# Scientific Memoirs/2/The Galvanic Circuit investigated Mathematically

Article XIII.

The Galvanic Circuit investigated Mathematically. By Dr. G. S. Ohm[1].

Preface.

I HEREWITH present to the public a theory of galvanic electricity, as a special part of electrical science in general, and shall successively, as time, inclination, and means permit, arrange more such portions together into a whole, if this first essay shall in some degree repay the sacrifices it has cost me. The circumstances in which I have hitherto been placed, have not been adapted either to encourage me in the pursuit of novelties, or to enable me to become acquainted with works relating to the same department of literature throughout its whole extent. I have therefore chosen for my first attempt a portion in which I have the least to apprehend competition. May the well-disposed reader receive the performance with the same love for the object as that with which it is sent forth.

The Author.

Berlin, May 1st, 1827.

Introduction.

The design of this Memoir is to deduce strictly from a few principles, obtained chiefly by experiment, the rationale of those electrical phænomena which are produced by the mutual contact of two or more bodies, and which have been termed Galvanic:—its aim is attained if by means of it the variety of facts be presented as unity to the mind. To begin with the most simple investigations, I have confined myself at the outset to those cases where the excited electricity propagates itself only in one dimension. They form, as it were, the scaffold to a greater structure, and contain precisely that portion, the more accurate knowledge of which may be gained from the elements of natural philosophy, and which, also, on account of its greater accessibility, may be given in a more strict form. To answer this especial purpose, and at the same time as an introduction to the subject itself, I give, as a forerunner of the compressed mathematical investigation, a more free, but not on that account less connected, general view of the process and its results.

Three laws, of which the first expresses the mode of distribution of the electricity within one and the same body, the second the mode of dispersion of the electricity in the surrounding atmosphere, and the third the mode of appearance of the electricity at the place of contact of two heterogeneous bodies, form the basis of the entire Memoir, and at the same time contain everything that does not lay claim to being completely established. The two latter are purely experimental laws; but the first, from its nature, is, at least partly, theoretical.

With regard to this first law, I have started from the supposition that the communication of the electricity from one particle takes place directly only to the one next to it, so that no immediate transition from that particle to any other situate at a greater distance occurs. The magnitude of the transition between two adjacent particles, under otherwise exactly similar circumstances, I have assumed as being proportional to the difference of the electric forces existing in the two particles: just as, in the theory of heat, the transition of caloric between two particles is regarded as proportional to the difference of their temperatures. It will thus be seen that I have deviated from the hitherto usual mode of considering molecular actions introduced by Laplace; and I trust that the path I have struck into will recommend itself by its generality, simplicity, and clearness, as well as by the light which it throws upon the character of former methods.

With respect to the dispersion of electricity in the atmosphere, I have retained the law deduced from experiments by Coulomb, according to which, the loss of electricity, in a body surrounded by air, in a given time, is in proportion to the force of the electricity, and to a coefficient dependent on the nature of the atmosphere. A simple comparison of the circumstances under which Coulomb performed his experiments, with those at present known respecting the propagation of electricity, showed, however, that in galvanic phænomena the influence of the atmosphere may almost always be disregarded. In Coulomb's experiments, for instance, the electricity driven to the surface of the body was engaged in its entire expanse in the process of dispersion in the atmosphere; while in the galvanic circuit the electricity almost constantly passes through the interior of the bodies, and consequently only the smallest portion can enter into mutual action with the air; so that, in this case, the dispersion can comparatively be but very inconsiderable. This consequence, deduced from the nature of the circumstances, is confirmed by experiment; in it lies the reason why the second law seldom comes into consideration.

The mode in which electricity makes its appearance at the place of contact of two different bodies, or the electrical tension of these bodies, I have thus expressed: when dissimilar bodies touch one another, they constantly maintain at the point of contact the same difference between their electroscopic forces.

With the help of these three fundamental positions, the conditions to which the propagation of electricity in bodies of any kind and form is subjected may be stated. The form and treatment of the differential equations thus obtained are so similar to those given for the propagation of heat by Fourier and Poisson, that even if there existed no other reasons, we might with perfect justice draw the conclusion that there exists an intimate connexion between both natural phænomena; and this relation of identity increases, the further we pursue it. These researches belong to the most difficult in mathematics, and on that account can only gradually obtain general admission; it is therefore a fortunate chance, that in a not unimportant part of the propagation of electricity, in consequence of its peculiar nature, those difficulties almost entirely disappear. To place this portion before the public is the object of the present memoir, and therefore so many only of the complex cases have been admitted as seemed requisite to render the transition apparent.

The nature and form commonly given to galvanic apparatus favours the propagation of the electricity only in one dimension; and the velocity of its diffusion combined with the constantly acting source of galvanic electricity is the cause of the galvanic phænomena assuming, for the most part, a character which does not vary with time. These two conditions, to which most frequently galvanic phænomena are subjected, viz. change of the electric state in a single dimension, and its independency of time, are however precisely the reasons why the investigation is brought to a degree of simplicity which is not surpassed in any branch of natural philosophy, and is altogether adapted to secure incontrovertibly to mathematics the possession of a new field of physics, from which it had hitherto remained almost totally excluded.

The chemical changes which so frequently occur in some, generally fluid, portions of a galvanic circuit, greatly deprive the result of its natural simplicity, and conceal, to a considerable extent, by the complications they produce, the peculiar progression of the phænomenon; they are the cause of an unexampled change of the phænomenon, which gives rise to so many apparent exceptions to the rule, frequently even to contradictions, in so far as the sense of this word is itself not in contradiction to nature. I have distinctly separated the consideration of such galvanic circuits in which no portion undergoes a chemical change, from those whose activity is disturbed by chemical action, and have devoted a separate part to the latter in the Appendix. This total separation of two parts forming a whole, and, as might appear, the less dignified position of the latter, will find in the following circumstance a sufficient explanation. A theory, which lays claim to the name of an enduring and fruitful one, must have all its consequences in accordance with observation and experiment. This, it seems to me, is sufficiently established with respect to the first of the parts above-mentioned, partly by the previous experiments of others, and partly by some performed by myself, which first made me acquainted with the theory here developed, and subsequently rendered me entirely devoted to it. Such is not the case with regard to the second part. A more accurate experimental verification is in this case almost entirely wanting, to undertake which I need both the requisite time and means; and therefore I have merely placed it in a corner, from which, if worth the trouble, it may be drawn hereafter, and may then also be further matured under better nursing.

By means of the first and third fundamental positions we obtain a distinct insight into the primary galvanic phænomenon in the following way. Imagine, for instance, a ring everywhere of equal thickness and homogeneous, having, at any one place, in its whole thickness, one and the same electrical tension, i.e. inequality in the electrical state of two surfaces situated close to each other, from which causes, when they have come into action, and the equilibrium is consequently disturbed, the electricity will, in its endeavour to re-establish itself, if its mobility be solely confined to the extent of the ring, flow off on both sides. If this tension were merely momentary, the equilibrium would very soon be re-established; but if the tension is permanent, the equilibrium can never be restored; but the electricity, by virtue of its expansive force, which is not sensibly restrained, produces in a space of time, the duration of which almost always escapes our senses, a state which comes nearest to that of equilibrium, and consists in this; that by the constant transmission of the electricity, a perceptible change in the electric condition of the parts of the body through which the current passes is nowhere produced. The peculiarity of this state, also occurring frequently in the transmission of light and heat, has its foundation in this; that each particle of the body situated in the circle of action receives in each moment just so much of the transmitted electricity from the one side as it gives off to the other, and therefore constantly retains the same quantity. Now since by reason of the first fundamental position the electrical transition only takes place directly from the one particle to the other, and is, under otherwise similar circumstances, determined according to its energy by the electrical difference of the two particles, this state must evidently indicate itself on the ring, uniformly excited in its entire thickness, and similarly constituted in all its parts, by a constant change of the electric condition, originating from the point of excitation, proceeding uniformly through the whole ring, and finally again returning to the place of excitation; whilst at this place itself, a sudden spring in the electric condition, constituting the tension, is, as was previously stated, constantly perceptible. In this simple separation or division of the electricity lies the key to the most varied phænomena.

The mode of separation of the electricity has been completely determined by the preceding observation; but the absolute force of the electricity at the various parts of the ring still remains uncertain. This property may be best conceived, by imagining the ring, without its nature being altered, opened at the point of excitation and extended in a straight line, and representing the force of the electricity at each point by the length of a perpendicular line erected upon it; that directed upwards may represent a positive electrical, but that downwards a negative electrical, state of the part. The line ${\displaystyle \mathrm {AB} }$ (Plate XXIV., fig. 1) may accordingly represent the ring extended in a straight line, and the lines ${\displaystyle \mathrm {AF} }$ and ${\displaystyle \mathrm {BG} }$ perpendicular to ${\displaystyle \mathrm {AB} }$ may indicate by their lengths the force of the positive electricities situated at the extremities ${\displaystyle \mathrm {A} }$ and ${\displaystyle \mathrm {B} }$. If now the straight line ${\displaystyle \mathrm {FG} }$ be drawn from ${\displaystyle \mathrm {F} }$ to ${\displaystyle \mathrm {G} }$, also ${\displaystyle \mathrm {FH} }$ parallel to ${\displaystyle \mathrm {AB} }$, the position of ${\displaystyle \mathrm {FG} }$ will give the mode of separation of the electricity, and the quantities ${\displaystyle \mathrm {BG} {\frac {\quad }{}}\mathrm {AF} }$ or ${\displaystyle \mathrm {GH} }$ the tension occurring at the extremities of the ring; and the force of the electricity at any other place ${\displaystyle \mathrm {C} }$, may easily be expressed by the length of ${\displaystyle \mathrm {CD} }$ drawn through ${\displaystyle \mathrm {C} }$ perpendicularly to ${\displaystyle \mathrm {AB} }$. But, from the nature of the galvanic excitation, merely the quantity of the tension or the length of the line ${\displaystyle \mathrm {GH} }$, therefore the difference of the lines ${\displaystyle \mathrm {AF} }$ and ${\displaystyle \mathrm {BG} }$, is determined, but not at all the absolute magnitudes of the lines ${\displaystyle \mathrm {AF} }$ and ${\displaystyle \mathrm {BG} }$; consequently the mode of separation may be represented quite as well by any other line parallel to the former, e.g. by ${\displaystyle \mathrm {IK} }$, for which the tension still constantly retains the same value expressed by ${\displaystyle \mathrm {KN} }$, because the ordinates situated at present below ${\displaystyle \mathrm {AB} }$ assume a relation opposed to their former one. Which of the infinitely numerous lines parallel to ${\displaystyle \mathrm {FG} }$ would express the actual state of the ring cannot be stated in general, but must in each case be separately determined from the circumstances which occur. Moreover, it is easily conceived that, as the position of the line sought is given, it would be completely determined for one single part of the ring by the determination of any one of its points, or, in other words, by the knowledge of the electric force. If, for instance, the ring lost all its electricity by abduction at the place ${\displaystyle \mathrm {C} }$, the line ${\displaystyle \mathrm {LM} }$ drawn through ${\displaystyle \mathrm {C} }$ parallel to ${\displaystyle \mathrm {FG} }$ would in this case express with perfect certainty the electrical state of the ring. This variability in the separation of the electricity is the source of the changeableness of the phænomenon peculiar to the galvanic circuit. I may further add, that it is evidently quite indifferent whether the position of the line ${\displaystyle \mathrm {FG} }$ with respect to that of ${\displaystyle \mathrm {AB} }$ be fixed; or whether the position of the line ${\displaystyle \mathrm {FG} }$ remain constantly the same, and the position of ${\displaystyle \mathrm {AB} }$ with respect to it be altered. The latter course is by far the more simple where the separation of the electricity assumes a more complex form.

