, are in direct proportion as the capacities of the substances for conducting electricity. Silver and copper wires therefore are the most advantageous.

From the latter observations we shall easily be able to deduce the capacity of the four metals for conducting, if we make a second similar observation, in which instead of bringing into the circle of the electric current two spirals of different metals, we make use of two of the already used copper spirals, and then place either of them on the armature, and determine the angle of deviation. Let this angle be called $\alpha$; and, for the other spirals, in the order in which they followed in the observation (therefore the copper spiral, with that of iron, platina, and brass), let these angles be designated by $\alpha ',\,\alpha ''\,$ and $\alpha '''$. Further, let the combined lengths of the wire of the multiplier of the conducting wires and that of the connecting wire of both spirals (all reduced to the diameter of the wire of the multiplier) be called $L$; but the lengths, which are equal in all the spirals, reduced also to the same diameter, be $\lambda$; we will further designate by $1,\,m',\,m'',\,m'''$, the conductive power of the metals in the above order, where that of the copper is also expressed by $1$.

If we take the general formula (A.), namely

$x=(L+l+\lambda )p\cdot \sin .{\frac {1}{2}}\alpha$,

we must here, since the wires are no longer of the same kind, substitute for the resistance $(L+l+\lambda )$, the resistances^[5]

$(L+\lambda ),\quad \left(L+{\frac {\lambda }{m'}}\right),\quad \left(L+{\frac {\lambda }{m''}}\right),\quad {\mbox{and}}\quad \left(L+{\frac {\lambda }{m'''}}\right),$

since they stand in inverse proportion to the capacities of conduction; we have therefore four equations (in which, according to the law just found, $x'=x''=x'''$ are ${}=x$),

$x=(L+\lambda )p\cdot \sin .{\frac {1}{2}}\alpha$
$x=\left(L+{\frac {\lambda '}{m'}}\right)p\cdot \sin .{\frac {1}{2}}\alpha '$
$x=\left(L+{\frac {\lambda ''}{m''}}\right)p\cdot \sin .{\frac {1}{2}}\alpha ''$
$x=\left(L+{\frac {\lambda '''}{m'''}}\right)p\cdot \sin .{\frac {1}{2}}\alpha ''',$

consequently, by division,

$1={\frac {L+\lambda }{L+{\frac {\lambda }{m'}}}}\cdot {\frac {\sin .{\frac {1}{2}}\alpha }{\sin .{\frac {1}{2}}\alpha '}}\quad {\mbox{or}}\quad L+{\frac {\lambda }{m'}}=(L+\lambda ){\frac {\sin .{\frac {1}{2}}\alpha }{\sin .{\frac {1}{2}}\alpha '}}$

$1={\frac {L+\lambda }{L+{\frac {\lambda }{m''}}}}\cdot {\frac {\sin .{\frac {1}{2}}\alpha }{\sin .{\frac {1}{2}}\alpha ''}}\quad {\mbox{or}}\quad L+{\frac {\lambda }{m''}}=(L+\lambda ){\frac {\sin .{\frac {1}{2}}\alpha }{\sin .{\frac {1}{2}}\alpha ''}}$

$1={\frac {L+\lambda }{L+{\frac {\lambda }{m'''}}}}\cdot {\frac {\sin .{\frac {1}{2}}\alpha }{\sin .{\frac {1}{2}}\alpha '''}}\quad {\mbox{or}}\quad L+{\frac {\lambda }{m''''}}=(L+\lambda ){\frac {\sin .{\frac {1}{2}}\alpha }{\sin .{\frac {1}{2}}\alpha '''}}$,

from which equations $m',\,m'',$ and $m'''$ may be found.

For our case, $L$ is ${}=849$ inches, $\lambda =84.1$, $\alpha =21^{\circ }\;52'$, $\alpha '=17^{\circ }\;36'$, $\alpha ''=15^{\circ }\;34'$, $\alpha '''=18^{\circ }\;20'$, hence we have

Capacity of conduction of	copper	= 1·00000,
———————————	iron or $m'$	= 0·27321,
———————————	platina or $m''$	= 0·18370,
———————————	brass or $m'''$	= 0·32106.

We might still find these values more exactly if the lengths of the wires were greater; but this investigation did not properly come within the scope of this paper, I therefore defer it till another occasion.

---

Consequences of the Laws already established in respect to the Construction of Electromotive Spirals.

In the following experiments I suppose the magnet for the production of the electric current to be given here, therefore the question is to determine those spirals of a certain metal which act most advantageously with this magnet and its cylindrical armature, which is likewise given. Further, I suppose the spirals, together with their free, not wound ends, to consist of one and the same wire; moreover, it is self-evident that every other property of the ends of the wires not belonging to the electromotive spirals may be reduced to those above mentioned, if we know the length, the diagonal, and the conducting power of the pieces of wire brought into the circle.