The conclusions just arrived at, which hold for a ring homogeneous throughout its whole extent, may easily be extended to a ring composed of any number of heterogeneous parts, if each part be of itself homogeneous and of the same thickness. I may here take as an example of this extension a ring composed of two heterogeneous parts. Let this ring be imagined as before open at one of its places of excitation and stretched out to form the right line ${\displaystyle \mathrm {ABC} }$ (fig. 2), so that ${\displaystyle \mathrm {AB} }$ and ${\displaystyle \mathrm {BC} }$ indicate the two heterogeneous parts of the ring. The perpendiculars ${\displaystyle \mathrm {AF} }$, ${\displaystyle \mathrm {BG} }$, will represent by their lengths the electrical forces present at the extremities of the part ${\displaystyle \mathrm {AB} }$; on the other hand, ${\displaystyle \mathrm {BH} }$ and ${\displaystyle \mathrm {CI} }$, those present at the extremities of the part ${\displaystyle \mathrm {BC} }$; accordingly ${\displaystyle \mathrm {AF} +\mathrm {CI} }$ or ${\displaystyle \mathrm {FK} }$ will represent the tension at the opened place of excitation, and ${\displaystyle \mathrm {GH} }$ the tension occurring at ${\displaystyle \mathrm {B} }$ at the point of contact. Now if we only bear in mind the permanent state of the circuit, the straight lines ${\displaystyle \mathrm {FG} }$ and ${\displaystyle \mathrm {HI} }$ will, from the reasons above mentioned, indicate by their position the mode of separation of the electricity in the ring; but whether the line ${\displaystyle \mathrm {AC} }$ will keep its place, or must be advanced further up or down, remains uncertain, and can only be found out in each distinct case by other separate considerations. If, for instance, the point ${\displaystyle \mathrm {O} }$ of the circuit is touched abductively, and thus deprived of all electricity, ${\displaystyle \mathrm {ON} }$ would disappear; and therefore the line ${\displaystyle \mathrm {LM} }$ drawn through ${\displaystyle \mathrm {N} }$ parallel with ${\displaystyle \mathrm {AC} }$ would in this case give the position of ${\displaystyle \mathrm {AC} }$ required. It is hence evident, how sometimes this, sometimes another, position of the line ${\displaystyle \mathrm {AC} }$ in the figure ${\displaystyle \mathrm {FGHI} }$, representing the separation of the electricity, may be the one suited to the circumstances; and herein we recognise the source of the variability of galvanic phænomena already mentioned.

It is, however, essentially requisite, in order to be able to judge thoroughly of the present case, to attend to a circumstance the mention of which has hitherto been purposely avoided, that the various considerations might be separated as distinctly as possible. The distances ${\displaystyle \mathrm {FK} }$ and ${\displaystyle \mathrm {GH} }$ are indeed given by the tensions existing at the two places of excitation, but the figure ${\displaystyle \mathrm {FGHI} }$ is not yet wholly determined by this alone. For instance, the points ${\displaystyle \mathrm {G} }$ and ${\displaystyle \mathrm {H} }$ might move down towards ${\displaystyle \mathrm {G} '}$ and ${\displaystyle \mathrm {H} '}$, so that ${\displaystyle \mathrm {G} '\mathrm {H} '}$ would equal ${\displaystyle \mathrm {GH} }$, giving rise to the figure ${\displaystyle \mathrm {FG} '\mathrm {H} '\mathrm {I} }$, which would indicate quite a different mode of separation of the electricity, although the individual tensions in it still retain their former magnitude. Consequently if that which has been stated with respect to the circuit of two members is to acquire a sense no longer subject to any arbitrary explanation, this uncertainty must be removed. The first fundamental law effects this in the following way:—For since the state of the ring alone, independent of the time, is regarded, each section must, as has already been stated, receive in every moment the same quantity of electricity from one side as it gives off to the other. This condition occasions upon such portions of the ring as have perfectly the same constitution at their various points, the constant and uniform change in the separation which is represented in the first figure by the straight line ${\displaystyle F\,G}$, and in the second by the straight lines ${\displaystyle F\,G}$ and ${\displaystyle H\,I}$. But when the geometrical or the physical nature of the ring changes in passing from one of its component parts to another, the reason of this constancy and uniformity no longer obtains; consequently the manner in which the several straight lines are combined into a complete figure must first be deduced from other considerations. To facilitate the object, I will separately consider the geometrical and physical difference of the single parts, each independently.

Let us first suppose that every section of the part ${\displaystyle B\,C}$ is ${\displaystyle m}$ times smaller than in the part ${\displaystyle A\,B}$, while both parts are composed of the same substance; the electric state of the ring, which is independent of time, and which requires that everywhere throughout the entire ring just as much electricity be received on one side as is given off from the other, can evidently only exist under the condition that the electric transition from one particle to the other in the same time within the portion ${\displaystyle B\,C}$ is ${\displaystyle m}$ times greater than in the portion ${\displaystyle A\,B}$; because it is only in this manner that the action in both parts can maintain equilibrium. But in order to produce this ${\displaystyle m}$ times greater transition of the electricity from element to element, the electrical difference of element to element within the portion ${\displaystyle B\,C}$ must, according to the first fundamental position, be ${\displaystyle m}$ times greater in the portion ${\displaystyle A\,B}$; or when this determination is transferred to the figure, the line ${\displaystyle H\,I}$ must sink ${\displaystyle m}$ times more on equal portions, or have an ${\displaystyle m}$ times greater "dip" than the fine ${\displaystyle F\,G}$. By the expression "dip" (Gefälle), is to be understood the difference of such ordinates which belong to two places distant one unit of length from each other. From this consideration results the following rule: The dips of the lines ${\displaystyle F\,G}$ and ${\displaystyle H\,I}$ in the portions ${\displaystyle A\,B}$ and ${\displaystyle B\,C}$, composed of like substance, will be inversely to each other as the areas of the sections of these parts. By this the figure ${\displaystyle F\,G\,H\,I}$ is now fully determined.

When the parts ${\displaystyle A\,B}$ and ${\displaystyle B\,C}$ of the ring have the same section but are composed of different substances, the transition of the electricity will then no longer be dependent solely on the progressive change of electricity in each part from element to element, but at the same time also on the peculiar nature of each substance. This difference in the distribution of the electricity, caused solely by the material nature of the bodies, whether it have its origin in the peculiar structure or in any other peculiar state of the bodies to electricity, establishes a distinction in the electrical conductibility of the various bodies; and even the present case may afford some information respecting the actual existence of such a distinction and give rise to its more accurate determination. In fact, since the ring composed of the two parts ${\displaystyle A\,B}$ and ${\displaystyle B\,C}$ differs from the homogeneous one only in this respect, that the two parts are formed of two different substances, a difference in the dip of the two lines ${\displaystyle F\,G}$ and ${\displaystyle H\,I}$ will make known a difference in the conductibility of the two substances, and one may serve to determine the other. In this way we arrive at the following position, supplying the place of a definition: In a ring consisting of two parts ${\displaystyle A\,B}$ and ${\displaystyle B\,C}$, of like sections but formed of different substances, the dips of the lines ${\displaystyle F\,G}$ and ${\displaystyle H\,I}$ are inversely as the conducting powers of the two parts. If we have once ascertained the conducting powers of the various substances, they may be employed to determine the dips of the lines ${\displaystyle F\,G}$ and ${\displaystyle H\,I}$ in every case that may occur. By this, then, the figure ${\displaystyle F\,G\,H\,I}$ is entirely determined. The determination of the conductibility from the separation of the electricity is rendered very difficult from the weak intensity of galvanic electricity, and from the imperfection of the requisite apparatus; subsequently we shall obtain a more easy means of effecting this purpose.

From these two particular cases we may now ascend in the usual way to the general one, where the two prismatic parts of the ring neither possess the same section nor are constituted of the same substance. In this case the dips of the two parts must be in the inverse ratio of the products of the sections and powers of conduction. We are hereby enabled to determine completely the figure ${\displaystyle F\,G\,H\,I}$ in every case, and also to distinguish perfectly the mode of electrical separation in the ring. All the peculiarities, hitherto considered separately, of the ring composed of two heterogeneous parts, may be summed up in the following manner: In a galvanic circuit consisting of two heterogeneous prismatic parts, there takes place in regard to its electrical state a sudden transition from the one part to the other at each point of excitation, forming the tension there occurring, and from one extremity of each point to the other a gradual and uniform transition; and the dips of these two transitions are inversely proportional to the products of the conductibilities and sections of each part.

Proceeding in this manner, we are able without much difficulty to inquire into the electrical state of a ring composed of three or more heterogeneous parts, and to arrive at the following general law: In a galvanic circuit consisting of any indefinite number of prismatic parts, there takes place in regard to its electrical state at each place of excitation a sudden transition, from one part to the other, forming the tension there prevailing, and within each part a gradual and uniform transition from the one extremity to the other; and the dips of the various transitions are inversely proportional to the products of the conductibilities and sections of each part. From this law may easily be deduced the entire figure of the separation for each particular case, as I will now show by an example.

Let ${\displaystyle A\,B\,C\,D}$ (fig. 3) be a ring composed of three heterogeneous parts, open at one of its places of excitation, and extended in a straight line. The straight lines ${\displaystyle F\,G}$, ${\displaystyle H\,I}$, ${\displaystyle K\,L}$ represent by their position the mode of separation of the electricity in each individual part of the ring, and the lines ${\displaystyle A\,F}$, ${\displaystyle B\,G}$, ${\displaystyle B\,H}$, ${\displaystyle C\,I}$, ${\displaystyle C\,K}$, and ${\displaystyle D\,E}$ drawn through ${\displaystyle A,\,B,\,C}$ and ${\displaystyle D}$ perpendicular to ${\displaystyle A\,D}$ such quantities that ${\displaystyle G\,H}$, ${\displaystyle K\,I}$ and ${\displaystyle L\,M}$ or ${\displaystyle D\,L-A\,F}$, show by their length the magnitude of the tensions occurring at the individual places of excitation. From the known magnitude of these tensions, and from the given nature of the single parts ${\displaystyle A\,B}$, ${\displaystyle B\,C}$, and ${\displaystyle C\,D}$, the figure of the electrical separation has to be entirely determined.

If we draw straight lines parallel to ${\displaystyle A\,D}$, through the points ${\displaystyle F,\,H}$ and ${\displaystyle K}$, meeting the line drawn through ${\displaystyle B,\,C}$ and ${\displaystyle D}$ perpendicular to ${\displaystyle A\,D}$, in the points ${\displaystyle F'}$, ${\displaystyle H'}$, ${\displaystyle K'}$, then according to what has already been demonstrated, the lines ${\displaystyle G\,F'}$, ${\displaystyle I\,H'}$ and ${\displaystyle L\,K'}$ are directly proportional to the lengths of the parts ${\displaystyle A\,B}$, ${\displaystyle B\,C}$ and ${\displaystyle C\,D}$, and inversely proportional to the products of the conductibility and section of the same part, consequently the relations of the lines ${\displaystyle G\,F'}$, ${\displaystyle I\,H'}$ and ${\displaystyle L\,K'}$ to each other are given. Further, that ${\displaystyle G\,F'+I\,H'+L\,K'=G\,H-K\,I+(D\,L-A\,F=L\,M)}$ is also known, as the tensions represented by ${\displaystyle G\,H}$, ${\displaystyle K\,I}$ and ${\displaystyle D\,L-A\,F}$ are given. From the given relations of the lines ${\displaystyle G\,F',I\,H',L\,K'}$ and their known sum, these lines may now be found individually; the figure ${\displaystyle F\,G\,H\,I\,K\,L}$ is evidently then entirely determined. But the position of this figure with respect to the line ${\displaystyle A\,D}$ remains from its very nature still undecided.

If we recollect, that proceeding in the same direction ${\displaystyle A\,D}$, the tensions represented by ${\displaystyle G\,H}$ and ${\displaystyle D\,L-A\,F}$ or ${\displaystyle L\,M}$ indicate a sudden sinking of the electric force at the respective places of excitation, that represented by ${\displaystyle I\,K}$ on the contrary a sudden rise of the force; and that tensions of the first kind are regarded and treated as positive quantities, while tensions of the latter kind are considered as negative quantities, we find the above example lead us to the following generally valid rule: if we divide the sum of all the tensions of the ring composed of several parts into the same number of portions which are directly proportional to the lengths of the parts and inversely proportional to the products of their conductibilities and their sections, these portions will give in succession the amount of gradation which must be assigned to the straight lines belonging to the single parts and representing the separation of the electricity; at the same time the positive sum of all the tensions indicates a general rise, on the contrary the negative sum of all the tensions a general depression of those lines.

I will now proceed to the determination of the electric force at any given position in every galvanic circuit, and here again I shall lay down as basis fig. 3. For this purpose let ${\displaystyle a}$, ${\displaystyle a'}$, ${\displaystyle a''}$ indicate the tensions existing at ${\displaystyle B}$, ${\displaystyle C}$, and between ${\displaystyle A}$ and ${\displaystyle D}$, so that in this case also ${\displaystyle a}$ and ${\displaystyle a''}$ represent additive, ${\displaystyle a'}$ on the contrary a subtractive line, and ${\displaystyle \lambda }$, ${\displaystyle \lambda '}$, ${\displaystyle \lambda ''}$ any lines which are directly as the lengths of the parts ${\displaystyle A\,B}$, ${\displaystyle B\,C}$, and ${\displaystyle C\,D}$, and inversely as the products of the conductibilities and sections of the same parts; further, let
 ${\displaystyle a+a'+a''=A}$
and
 ${\displaystyle \lambda +\lambda '+\lambda ''=L}$

then according to the law just ascertained

 ${\displaystyle GF'}$ is a fourth proportional to ${\displaystyle L}$, ${\displaystyle A}$ and ${\displaystyle \lambda }$ ${\displaystyle IH'}$ a fourth proportional to ${\displaystyle L}$, ${\displaystyle A}$ and ${\displaystyle \lambda '}$ ${\displaystyle LK'}$ a fourth proportional to ${\displaystyle L}$, ${\displaystyle A}$ and ${\displaystyle \lambda ''}$.

Draw the line ${\displaystyle FM}$ through ${\displaystyle F}$ parallel to ${\displaystyle AD}$, regard this line as the axis of the abscissæ, and erect at any given points ${\displaystyle X}$, ${\displaystyle X'}$, ${\displaystyle X''}$ the ordinates ${\displaystyle XY}$, ${\displaystyle X'Y'}$, ${\displaystyle X''Y''}$, we obtain their respective values, thus:

In the first place we have, since ${\displaystyle AB=FF''}$
 ${\displaystyle AB:GF'=FX:XY,}$
whence follows:
 ${\displaystyle XY={\frac {FX\cdot GF'}{AB}}}$,
or if we substitute for ${\displaystyle GF'}$ its value ${\displaystyle {\frac {A\cdot \lambda }{L}}}$
 ${\displaystyle XY={\frac {A}{L}}\centerdot {\frac {FX\cdot \lambda }{AB}}}$.
If now ${\displaystyle x}$ represent a line such that
 ${\displaystyle AB:FX=\lambda :x}$,
then
 ${\displaystyle XY={\frac {A}{L}}\cdot x}$.
Secondly, since ${\displaystyle BC}$ and ${\displaystyle F'X'}$ are equal to the lines drawn through ${\displaystyle I}$ and ${\displaystyle Y'}$ to ${\displaystyle GH}$ parallel to ${\displaystyle AD}$
 ${\displaystyle BC:IH'=F'X':F'H-X'Y'}$,
whence
 ${\displaystyle -X'Y'={\frac {IH'\cdot F'X'}{BC}}-F'H}$
or, since ${\displaystyle F'H=GH-GF'}$
 ${\displaystyle -X'Y'={\frac {IH\cdot F'X'}{BC}}+GF'-a}$.
If now for ${\displaystyle IH'}$ and ${\displaystyle GF'}$ we substitute their values ${\displaystyle {\frac {A\cdot \lambda '}{L}}}$ and ${\displaystyle {\frac {A\cdot \lambda }{L}}}$, we obtain
 ${\displaystyle -X'Y'={\frac {A}{L}}\left(\lambda +{\frac {F'X'\cdot \lambda }{BC}}\right)-a}$;
and if by ${\displaystyle x'}$ we represent a line such that
 ${\displaystyle BC:F'X'=\lambda ':x'}$,
then
 ${\displaystyle -X'Y'={\frac {A}{L}}(\lambda +x')-a}$.
Thirdly, since ${\displaystyle CD=KK'}$ and ${\displaystyle F''X''}$ is equal to the part of ${\displaystyle KK'}$ which extends from ${\displaystyle K}$ to the line ${\displaystyle X''Y''}$, we have
 ${\displaystyle CD:LK'=F''X'':X''Y''-KF''}$,
whence
 ${\displaystyle X''Y''={\frac {LK'\cdot F''X''}{CD}}+KF''}$,
or, since ${\displaystyle KF''=KI+IH'-F'H}$ and ${\displaystyle F'H=GH-GF'}$,
 ${\displaystyle X''Y''={\frac {LK'\cdot F''X''}{CD}}+IH'+GF-(a+a')}$.
If now for ${\displaystyle LK'}$, ${\displaystyle IH'}$, ${\displaystyle GF'}$ we substitute their values
 ${\displaystyle {\frac {A\cdot \lambda ''}{L}},\quad {\frac {A\cdot \lambda '}{L}},\quad {\frac {A\cdot \lambda }{L}},\quad }$we obtain
 ${\displaystyle X''Y''={\frac {A}{L}}\left(\lambda +\lambda '+{\frac {F''X''\cdot \lambda ''}{CD}}\right)-(a+a')}$;
and if by ${\displaystyle x''}$ we represent a line such that
 ${\displaystyle CD:F''X''=\lambda '':x''}$
we have
 ${\displaystyle X''Y''={\frac {A}{L}}(\lambda +\lambda '+x'')-(a+a')}$.

These values of the ordinates, belonging to the three distinct parts of the circuit and different in form from each other, may be reduced as follows to a common expression. For if ${\displaystyle F}$ is taken as the origin of the abscissæ, ${\displaystyle FX}$ will be the abscissa corresponding to the ordinate ${\displaystyle XY}$ which belongs to the homogeneous part ${\displaystyle AB}$ of the ring, and ${\displaystyle x}$ will represent the length corresponding to this abscissa in the reduced proportion of ${\displaystyle AB:\lambda }$. In like manner ${\displaystyle FX'}$ is the abscissa corresponding to the ordinate ${\displaystyle X'Y'}$ which is composed of the parts ${\displaystyle FF'}$ and ${\displaystyle F'X'}$ belonging to the homogeneous portions of the ring, and ${\displaystyle \lambda }$, ${\displaystyle x'}$ are the lengths reduced in the proportions of ${\displaystyle AB:\lambda }$ and ${\displaystyle BC:\lambda '}$ corresponding to these parts. Lastly ${\displaystyle FX''}$ is the abscissa corresponding to the ordinate ${\displaystyle X''Y''}$, which is composed of the parts ${\displaystyle FF'}$, ${\displaystyle F'F''}$, ${\displaystyle F'X''}$ belonging to the homogeneous portions of the ring, and ${\displaystyle \lambda }$, ${\displaystyle \lambda '}$, ${\displaystyle x''}$ are the lengths reduced in the proportions of ${\displaystyle AB:\lambda }$, ${\displaystyle BC:\lambda '}$, ${\displaystyle CD:\lambda ''}$. If in consequence of this consideration we call the values ${\displaystyle x}$, ${\displaystyle \lambda +x'}$, ${\displaystyle \lambda +\lambda '+x''}$ reduced abscisscæ and represent them generally by ${\displaystyle y}$, we obtain
 {\displaystyle {\begin{aligned}&XY={\frac {A}{L}}\cdot y\\-{}&X'Y'={\frac {A}{L}}\cdot y-a\\&X''Y''={\frac {A}{L}}\cdot y-(a+a')\end{aligned}}},
and it is evident that ${\displaystyle L}$ is the same in reference to the whole length ${\displaystyle AD}$ or ${\displaystyle FM}$ as ${\displaystyle y}$ is to the lengths ${\displaystyle FX}$, ${\displaystyle FX'}$, ${\displaystyle FX''}$, on account of which ${\displaystyle L}$ is termed the entire reduced length of the circuit. Further, if we consider that for the abscissa corresponding to the ordinate ${\displaystyle XY}$ the tension has experienced no abrupt change, but that for the abscissa corresponding to the ordinate ${\displaystyle X'Y'}$ the tension has experienced the abrupt changes ${\displaystyle a}$, ${\displaystyle a'}$; and if we represent generally by ${\displaystyle O}$ the sum of all the abrupt changes of the tensions for the abscissa corresponding to the ordinate ${\displaystyle y}$, then all the values found for the various ordinates are contained in the following expression:
 ${\displaystyle {\frac {A}{L}}\cdot y-O}$.
But these ordinates express, when an arbitrary constant, corresponding to the length ${\displaystyle AF}$, is added to them, the electric forces existing at the various parts of the ring. If therefore we represent the electric force at any place generally by ${\displaystyle u}$ we obtain the following equation for its determination:
 ${\displaystyle u={\frac {A}{L}}y-O+c}$,