It is easy to see that by increasing the convolutions ad infinitum we do not also increase the strengths of the current ad infinitum. In the first place,—the number of convolutions of a given wire is limited by the length of the cylindrical armature; therefore the further increase of the number of convolutions can only be made by several series of convolutions placed one above another. Let the electromotive power of a series of convolutions which the length of the armature occupies ${}=\phi$; the length of the wire of all these convolutions, or, in this case, on account of the diameter of the wire being equal throughout, the resistance it offers ${}=\alpha$; let the length of the necessarily free ends of the wires together ${}=\beta$, the power therefore of the current of this first series of convolutions is

$\mu _{1}={\frac {\phi }{\alpha +\beta }};$

let $\gamma$ be the piece of the second series of convolutions by which its length, on account of its necessarily greater diameter, is greater than the length $\alpha$ of the first series, the power of the current from these two series is

$\mu _{2}={\frac {2\phi }{2\alpha +\gamma +\beta }},$

and in the same manner

$\mu _{3}={\frac {3\phi }{3\alpha +\gamma +\delta +\beta }},$

where $\delta$ designates the quantity by which the first series is surpassed in length by the third. If now the second series of convolutions does not add to the strength of the current, we put $\mu _{1}=\mu _{2}$, therefore

${\frac {\phi }{\alpha +\beta }}={\frac {2\phi }{2\alpha +\beta +\gamma }},$

whence we have

$\beta =\gamma ,$

i. e. as soon as the length of the free ends is only equal to the difference between the lengths of the second series of convolutions and those of the first, the second series would then add nothing to the strength of the current. In order to see what three series would do in this case, let us put $\beta =\gamma$ in the expression for $\mu$, and we obtain

$\mu _{3}={\frac {3\phi }{3\alpha +2\beta +\delta }};$

$\delta$ however is now greater than $\gamma$ or $\beta$, we therefore put $\delta =\beta +\mu$, where $\mu$ expresses a positive magnitude; we obtain by this

$\mu _{3}={\frac {3\phi }{3\alpha +3\beta +\mu }}={\frac {\phi }{(\alpha +\beta )+{\frac {\mu }{3}}}}.$

This last expression for $\mu _{3}$ is evidently smaller than ${\frac {\phi }{\alpha +\beta }}$, consequently three series of convolutions would only weaken the action of one or two series (which actions have been here assumed as equal).

In the same manner we find, if three series have not a more powerful action than two

$\delta ={\frac {1}{2}}(\gamma =\beta ),$

i. e. this happens when the length of the free ends is half as great as half the sum of the differences between the length of the first series and the lengths of the second and third series.

Having thus proved that by increasing the series of convolutions we never obtain a maximum of the electric current, and therefore that a greater increase would only do harm, we proceed to the general consideration of the subject.

We therefore suppose the convolutions of a series of the bespun metallic wire to lie thick on one another. Let the length of the space on which the convolutions may be wound up be ${}=\alpha$, the thickness of the wire ${}=\beta$; let the thickness of the wire covered with silk surpass the thickness of the uncovered wire by the excess $\beta$, so that it be ${}=b+\beta$,

the length of a convolution be ${}=c$, the lengths of the free ends of the wire ${}=m$; the number of convolutions then which can be wound in one series upon the armature is ${}={\frac {\alpha }{b+\beta }}$ and the length of the wire of these convolutions ${}={\frac {\alpha }{b+\beta }}\cdot c$, and the whole length which the electricity has to run through for one series of convolutions

${}={\frac {\alpha }{(b+\beta )}}c+m.$

If we assume the resistance offered by a wire of the same substance, whose length ${}=1$, and whose thickness ${}=1$, as unity, the resistance for one series of convolutions becomes

${\frac {={\frac {\alpha }{b+\beta }}c+m}{b^{2}}}={\frac {ac+(b+\beta )m}{b^{2}(b+\beta )}}.$

Further, let the electromotive power produced in one convolution, which, according to the second and third of our laws above proved, remains the same for every magnitude of the convolutions and for every thickness of the wire, be called $f$; the electromotive power produced in a series of convolutions is therefore, according to the first of the above laws,

${}={\frac {\alpha }{b+\beta }}\cdot f,$

and consequently the power of the electric current for a series of convolutions, or