in which ${\displaystyle c}$ represents an arbitrary constant. This equation is generally true, and may be thus expressed in words: The force of the electricity at any place of a galvanic circuit composed of several parts, is ascertained by finding the fourth proportional to the reduced length of the entire circuit, the reduced length of the part belonging to the abscissa, and the sum of all the tensions, and by increasing or diminishing the difference between this quantity and the sum of all the abrupt changes of tension for the given abscissa by an arbitrary quantity which is constant for all parts of the circuit.

When the determination of the electric force at each place of the circuit has been effected, it only remains to determine the magnitude of the electric current. Now in a galvanic circuit of the kind hitherto mentioned, the quantity of electricity passing through a section of it in a given time is everywhere the same, because at all places and in each moment the same quantity in the section leaves it on the one side as enters it from the other, but in different circuits this quantity may be very different: therefore, in order to compare the actions of several galvanic circuits inter se, it is requisite to have an accurate determination of this quantity, by which the magnitude of the current in the circuit is measured. This determination may be deduced from figure 3 in the following manner. It has already been shown that the force of the electric transition in each instant from one element to the adjacent one is given by the electric difference between the two existing at that time, and by a magnitude dependent upon the kind and form of the particles of the body, viz. the conductibility of the body. But the electrical difference of the elements in the part ${\displaystyle BC}$, for instance, reduced to a constant unit of distance, will be expressed by the dip of the line ${\displaystyle HI}$ or by the quotient ${\displaystyle {\frac {IH'}{BC}}}$; if, therefore, we now indicate by ${\displaystyle \chi }$ the magnitude of the conductibility of the part ${\displaystyle BC}$,
 ${\displaystyle {\frac {\chi \cdot IH'}{BC}}}$
will express the force of the transition from element to element, or the intensity of the current in the part ${\displaystyle BC}$; consequently if ${\displaystyle \omega }$ represent the magnitude of the section in the part ${\displaystyle BC}$, the quantity of electricity passing in each instant from one section to the adjacent one, or the magnitude of the current, will be expressed by
 ${\displaystyle {\frac {\chi \cdot \omega \cdot IH'}{BC}}}$;
or if ${\displaystyle S}$ represent this magnitude of the current, we have
 ${\displaystyle S={\frac {\chi \cdot \omega \cdot IH'}{BC}}}$,
and if we substitute for ${\displaystyle IH'}$ its value ${\displaystyle {\frac {A\cdot \lambda '}{L}}}$
 ${\displaystyle S={\frac {A}{L}}\cdot {\frac {\chi \cdot \omega \cdot \lambda '}{BC}}}$.

Hitherto the letters ${\displaystyle \lambda }$, ${\displaystyle \lambda '}$, ${\displaystyle \lambda ''}$ have represented lines which are proportional to the quotients formed of the lengths of the parts ${\displaystyle AB}$, ${\displaystyle BC}$, ${\displaystyle CD}$, and the products of their conductibilities and their sections. If we restrict for the present this determination, which leaves the absolute magnitude of the lines ${\displaystyle \lambda ,\lambda ',\lambda ''}$ uncertain, so that the magnitudes ${\displaystyle \lambda ,\lambda ',\lambda ''}$ shall not be merely proportional to the said quotients, but shall be likewise equal to them, and henceforth vary this limitation in accordance with the meaning of the expression "reduced lengths," the first of the two preceding equations becomes
 ${\displaystyle S={\frac {IH'}{\lambda '}}}$,
which gives the following generally: The magnitude of the current in any homogeneous portion of the circuit is equal to the quotient of the difference between the electrical forces present at the extremities of this portion divided by its reduced length. This expression for the forces of the current will be continued to be employed subsequently. The second of the former equations passes, by the adopted change, into
 ${\displaystyle S={\frac {A}{L}}}$,

which is generally true, and already reveals the equality of the force of the current at all parts of the circuit; in words it may be thus expressed: The force of the current in a galvanic circuit is directly as the sum of all the tensions, and inversely as the entire reduced length of the circuit, bearing in mind that at present by reduced length is understood the sum of all the quotients obtained by dividing the actual lengths corresponding to the homogeneous parts by the product of the corresponding conductibilities and sections.

From the equation determining the force of the current in a galvanic circuit in conjunction with the one previously found, by which the electric force at each place of the circuit is given, may be deduced with ease and certainty all the phænomena belonging to the galvanic circuit. The former I had already some time ago derived from manifoldly varied experiments[2] with an apparatus which allows of an accuracy and certainty of measurement not suspected in this department; the latter expresses all the observations pertaining to it, which already exist in great number, with the greatest fidelity, which also continues where the equation leads to results no longer comprised in the circle of previously published experiments. Both proceed uninterruptedly hand in hand with nature, as I now hope to demonstrate by a short statement of their consequences; at the same time I consider it necessary to observe, that both equations refer to all possible galvanic circuits whose state is permanent, consequently they comprise the voltaic combination as a particular case, so that the theory of the pile needs no separate comment. In order to be distinct, I shall constantly, instead of employing the equation ${\displaystyle u={\frac {A}{L}}y-O+c}$, only take the third figure, and therefore will merely remark here, once for all, that all the consequences drawn from it hold generally.

In the next place, the circumstance that the separation of the electricity, diffusing itself over the galvanic circuit, maintains at the different places a permanent and unchangeable gradation, although the force of the electricity is variable at one and the same place, deserves a closer inspection. This is the reason of that magic mutability of the phænomena which admits of our predetermining at pleasure the action of a given place of the galvanic circuit on the electrometer, and enables us to produce it instantly. To explain this peculiarity I will return to figure 3. Since the figure of separation ${\displaystyle FGHIKL}$, is always wholly determined from the nature of any circuit; but its position with respect to the circuit ${\displaystyle AD}$, as was seen, is fixed by no inherent cause, but can assume any change produced by a movement common to all its points in the direction of the ordinates, the electrical condition of each point of the circuit expressed by the mutual position of the two lines, may be varied constantly, and at will, by external influences. When, for example, ${\displaystyle AD}$ is at any time the position representing the actual state of the circuit, so that, therefore, the ordinate ${\displaystyle SY''}$ expresses by its length the force of the electricity at the place of the circuit to which that ordinate belongs, then the electrical force corresponding to the point ${\displaystyle A}$, at the same time will be represented by the line ${\displaystyle AF}$. If now the point ${\displaystyle A}$ be touched abductively, and thus be entirely deprived of all its force, the line ${\displaystyle AD}$ will be brought into the position ${\displaystyle FM}$, and the force previously existing in the point ${\displaystyle S}$ will be expressed by the length ${\displaystyle X''Y''}$; this force, therefore, has suddenly undergone a change, corresponding to the length ${\displaystyle SX''}$. The same change would have occurred if the circuit had been touched abductively at the point ${\displaystyle Z}$, because the ordinate ${\displaystyle ZW}$ is equal to that of ${\displaystyle AF}$. If the circuit were touched at the place where the two parts ${\displaystyle AB}$ and ${\displaystyle BC}$ join, but so that the contact was made within the part ${\displaystyle BC}$, we should have to imagine ${\displaystyle AD}$ advanced to ${\displaystyle NO}$; the electrical force at the point ${\displaystyle S}$ would in this case be increased to the force indicated by ${\displaystyle TY''}$. But if the contact took place, still at the same point, viz. where the parts ${\displaystyle AB}$ and ${\displaystyle BC}$ touch each other, but within the part ${\displaystyle AB}$, the line ${\displaystyle AD}$ would be moved to ${\displaystyle PQ}$, and the force belonging to the point ${\displaystyle S}$ would sink to the negative force expressed by ${\displaystyle UY''}$. If, lastly, the pile had been touched abductively at the point ${\displaystyle D}$, we should have prescribed for the line ${\displaystyle AD}$ the position ${\displaystyle RL}$, and the electrical force at the point ${\displaystyle S}$ would have assumed the negative force indicated by ${\displaystyle VY''}$. The law of these changes is obvious, and may be expressed generally thus: each place of a galvanic circuit undergoes mediately, in regard to its outwardly acting electrical force, the same change which is produced immediately at any other place of the circuit by external influences.

Since each place of a galvanic circuit undergoes, of itself, the same change to which a single place was compelled, the change in the quantity of electricity, extending over the whole circuit, is proportional, on the one hand, to the sum of all the places, i. e. to the space over which the electricity is diffused in the circuit, and moreover, to the change in the electric force produced at one of these places. From this simple law result the following distinct phænomena. If we call ${\displaystyle r}$ the space over which the electricity is diffused in the galvanic circuit, and imagine this circuit touched at any one place by a non-conducting body, and designate by ${\displaystyle u}$ the electric force at this place before contact, by ${\displaystyle u}$ that after contact, the change produced in the force at this place is ${\displaystyle u_{1}-u}$; consequently the change of the whole quantity of electricity in the circuit is ${\displaystyle (u_{1}-u)r}$. If, now, we suppose that the electricity in the touched body is diffused over the space ${\displaystyle R}$, and is at all places of equal strength, and, at the same time, that at the place of contact itself the circuit and the body possess the same electric force, viz. ${\displaystyle u}$, it is evident ${\displaystyle uR}$ will be the quantity of electricity imparted the body, and
 ${\displaystyle (u_{1}-u)r=uR}$,
whence we obtain
 ${\displaystyle u={\frac {u_{1}r}{r+R}}}$

The intensity of the electricity received by the body will, therefore, be the more nearly equal to that which the circuit possessed at the place of contact before being touched, the smaller ${\displaystyle R}$ is with respect to ${\displaystyle r}$; it will amount to the half when ${\displaystyle R=r}$, and become weaker, as ${\displaystyle R}$ becomes greater in comparison with ${\displaystyle r}$. Since these changes are merely dependent on the relative magnitude of the spaces ${\displaystyle r}$ and ${\displaystyle R}$, and not at all on the qualitative nature of the circuit, they are merely determined by the dimensions of the circuit, nay, even by foreign masses brought into conducting connexion with the circuit. If we connect this fact with the theory of the condensor, we arrive at an explanation of all the relations of the galvanic circuit to the condensor, noticed by Jäger[3], which is perfectly surprising. I content myself with regard to this point to refer to the memoir itself, to give room here for the insertion of some new peculiarities of the galvanic circuit.

The mode of separation of the electricity, within a homogeneous part of the circuit, is determined by the magnitudes of the dips of the lines ${\displaystyle FG}$, ${\displaystyle HI}$, ${\displaystyle KL}$, (fig. 3,) and there again by the magnitudes of the ratios ${\displaystyle {\frac {GF'}{AB}}}$, ${\displaystyle {\frac {IH'}{BC}}}$, ${\displaystyle {\frac {LK'}{CD}}}$. But, as was already shown,
 ${\displaystyle GF'={\frac {A}{L}}\cdot \lambda ,\qquad IH'={\frac {A}{L}}\cdot \lambda ',\qquad LK'={\frac {A}{L}}\cdot \lambda ''}$;
hence it may be seen, without much trouble, that the magnitude of the dip of the line corresponding to any part of the circuit, and representing the separation of the electricity, is obtained by multiplying the value ${\displaystyle {\frac {A}{L}}}$ by the ratio of the reduced to the actual length of the same part. If, therefore, ${\displaystyle (\lambda )}$ represent the reduced length of any homogeneous part of the circuit and ${\displaystyle (l)}$ its actual length, the magnitude of the dip of the straight line belonging to this part, and representing the separation of the electricity, is
 ${\displaystyle {\frac {A}{L}}\centerdot {\frac {(\lambda )}{(l)}}}$,
which expression, if we designate by ${\displaystyle (\chi )}$ the conductibility, and by ${\displaystyle (\omega )}$ the section of the same part, may also be written thus:
 ${\displaystyle {\frac {A}{L}}\centerdot {\frac {(\lambda )}{(\chi )(\omega )}}}$.

This expression leads to a more detailed knowledge of the separation of the electricity in a galvanic circuit. For since ${\displaystyle A}$ and ${\displaystyle L}$ designate values which remain identical for each part of the same circuit, it is evident that the dips in the separate homogeneous parts of a circuit are to one another inversely as the products of the conductibility, and the section of the part. If consequently a part of the circuit surpasses all others from the circumstance, that the product of its conductibility and its section is far smaller than in the others, it will be the most adapted to reveal, by the magnitude of its dip, the differences of the electric force at its various points. If its actual length is, at the same time, not inferior to those of the other parts, its reduced length will far surpass those of the other parts; and it is easily conceived that such a relation between the various parts can be brought about, that even its reduced length may remain far greater than the sum of the reduced lengths of all the other parts. But in this case the reduced length of this one part is nearly equal to the reduced lengths of the entire circuit, so that we may substitute, without committing any great error, ${\displaystyle {\frac {(l)}{(\chi )(\omega )}}}$ for ${\displaystyle L}$, if ${\displaystyle (l)}$ represent the actual length of the said part, ${\displaystyle (\chi )}$ its conductibility, and ${\displaystyle (\omega )}$ its section; but then the dip of this part changes nearly into
 ${\displaystyle {\frac {A}{(l)}}}$,

whence it follows that the difference of the electrical forces at the extremities of this part is nearly equal to the sum of all the tensions existing in the circuit. All the tensions seem, as it were, to tend towards this one part, causing the electrical separation to appear in it with otherwise unusual energy, when all the tensions, or, at least, the greater part in number and magnitude, are of the same kind. In this way the scarcely perceptible gradation in the separation of the electricity, in a closed circuit, may be rendered distinctly evident, which, otherwise, would not be the case without a condenser, on account of the weak intensity of galvanic forces. This remarkable property of galvanic circuits, representing, as it were, their entire nature, had already been noticed long ago in various bad conducting bodies, and its origin sought for in their peculiar constitution[4]; I have, however, enumerated in a letter to the editor of the Annalen der Physik[5], the conditions under which this property of the galvanic circuit may be observed, even in the best conductors, the metals; and the necessary precautions, founded on experience, by which the success of the experiment is assured, described in it, are in perfect accordance with the present considerations.

The expression ${\displaystyle {\frac {A}{L}}\centerdot {\frac {(\lambda )}{(l)}}}$ denoting the dip of any portion of the circuit, vanishes when ${\displaystyle L}$ is indefinitely great, while ${\displaystyle A}$ and ${\displaystyle {\frac {(\lambda )}{(l)}}}$ retain finite values. Consequently, if ${\displaystyle L}$ assumes an indefinitely great value, while ${\displaystyle A}$ remains finite, the dip of the straight lines representing the separation of the electricity, in all such parts of the circuit, whose reduced length has a finite ratio to the actual length, vanishes, or what comes to the same thing, the electricity is of equal force at all places of each such part. Now, since ${\displaystyle L}$ represents the sum of the reduced lengths of all the parts of the circuit, and these reduced lengths evidently can only assume positive values, ${\displaystyle L}$ becomes indefinite as soon as one of the reduced lengths assumes an infinite value. Further, since the reduced length of any part represents the quotient obtained by dividing the actual length by the product of the conductibility and the section of the same part, it becomes infinite when the conductibility of this part vanishes, i. e. when this part is a non-conductor of electricity. When, therefore, a part of the circuit is a non-conductor, the electricity expands uniformly over each of the other parts, and its change from one part to the other is equal to the whole tension there situated. This separation of the electricity, relative to the open circuit, is far more simple than that in the closed circuit, which has hitherto formed the object of our consideration, as is geometrically represented by the lines ${\displaystyle FG}$, ${\displaystyle HI}$, ${\displaystyle KL}$, (fig. 3) taking a position parallel to ${\displaystyle AD}$. It distinctly demonstrates that the difference between the electrical forces, occurring at any two places of the circuit, is equal to the sum of all the tensions situated between these two places, and consequently increases or decreases exactly in the same proportion as this sum. When, therefore, one of these places is touched abductively, the sum of all the tensions, situated between the two, makes its appearance at the other place, at the same time the direction of the tensions must always be determined by an advance from the latter place. All the experiments on the open pile, with the help of the electroscope, instituted at such length by Ritter, Erman, and Jäger, and described in Gilbert's Annalen[6], are expressed in this last law.

All the electroscopic actions of a galvanic circuit of the kind, described at the outset, have been above stated; I therefore pass at present to the consideration of the current originating in the circuit, the nature of which, as explained above, is expressed at every place of the circuit by the equation
 ${\displaystyle S={\frac {A}{L}}}$.

Both the form of this equation, as well as the mode by which we arrive at it, show directly that the magnitude of the current in such a galvanic circuit remains the same at all places of the circuit, and is solely dependent on the mode of separation of the electricity, so that it does not vary, even though the electric force at any place of the circuit be changed by abductive contact, or in any other way. This equality of the current at all places of the circuit has been proved by the experiments of Becquerel[7], and its independency of the electric force at any determinate place of the circuit by those of G. Bischof[8]. An abduction or adduction does not alter the current of the galvanic circuit so long as they only act immediately on a single place of the circuit; but if two different places were acted upon contemporaneously, a second current would be formed, which would necessarily, according to circumstances, more or less change the first.

The equation
 ${\displaystyle S={\frac {A}{L}}}$
shows that the current of a galvanic circuit is subjected to a change, by each variation originating either in the magnitude of a tension or in the reduced length of a part, which latter is itself again determined, both by the actual length of the part, as well as by its conductibility and by its section. This variety of change may be limited, by supposing only one of the enumerated elements to be variable, and all the remainder constant. We thus arrive at distinct forms of the general equation corresponding to each particular instance of the general capability of change of a circuit. To render the meaning of this phrase evident by an example, I will suppose that in the circuit only the actual length of a single part is subjected to a continual change; but that all the other values denoting the magnitude of the current remain constantly the same, and, consequently, also in its equation. If we designate by ${\displaystyle x}$ this variable length, and the conductibility corresponding to the same part by ${\displaystyle \chi }$, its section by ${\displaystyle \omega }$, and the sum of the reduced lengths of all the others by ${\displaystyle \Lambda }$, so that ${\displaystyle L=\Lambda +{\frac {x}{\chi \cdot \omega }}}$ , then the general expression for the current changes into the following:
 ${\displaystyle S={\frac {A}{\Lambda +{\frac {x}{\chi \cdot \omega }}}}}$;
or if we multiply both the numerator and denominator by ${\displaystyle \chi \omega }$, and substitute ${\displaystyle a}$ for ${\displaystyle \chi \omega A}$, and ${\displaystyle b}$ for ${\displaystyle \chi \omega \Lambda }$ into the following:
 ${\displaystyle S={\frac {a}{b+x}}}$,

where ${\displaystyle a}$ and ${\displaystyle b}$ represent two constant magnitudes, and ${\displaystyle x}$ the variable length of a portion of the circuit fully determined with respect to its substance and its section. This form of the general equation, in which all the invariable elements have been reduced to the smallest number of constants, is that which I had practically deduced from experiments to which the theory here developed owes its origin[9]. The law which it expresses relative to the length of conductors, differs essentially from that which Davy formerly, and Becquerel more recently, were led to by experiments; it also differs very considerably from that advanced by Barlow, as well as from that which I had previously drawn from other experiments, although the two latter come much nearer to the truth. The first, in fact, is nothing more than a formula of interpolation, which is valid only for a relatively very short variable part of the entire circuit, and, nevertheless, is still applicable in very different possible modes of conduction, which is already evident, from its merely admitting the variable portion of the circuit, and leaving out of consideration all the other part; but all partake in common of this evil, that they have admitted a foreign source of variability, produced by the chemical change of the fluid portion of the circuit, of which I shall speak more fully hereafter. I have already treated, in other places, more at length of the relations of the various forms of the law to one another.