$p_{1}={\frac {\alpha b^{2}f}{\alpha c+(b+\beta )m}}.$

We must now for our purpose express the length of a convolution or $c$ in terms of the diameter of the cylindrical armature, and the thickness of the wire and its silky envelope. We have however the semi-diameter of a convolution, if half the thickness of the iron cylinders is ${}=q$,

for the first		series	${}=q+{\frac {b+\beta }{2}}$
——	second	———	${}=q+{\frac {3}{2}}(b+\beta )$
——	third	———	${}=q+{\frac {5}{2}}(b+\beta )$
——	nth	———	${}=q+{\frac {2n-1}{2}}(b+\beta )$

whence we have the length of a convolution or

$c$	for the first		series	${}={\big (}2q+(b+\beta ){\big )}\pi$
$c$	——	second	———	${}={\big (}2q+3(b+\beta ){\big )}\pi$
$c$	——	third	———	${}={\big (}2q+5(b+\beta ){\big )}\pi$
$c$	——	nth	———	${}={\big (}2q+(2n-1)(b+\beta ){\big )}\pi .$

If we substitute the first value of $c$ in the equation for $m$, we obtain

$p_{1}={\frac {\alpha b^{2}f}{\alpha \pi {\big (}2q+(b+\beta ){\big )}+(b+\beta )m}}.$

The electromotive power is for two series of convolutions with regard to the above law, No. 2,

${}=2{\frac {\alpha }{b+\beta }}\cdot f;$

but the resistance is equal to that of the two series of convolutions, together with the piece $m$, therefore

${}={\frac {{\frac {\alpha }{b+\beta }}(2q+b+\beta )\pi +{\frac {\alpha }{b+\beta }}(2q+3(b+\beta ))\pi +m}{b^{2}}}$

${}={\frac {\alpha \pi (4q+4(b+\beta ))+m(b+\beta )}{b^{2}(b+\beta )}},$

consequently the power of the current for two series, or

${\begin{aligned}p_{2}&={\frac {2\cdot {\frac {\alpha }{b+\beta }}\cdot f}{\frac {\alpha \pi (4q+4(b+\beta ))+m(b+\beta )}{b^{2}(b+\beta )}}}\\&={\frac {2\alpha b^{2}f}{\alpha \pi (4q+4(b+\beta ))+m(b+\beta )}}.\end{aligned}}$

In the same manner we find Pi = P^ = ^" ~ « ,r (2 nq + n"- (6 + |3) ) + /« (b + ^) ' " ^ '^

If I differentiate this general expression of the power of the current for $n$ series of convolutions in regard to $n$, I obtain

${\frac {d\cdot p_{n}}{d\,n}}=ab^{2}f{\frac {a\pi (2nq+n^{2}(b+\beta ))+m(b+\beta )-a\pi n(2q+2n(b+\beta ))}{\left[a\pi (2nq+n^{2}(b+\beta ))+m(b+\beta )\right]^{2}}}$

If I put this expression ${}=0$, we have after some reductions

$m-a\pi n^{2}=0,$

consequently

$n={\sqrt {\left({\frac {m}{a\pi }}\right)}}.$

I take here the positive sign of the root, because $n$ according to its nature cannot be negative, and $m,\,\alpha ,\,\pi ,$ are all three positive.

If we further develope ${\frac {d^{2}p_{2}}{d\,n^{2}}}$ and put in the expression found this value of $n={\sqrt {\left({\frac {m}{a\pi }}\right)}}$ we obtain a negative magnitude; consequently this value of $n$ represents a maximum of the current.

From the discovered value of $n$ for the maximum of the current, we can infer

1. That the maximum of the action of the magnet on our spirals, for every thickness of the wire, is attained by the same number of series of convolutions; for $n$ is independent of $b+\beta$.

2. That the longer the free ends of the spirals are, or the greater $m$ is, the greater is the number of the series of convolutions required in order to attain the maximum of the action.

3. That the longer the space $\alpha$ is on which the convolutions may be wound round in one series, the less number of series of convolutions are necessary to produce the greatest current.

4.. That the maximum is independent of $q$, i. e. that it is quite indifferent for the number of series of convolutions necessary to the attainment of the maximum, whether they are immediately wound round the cylinder of iron, or round another cylinder which is placed on the other one.

If we put the above found value $n={\sqrt {\left({\frac {m}{a\pi }}\right)}}$ in the general expression of the power which is contained in the equation (D.), we obtain after some reductions, as the expression for the maximum which is attained by the current,

$p_{(\mathrm {maximum} )}={\frac {b^{2}f}{2\left(\pi q+(b+\beta ){\sqrt {\left({\frac {\pi m}{\alpha }}\right)}}\right)}}$

(E.)

This expression again shows:

1. That the maximum of the current stands in direct proportion to $f$, i. e., to the power of the magnet, or rather to the strength of the magnetism which is produced in the armature by the placing on of the magnet, and which again vanishes.

2. The maximum is more powerful for a thick wire than for a slender one, for we can bring its expression to the form

${\frac {f}{{\frac {A}{b^{2}}}+{\frac {B}{b}}}},$

which shows that the whole expression increases with the increase of $b$.