From the numerous separate peculiarities of the galvanic circuit resulting from the general equation
 ${\displaystyle S={\frac {A}{L}}}$,
I will here merely mention a few. It is immediately evident that a change in the arrangement of the parts has no influence on the magnitude of the current if the sum of the tensions be not affected by it. Nor is the magnitude of the current altered, when the sum of the tensions, and the entire reduced length of the circuit, change in the same proportion; consequently a circuit, the sum of whose tensions is very small in comparison to that of another circuit, may still produce a current, which, in energy, may be equal to that in the other circuit, when merely that which it loses in force of tensions is replaced by a shortening of its reduced length. In this circumstance is the source of the peculiar difference between thermo- and hydro-circuits. In the former only metals occur as parts of the circuit; in the latter, besides the metals, aqueous fluids, whose power of conduction, in comparison to that of the metals, is exceedingly small; on which account the reduced lengths of the fluid surpass, beyond all proportion, those of the metallic parts, with in all respects equal dimensions, and even remain considerably greater when diminished by shortening their actual lengths, and increasing their sections, so long, at least, as this diminution is not carried too far. And thence it is that the reduced length of the thermo-circuit is, in general, far smaller than that of the hydro-circuit, whence we may infer a tension smaller in the same proportion in the former, although the magnitude of the current, in the thermo-circuit, cedes in nothing to that in the hydro-circuit. The great difference between a thermo- and hydro-circuit, both of which produce a current of the same energy, is evident when the same change is made on both, as will be shown in the following consideration. Let the reduced length of a thermo-circuit be ${\displaystyle L}$, and the sum of its tensions ${\displaystyle A}$, the reduced length of an hydro-circuit ${\displaystyle mL}$, and the sum of its tensions ${\displaystyle mA}$, then the magnitude of the current in the former is expressed by ${\displaystyle {\frac {A}{L}}}$, in the latter by ${\displaystyle {\frac {mA}{mL}}}$, and is consequently the same in both circuits. But this equality of the current no longer exists if the same new part ${\displaystyle \lambda }$ of the reduced length be introduced into both, for then the magnitude of the current is in the first
 ${\displaystyle {\frac {a}{L+\lambda }}}$,
in the second
 ${\displaystyle {\frac {mA}{mL+\lambda }}.}$

If we connect with this determination an evaluation, even if merely superficial, of the quantities ${\displaystyle m}$, ${\displaystyle L}$, and ${\displaystyle \lambda }$, we shall readily be convinced that in cases where the simple hydro-circuit can still produce in the part ${\displaystyle \lambda }$ actions of heat or chemical decomposition, the simple thermo-circuit may not possess the hundredth, and in some cases not the thousandth part of the requisite force, whence the absence of similar effects in it is easily to be understood. We are also able to understand why a diminution of the reduced lengths of the thermo-circuit (by increasing, for instance, the section of the metals constituting it) cannot give rise to the production of those effects, although the magnitude of its current may be increased by this means to a higher degree than in the hydro-circuit producing such effects. This difference in the conductibility of metallic bodies and aqueous fluids, is the cause of a peculiarity noticed with respect to hydro-circuits, which it is here, perhaps, the proper place to mention. Under the usual circumstances, the reduced length of the fluid portion is so large, in comparison to that of the metallic portion, that the latter may be overlooked, and the former alone taken instead of the reduced length of the entire circuit; but then the magnitude of the current in circuits which have the same tension is in the inverse ratio to the reduced length of the fluid portion. Consequently, if merely such circuits are compared in which the fluid parts have the same actual lengths and the same conductibilities, then the magnitude of the current in these circuits is in direct ratio to the section of the fluid portion. However, it must not be overlooked, that a more complex definition must take the place of this simple one when the reduced length of the metallic portion can no longer be regarded as evanescent towards that of the fluid, which case occurs whenever the metallic portion is very long and thin, or the fluid portion is a good conductor, and with unusually large terminal surfaces.

From the equation
 ${\displaystyle S={\frac {A}{L}}}$

we can easily perceive that, when a portion is taken from the galvanic circuit, and is replaced by another, and after this change the sum of the tensions as well as the energy of the current still remains perfectly the same, these two parts have the same reduced length, consequently their actual lengths are as the products of their conductibilities and sections. The actual lengths of such parts are therefore, when they have like sections, as their conductibilities, and when they have like conductibilities as their sections. By the first of these two relations we are enabled to determine the conductibilities of various bodies in a far more advantageous manner than by the previously mentioned process, and it has already been employed by Becquerel and myself for several metals[10]. The second relation may serve to demonstrate experimentally the independence of the effect on the form of the section, as has previously been done by Davy, and recently by myself[11].

In the voltaic pile, the sum of the tensions, and the reduced length of the simple circuit, is repeated as frequently as the number of elements of which it consists expresses. If, therefore, we designate by ${\displaystyle A}$ the sum of all the tensions in the simple circuit, by ${\displaystyle L}$ its reduced length, and by ${\displaystyle n}$ the number of elements in the pile, the magnitude of the current in the closed pile is evidently
 ${\displaystyle {\frac {n\mathrm {A} }{n\mathrm {L} }}}$,
while in the simple closed circuit it is
 ${\displaystyle {\frac {\mathrm {A} }{\mathrm {L} }}}$.
If we now introduce into the simple circuit, as well as into the pile, one and the same new part ${\displaystyle \Lambda }$ of the reduced length, upon which the current is to act, the magnitude of the current thus altered in the simple circuit will be
 ${\displaystyle {\frac {\mathrm {A} }{\mathrm {L} +\Lambda }}}$,
and in the voltaic pile
 ${\displaystyle {\frac {n\mathrm {A} }{n\mathrm {L} +\Lambda }},\quad {\text{or}}\;{\frac {\mathrm {A} }{\mathrm {L} +{\frac {\Lambda }{n}}}}}$.
It is hence evident that the current is constantly greater in a voltaic pile than in the simple circuit, but it is merely imperceptibly greater so long as ${\displaystyle \Lambda }$ is very small in comparison with ${\displaystyle \mathrm {L} }$; on the contrary, this increase approximates the nearer to ${\displaystyle n}$ times, the greater ${\displaystyle \Lambda }$ becomes to ${\displaystyle n\mathrm {L} }$, and consequently the more so in comparison with ${\displaystyle \mathrm {L} }$. Besides this mode of increasing the magnitude of the galvanic current, there is a second one, which consists in shortening the reduced lengths of the simple circuit, which may be effected by increasing its section, or placing several simple circuits by the side of each other, and connecting them in such a way that together they only form one single simple circuit. If we now retain the same signs, so that
 ${\displaystyle {\frac {\mathrm {A} }{\mathrm {L} +\Lambda }}}$
again denotes the magnitude of the current in one element, then, in the above-mentioned combination of ${\displaystyle n}$ elements into a single circuit, the magnitude of the current is evidently
 ${\displaystyle {\frac {\mathrm {A} }{{\frac {\mathrm {L} }{n}}+\Lambda }},\quad {\text{or}}\;{\frac {n\mathrm {A} }{\mathrm {L} +n\Lambda }}}$,

which indicates a slight increase in the action of the new combination when ${\displaystyle \Lambda }$ is very great in comparison with ${\displaystyle \mathrm {L} }$; on the contrary, a very powerful one when ${\displaystyle \Lambda }$ is very small in comparison with ${\displaystyle {\frac {\mathrm {L} }{n}}}$, and consequently the more so in comparison with ${\displaystyle \mathrm {L} }$. It hence follows that the one combination is most active in those cases where the other ceases to be so, and vice versâ. If therefore we are in possession of a certain number of simple circuits intended to act upon the portion whose reduced length is ${\displaystyle \Lambda }$, much depends on the way in which they are placed, in order to produce the greatest effect of current; whether all be side by side, or all in succession, or whether part be placed by the side of each other, and part in series. It may be mathematically shown that it is most advantageous to form them into a voltaic combination, of so many equal parts, that the square of this number be equal to the quotient ${\displaystyle {\frac {\Lambda }{L}}}$. When ${\displaystyle {\frac {\Lambda }{L}}}$ is equal to, or smaller than ${\displaystyle \Lambda }$, they had best be arranged by the side of each other, and in succession when ${\displaystyle {\frac {\Lambda }{L}}}$ is equal to, or larger than the square of the number of all the elements. We see in this determination the reason why in most cases a simple circuit, or at least a voltaic combination of only a few simple circuits, is sufficient to produce the greatest effect. If we bear in mind, that since the quantity of the current is the same at all places of the circuit, its intensity at the various places must be in inverse proportion to the magnitude of the section there situated, and if we grant that the magnetic and chemical effects, as well as the phænomena of light and heat in the circuit, are direct indications of the electrical current, and that their energy is determined by that of the current itself, then a detailed analysis of the current, here indicated merely in outline, will lead to the perfect explanation of the numerous and partially enigmatical anomalies observed in the galvanic circuit, in as far as we are justified in considering the physical nature of the circuit as invariable[12]. Those great differences which are frequently met with in the statements of various observers, and which are not consequences of the dimensions of their different apparatus, have undoubtedly their origin in the double capability of change of the hydro-circuits, and will therefore cease when this circumstance is taken into consideration on a repetition of the experiments.

The remarkable variability in the circle of action of one and the same multiplier in various circuits, and of different multipliers in the same circuit, is completely explained by the preceding consideration. For if we denote by ${\displaystyle A}$ the sum of the tensions, and by ${\displaystyle L}$ the reduced length of any galvanic circuit,
 ${\displaystyle {\frac {A}{L}}}$
expresses the magnitude of its current. If we now imagine a multiplier of ${\displaystyle n}$ similar convolutions each of the reduced length ${\displaystyle \lambda }$,
 ${\displaystyle {\frac {A}{L+n\lambda }}}$
indicates the magnitude of the current when the multiplier is brought into the circuit as an integral part. Moreover, if we grant, for the sake of simplicity, that each of the ${\displaystyle n}$ convolutions exerts the same action on the magnetic needle, the action of the multiplier on the magnetic needle is evidently
 ${\displaystyle {\frac {nA}{L+n\lambda }}}$,
when the action of an exactly similar coil of the circuit, without the multiplier on the needle, is taken as
 ${\displaystyle {\frac {A}{L}}}$.
Hence it follows directly that the action on the magnetic needle is augmented or weakened by the multiplier, according as ${\displaystyle nL}$ is greater or smaller than ${\displaystyle L+n\lambda }$, i. e., according as ${\displaystyle n}$ times the reduced length of the circuit without the multiplier is greater or smaller than the reduced length of the circuit with the multiplier. Further, a mere glance at the expression by which the action of the multiplier on the needle has been determined, will show that the greatest or smallest action occurs as soon as ${\displaystyle L}$ may be neglected with reference to ${\displaystyle n\lambda }$, and is expressed by
 ${\displaystyle {\frac {A}{\lambda }}}$.
If we compare this extreme action of the multiplier with that which a perfectly similarly constructed convolution of the circuit without the multiplier produces, we perceive that both are in the same ratio to one another as the reduced lengths ${\displaystyle L}$ and ${\displaystyle \lambda }$, which relation may serve to determine one of the values when the others are known. The expression found for the extreme action of the multiplier shows that it is proportional to the tension of the circuit, and independent of its reduced length; consequently the extreme action of the same multiplier may serve not merely to determine the tensions in various circuits, but it also indicates that the extreme action may be also augmented to the same degree as the sum of the tensions is increased, which may be effected by forming a voltaic combination with several simple circuits. If we represent the actual length of a coil of the multiplier by ${\displaystyle l}$, its couductibility by ${\displaystyle \chi }$, and its section by ${\displaystyle \omega }$, so that ${\displaystyle \lambda ={\frac {l}{\chi \cdot \omega }}}$ , the expression for the extreme action of the multiplier is converted into
 ${\displaystyle \chi \cdot \omega \cdot {\frac {A}{l}}}$,

from which it will be seen that the extreme action of two multipliers of different metals, constructed of wire of the same thickness, are in the same ratio to each other as the conductibilities of these metals, and that the extreme actions of two multipliers formed of wire of the same metal, are in the same proportion to each other as the sections of the wires. All these various peculiarities of the multiplier I have shown to be founded on experience, partly on experiments performed by other persons, and partly on those by myself[13]. The most recent experiments made on this subject on thermo-circuits, have, in a different, and, in a certain sense, opposite manner, already afforded the conclusion deduced above from an equation of the reduced lengths, that the sum of the tensions in a thermo-circuit is far weaker than in the ordinary hydro-circuits; and a preliminary comparison has convinced me, that, with respect to the heating effects, if they are to be predicted with certainty, a voltaic combination of some hundred well-chosen simple thermo-circuits is requisite, and for chemical effects of some energy a far greater apparatus. Experiments, which place this prediction beyond doubt, will afford a new and not unimportant confirmation to the theory here propounded.

The previous considerations are also sufficient to indicate the process which is carried on when the galvanic circuit is divided at any place into two or more branches. For this purpose I call attention to the circumstance, that at the time we found the equation ${\displaystyle S={\frac {A}{L}}}$, we also obtained the rule that the magnitude of the current in any homogeneous part of the galvanic circuit is given by the quotient of the difference between the electrical forces existing at the extremities of the portion and its reduced length. It is true, this rule was only advanced above for the case in which the circuit nowhere divides into several branches; but a very simple consideration, analogous to the one then made, derived from the equality of the abducted and adducted quantity of electricity in all sections of each prismatic part, is sufficient to prove that the same rule holds good for every single branch in case of a division of the circuit. Let us suppose that the circuit be divided, for instance, into three branches, whose reduced lengths are ${\displaystyle \lambda }$, ${\displaystyle \lambda '}$, ${\displaystyle \lambda ''}$; and, moreover, that at each of these places the undivided circuit and the single branches possess equal electrical force, and consequently no tension occurs there, and designate by ${\displaystyle \alpha }$. the difference between the electrical forces at these two places; then, according to the above rule, the magnitude of the current in each of the three branches is
 ${\displaystyle {\frac {\alpha }{\lambda }},\quad {\frac {\alpha }{\lambda '}},\quad {\frac {\alpha }{\lambda ''}};}$
whence it directly follows that the currents in the three branches are inversely as their reduced lengths; so that each separate one may be found when the sum of all three together is known. But the sum of all three is evidently equal to the magnitude of the current at any other place of the non-divided portion of the circuit, for otherwise the permanent state of the circuit, which is still constantly supposed, would not be maintained. If we connect with this the conclusion resulting from the above considerations, namely, that the magnitude of the current, and the nature of each homogeneous part of the circuit, give the dip of the corresponding straight line, representing the separation of the electricity, we are certain that the figure of the separation belonging to the non-divided portion of the circuit must remain the same so long as the current in it retains the same magnitude, and vice versâ; whence it follows that the variability of the current in the non-divided portion necessarily supposes that the difference between the electrical forces at the extremities of this portion is constant. If we now imagine, instead of the separate branches, a single conductor of the reduced length ${\displaystyle A}$ brought into the circuit which does not at all alter the magnitude of its current and its tensions, then, according to what has just been stated, the difference between the electrical forces at its extremities must still always remain ${\displaystyle \alpha }$, and consequently be
 ${\displaystyle {\frac {\alpha }{\Lambda }}={\frac {\alpha }{\lambda }}+{\frac {\alpha }{\lambda '}}+{\frac {\alpha }{\lambda ''}}}$,
or
 ${\displaystyle {\frac {1}{\Lambda }}={\frac {1}{\lambda }}+{\frac {1}{\lambda '}}+{\frac {1}{\lambda ''}}}$,
which equation serves to determine the value of ${\displaystyle \Lambda }$. But if this value is known, and we call ${\displaystyle A}$ the sum of all the tensions in the circuit, and ${\displaystyle L}$ the reduced length of the non-divided portion of the circuit, we obtain, as is known, for the magnitude of the current in the last-mentioned circuit
 ${\displaystyle {\frac {A}{L+\Lambda }}}$,
which is equal to the sum of the currents occurring in the separate branches. Now since it has already been proved that the currents in the separate branches are in inverse proportion to one another as the reduced lengths of these branches, we obtain for the magnitude of the current in the branch whose reduced length is ${\displaystyle \lambda }$,
 ${\displaystyle {\frac {A}{L+\Lambda }}\centerdot {\frac {\Lambda }{\lambda }}}$;
in the branch whose reduced length is ${\displaystyle \lambda '}$,
 ${\displaystyle {\frac {A}{L+\Lambda }}\centerdot {\frac {\Lambda }{\lambda '}}}$;
and in the branch whose reduced length is ${\displaystyle \lambda ''}$,
 ${\displaystyle {\frac {A}{L+\Lambda }}\centerdot {\frac {\Lambda }{\lambda ''}}}$.

This remote, and hitherto but slightly noticed peculiarity of the galvanic circuit, I have also found to be perfectly confirmed by experiment[14].

I herewith conclude the consideration of such galvanic circuits which have already attained the permanent state, and which neither suffer modifications by the influence of the surrounding atmosphere, nor by a gradual change in their chemical composition. But from this point the simplicity of the subject decreases more and more, so that the previous elementary treatment soon entirely disappears. With respect to those circuits on which the atmosphere exercises some influence, and whose condition varies with time, without this change originating in a progressive chemical transformation of the circuit, and is thus distinguished from all the others by the magnitude of its current being different at different places,—I have been content, with respect to each of these, always to treat of only the most simple case, as they but rarely occur in nature, and in general appear to be of less interest. I have adopted this plan the more willingly, as I intend to return to this subject at some future time. But with regard to that modification of galvanic circuits which is produced by a chemical change in the circuit, proceeding first from the current, and then again reacting on it, I have devoted separate attention to it in the Appendix. The course adopted is founded on a vast number of experiments performed on this subject, the communication of which, however, I omit, because they appear to be capable of being far more accurately determined than I was able to do at that time, from failing to attend to several elements in operation; nevertheless, I consider it proper to mention the circumstance in this place, in order that the careful manner with which I advance in the inquiry, and which I consider to be due to truth, may not operate more than is just against its reception.

I have sought for the source of the chemical changes caused by the current, in the above-described peculiar mode of separation of the electricity of the circuit; and, I can scarcely doubt, have at least found the main cause. It is immediately evident that each disk belonging to a section of a galvanic circuit, which obeys the electric attractions and repulsions and does not oppose their movement, must in the closed circuit be propelled always towards one side only, as these attractions and repulsions, in consequence of the continually varying electric force are different at the two sides; and it is mathematically demonstrable that the force with which it is driven to the one side, is in the ratio compounded of the magnitude of the electric current and of the electric force in the disk. It is true, however, that merely a change of position in space would be immediately produced by this. But if this disk be regarded as a compound body, the constituent parts of which, according to electro-chemical views, are distinguished by a difference in their electrical relation to one another, it thence directly follows that this one-sided pressure on the various constituent parts would in most cases act with unequal force, and sometimes even in contrary direction, and must thus excite a tendency in them to separate from one another. From this consideration results a distinct activity of the circuit, tending to produce a chemical change in its parts, which I have termed its decomposing force, and I have endeavoured to determine its magnitude for each particular case. This determination is independent of the mode, in which the electricity may be conceived to be asssociated with the atoms.[15] Granting, which seems to be most natural, that the electricity is diffused proportionately to the mass over the space which these bodies then occupy, a complete analysis will show that the decomposing force of the circuit is in direct proportion to the energy of the current, and, moreover, that it depends on a coefficient, to be derived from the nature of the constituent parts and their chemical equivalents. From the nature of this decomposing force of the circuit, which is of equal energy at all places of an homogeneous portion, it directly follows, that when it is capable of overcoming, under all circumstances, the reciprocal connexion of the constituent parts, the separation and abduction of the constituents to both sides of the circuit are limited solely by mechanical obstacles; but if the connexion of the constituent parts inter se, either immediately at the commencement everywhere, or in the course of the action anywhere, overcome the decomposing force of the circuit, then from that time no further movement of the elements can take place. This general description of the decomposing force is in accordance with the experiments of Davy and others.

There is a peculiar state which seems to be produced in most cases of the separation of the two elements of a chemically compound liquid, which is especially worthy of attention, and which is caused in the following manner. When the decomposition is confined solely to a limited portion of the circuit, and the constituent parts of the one kind are propelled towards the one side of this part, and the constituent parts of the other kind to its opposite side, then, for this very reason, a natural limit is prescribed to the action; for the constituent part preponderating on the one side of any disk, within the portion in the act of decomposition, will, by force of its innate repulsive power, constantly oppose the movement of a similar constituent to the same side, so that the decomposing force of the circuit has not merely to overcome the constant connexion of the two constituents inter se, but also this reaction of each constituent on itself. It is hence evident that a cessation in the chemical change must occur, if at any time there arises an equilibrium between the two forces. This state, founded on a peculiar chemical and permanent separation of the constituents of the portion of the circuit in the act of decomposition, is the very one from which I started, and whose nature I have endeavoured to determine as accurately as possible in the Appendix. Even the mere description of the mode of origin of this highly remarkable phænomenon shows that at the extremities of the divided portion no natural equilibrium can occur, on which account the two constituents must be retained at these two places by a mechanical force, unless they pass over to the next parts of the circuit, or, where the other circumstances allow, separate entirely from the circuit. Who would not recognise in this plain statement all the chief circumstances hitherto observed of the external phænomenon in chemical decompositions by the circuit?