3. The maximum decreases with $q$, i. e. it becomes so much the smaller according to the greatness of the cylinder on which the first series of convolutions is wound, it being assumed that the armature does not on that account become greater.

4. It becomes smaller with the increase of $m$, i. e. the greater the free connecting ends of the spirals are, the smaller is the ultimate attainable maximum of the current.

5. Finally, the maximum increases when $\alpha$ increases, i. e. when the space of the armature upon which a series of convolutions can be wound becomes greater.

We shall consider the power of the current of a single convolution wound round the armature to be the same as $m$, then as soon as we put in the general expression (D.) for the current $n=1$, and $\alpha =b+\beta$, we find

$p_{\mathrm {(a\,convolution)} }={\frac {b^{2}f}{\pi (2q+b+\beta )+{}}}m.$

If we divide the expression for the maximum of the current (E.) by this, we may designate the quotients as the maximum of increase, and find that

the maximum of the increase is

${\frac {2q+(b+\beta )+{\frac {m}{\pi }}}{2q+2(b+\beta ){\sqrt {\left({\frac {m}{\alpha \pi }}\right)}}}}$

(F.)

If I propose to find, for instance, with how many series of convolutions I attain the maximum of the current for my magnet and armature, when I take a length of 850 English inches for the wire of the multiplier and the connecting wires together, I have

$\alpha =1\cdot 6,\quad b+\beta =0\cdot 065\quad {\mbox{(wire No. 4)}}\quad q=0\cdot 335,\quad m=850.$

The formula $n={\sqrt {\frac {m}{\alpha \pi }}}$ gives for $n=13\cdot 07$, and the formula (F.) gives the maximum of increase ${}=114\cdot 8$. We shall obtain therefore the maximum of the current at somewhat about thirteen series of convolutions, and the current then becomes about 115 times stronger than when produced by one convolution.

We will here separately consider the case in which $m=0$, i. e. where the spirals have no free ends, but close in themselves. If we put $m=0$ in the expression of the current for one convolution, for one series of convolutions, and for $n$ series of convolutions, we shall then obtain

for a single convolution ${}={\frac {b^{2}f}{2q\pi +\pi (b+\beta )}}$

for a series of convolutions ${}={\frac {b^{2}f}{2q\pi +\pi (b+\beta )}}$

for $n$ series of convolutions ${}={\frac {b^{2}f}{2q\pi +n\pi (b+\beta )}}$

whence it follows that here the current in one convolution is just as strong as in a series consisting of any number of convolutions; and that in both these cases it is stronger than when several series of convolutions cross one another (for $n$ is quite a positive number). The expression of the current for a convolution may moreover be exhibited thus

${\frac {f}{\frac {(2q+b+\beta )\pi }{b^{2}}}}$

i. e. it is equal to the electromotive power, divided by the resistance offered by a convolution; and in effect it is evident that in this case of $m=0$ a series of convolutions must act just in the same manner as a single convolution; for with the increase of the number of convolutions the electromotive power and the resistance become increased in the same proportion, consequently the quotient of the one by the other, or the electric current remains unchanged. It is also now evident that in effect a second series of convolutions can only weaken the current, since in the second series the electromotive power increases as in the first, with the increase of the number of convolutions; while, on the contrary, the resistance is greater in the two series than double the same in one series, on account of the enlarged diameter.

But there is one phænomenon of electro-magnetism to which all the above positions however cannot yet be applied, namely, to the production of the spark. This occurs then only, when the metallic conductor of the current is disturbed at some place; there enters therefore into the circular passage of the current an intermediate conductor, whose length is almost indefinitely small, but whose resistance is almost indefinitely great. We must therefore, in order to apply the above-developed formulæ, first be in a condition to reduce this intermediate conductor to a certain length of wire, with the diameter of the wire given, and thus to determine $m$;—but for this reduction we are yet in want of the data.

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1. Translated from the German by Mr. W. Francis.
2. In this repetition I obtained the spark beautifully by means of a spiral of a wire 70 feet in length and 0·044 inch thick. The apparatus was formed after the one described by Nobili, so that the horse-shoe magnet (of 22 lbs. lifting power) caused of itself the closing of the current.
3. In this memoir the measures are always expressed in English inches, except when otherwise remarked.
4. After I had made the above experiment I saw from No. 5 of Poggendorff's Annalen, which I had then just received, that this last law had already been demonstrated, although in a different way, by Faraday. My experiment may therefore serve to confirm it.
5. In the following expressions I consider the resistance $l$ jointly with that of $L$, since in the multiplier last employed the conducting wires consisted of one piece with the wire of the multiplier, therefore $L+\lambda$ must always remain constant.