If the current, and, at the same time, the decomposing force, be suddenly interrupted, the separated constituents gradually return to their natural equilibrium; but tend to re-assume immediately the relinquished state, if the current is re-established. During this process, both the conductibility, and the mode of excitation between the elements of the portion in the act of decomposition, obviously vary with their chemical nature; but this necessarily produces a constant change in the electrical separation, and in the magnitudes of the current in the galvanic circuit dependent thereon, which only finds its natural limits in the permanent state of the electrical separation. For the accurate determination of this last stage of the electric current it is requisite to be acquainted with the law which governs the conductibility and force of excitation of the variable mixtures, formed of two different liquids. Experiment has hitherto afforded insufficient data for this purpose, I have therefore given the preference to a theoretical supposition, which will supply its place until the true law is discovered. With the help of this law, which is not altogether imaginary, I now arrive at the equations which make known for each case all the individual circumstances constituting the permanent state of the chemical separation in the galvanic circuit; I have, however, neglected the further use of them, as the present state of our experimental knowledge in this respect did not appear to me to repay the requisite trouble. Nevertheless, in order to compare in their general features the results of this examination with what has hitherto been supplied by experiments, I have fully carried out one particular case, and have found that the formula represents very satisfactorily the kind of wave of the force, as I have above described it[16].

Having thus given a slight outline of the contents of this Memoir, 1 will now proceed to the fundamental investigation of the individual points.

SCIENTIFIC MEMOIRS.

VOL. II.—PART VIII.

Article XIII. continued.

The Galvanic Circuit investigated Mathematically. By Dr. G. S. Ohm[17].

The Galvanic Circuit.

A. General observations on the diffusion of electricity.

1. A property of bodies, called into activity under certain circumstances, and which we call electricity, manifests itself in space, by the bodies which possess it, and which on that account are termed electric, either attracting or repelling one another.

In order to investigate the changes which occur in the electric condition of a body ${\displaystyle A}$ in a perfectly definite manner, this body is each time brought, under similar circumstances, into contact with a second moveable body of invariable electrical condition, called the Electroscope, and the force with which the electroscope is repelled or attracted by the body is determined. This force is termed the electroscopic force of the body ${\displaystyle A}$; and to distinguish whether it is attractive or repulsive we place before the expression for its measure the sign + in the one case, and – in the other.

The same body ${\displaystyle A}$ may also serve to determine the electroscopic force in various parts of the same body. For this purpose we take the body ${\displaystyle A}$ of very small dimensions, so that when we bring it into contact with the part to be tested of any third body, it may from its smallness be regarded as a substitute for this part; then its electroscopic force, measured in the way described, will, when it happens to be different at the various places, make known the relative difference with regard to electricity between these places.

The intention of the preceding explanations is to give a simple and determinate signification to the expression "electroscopic force"; it does not come within the limits of our plan to take notice either of the greater or less practicability of this process, nor to compare inter se the various possible modes of proceeding for the determination of the electroscopic force.

2. We perceive that the electroscopic force moves from one place to another, and from one body to another, so that it does not merely vary at different places at the same time, but also at a single place at different times. In order to determine in what manner the electroscopic force is dependent upon the time when it is perceived, and on the place where it is elicited, we must set out from the fundamental laws to which the exchange of electroscopic force occurring between the elements of a body is subject.

These fundamental laws are of two kinds, either borrowed from experiment, or, where this is wanting, assumed hypothetically. The admissibility of the former is beyond all doubt, and the justness of the latter is distinctly evident from the coincidence of the results deduced from calculation with those which actually occur; for since the phænomenon with all its modifications is expressed in the most determinate manner by calculation, it follows, since no new uncertainties arise and increase the earlier ones during the process, that an equally perfect observation of nature must in a decisive manner either confirm or refute its statements. This in fact is the chief merit of mathematical analysis, that it calls forth, by its never-vacillating expressions, a generality of ideas, which continually excites to renewed experiments, and thus leads to a more profound knowledge of nature. Every theory of a class of natural phænomena founded upon facts, which will not admit of analytical investigation in the form of its exposition, is imperfect; and no reliance is to be placed upon a theory developed in ever so strict a form, which is not confirmed to a sufficient extent by observation. So long, therefore as not even one portion of the effects of a natural force has been observed with the greatest accuracy in all its gradations, the calculation employed in its investigation only treads on uncertain ground, as there is no touchstone for its hypotheses, and in fact it would be far better to wait a more fit time; but when it goes to work with the proper authority, it enriches, at least in an indirect manner, the field it occupies with new natural phænomena, as universal experience shows. I have thought it necessary to premise these general remarks, as they not only serve to throw more light on what follows, but also because they explain the reason why the galvanic phænomena have not long since been mathematically treated with greater success, although, as we shall subsequently find, the requisite course has been already earlier pursued in another, apparently less prepared, branch of Physics.

After these reflections we will now proceed to the establishment of the fundamental laws themselves.

3. When two electrical elements, ${\displaystyle E}$ and ${\displaystyle E'}$, of equal magnitude, of like form and similarly placed with respect to each other, but unequally powerful, are situated at the proper distance from each other, they exhibit a mutual tendency to attain electric equilibrium, which is apparent in both constantly and uninterruptedly approaching nearer to the mean of their electric state, until they have actually attained it. That is to say, both elements reciprocally change their electric state so long as a difference continues to exist between their electroscopic forces; but this change ceases as soon as they have both attained the same electroscopic force. Consequently this change of the electric difference of the elements is so dependent that the one disappears at the same time with the other. We now suppose that the change, effected in an extremely short instant of time in both elements, is proportional to the difference of their cotemporaneous electroscopic force and the magnitude of the instant of time; and without yet attending to any material distinctions of the electricity, it is always to be understood that the forces designated by + and — are to be treated exactly as opposite magnitudes. That the change is effected accurately according to the difference of the forces, is a mathematical supposition, the most natural because it is the most simple; all the rest is given by experiment. The motion of electricity is effected in most bodies so rapidly that we are seldom able to determine its changes at the various places, and on that account we are not in a condition to discover by observation the law according to which they act. The galvanic phænomena, in which such changes occur in a constant form, are therefore of the highest importance for testing this assumption: for if the conclusions drawn from the supposition are thoroughly confirmed by those phænomena, it is admissible, and may then be applied without any further consideration to all analogous researches, at least within the same limits of force.

We have assumed, in accordance with the observations hitherto made, that when by any two exteriorly like constituted elements, whether they be of the same or of different matter, a mutual change in their electrical state is produced, the one loses just so much force as the other gains. Should it hereafter be shown by experiments that bodies exhibit a relation similar to that which in the theory of heat is termed the capacity of bodies, the law we have established will have to undergo a slight alteration, which we shall point out in the proper place.

4. When the two elements ${\displaystyle E}$ and ${\displaystyle E'}$ are not of equal magnitude, it is still allowed to regard them as sums of equal parts. Granting that an element ${\displaystyle E}$ consist of ${\displaystyle m}$ perfectly equal parts, and the other ${\displaystyle E'}$ of ${\displaystyle m'}$ exactly similar parts, then, if we imagine the elements ${\displaystyle E}$ and ${\displaystyle E'}$ exceedingly small in comparison with their mutual distance, so that the distances from each part of the one to each part of the other element are equal, the sum of the actions of all the ${\displaystyle m'}$ parts of the element ${\displaystyle E'}$ on a part of ${\displaystyle E}$ will be ${\displaystyle m'}$ times that which a single part exerts, and the sum of all the actions of the element ${\displaystyle E'}$ on all the ${\displaystyle m}$ parts of ${\displaystyle E}$ will be ${\displaystyle mm'}$ times that which a part of ${\displaystyle E'}$ exerts on a part of ${\displaystyle E}$. It is hence evident, that in order to ascertain the mutual actions of dissimilar elements on each other, they must be taken as proportional not merely to the difference of their electroscopic forces and their duration, but also to the product of their relative magnitudes. We shall in future term the sum of the electroscopic actions, referred to the magnitude of the elements—by which therefore we have to understand the force multiplied by the magnitude of the space over which it is diffused, in the case where the same force prevails at all places in this space—the quantity of electricity, without intending to determine anything thereby with respect to the material nature of electricity. The same observation is applicable to all figurative expressions introduced, without which, perhaps for good reasons, our language could not exist.

In cases where the elements cannot be regarded as evanescent in comparison with their relative distances a function, to be determined separately for each given case from their dimensions and their mean distance, must be substituted for the product of the magnitudes of the two elements, and which we will designate where it is employed by ${\displaystyle F}$.

5. Hitherto we have taken no notice of the influence of the mutual distance of the elements between which an equalization of their electric state takes place, because as yet we have only considered such elements as always retained the same relative distance. But now the question arises, whether this exchange is directly effected only between adjacent elements, or if it extends to others more distant, and how on the one or the other supposition is its magnitude modified by the distance? Following the example of Laplace, it is customary in cases where molecular actions at the least distance come into play, to employ a particular mode of representation, according to which a direct mutual action between two elements separated by others, still occurs at finite distances, which action, however, decreases so rapidly, that even at any perceptible distance, be it ever so minute, it has to be considered as perfectly evanescent. Laplace was led to this hypothesis, because the supposition that the direct action only extended to the next element produced equations, the individual members of which were not of the same dimension relatively to the differentials of the variable quantities[18],—a non-uniformity which is opposed to the spirit of the differential calculus. This apparent unavoidable disproportion between the members of a differential equation, belonging nevertheless necessarily to one another, is too remarkable not to attract the attention of those to whom such inquiries are of any value; an attempt therefore to add something to the explanation of this ænigma will be the more proper in this place, as we gain the advantage of rendering thereby the subsequent considerations more simple and concise. We shall merely take as an instance the propagation of electricity, and it will not be difficult to transfer the obtained results to any other similar subject, as we shall subsequently have occasion to demonstrate in another example.

6. Above all, it is requisite that the term goodness of conduction be accurately defined. But we express the energy of conduction between two places by a magnitude which, under otherwise similar circumstances, is proportional to the quantity carried over in a certain time from one place to the other multiplied by the distance of the two places from each other. If the two places are extended, then we have to understand by their distance the straight line connecting the centres of the dimensions of the two places. If we transfer this idea to two electric elements, ${\displaystyle E}$ and ${\displaystyle E'}$, and call ${\displaystyle s}$ the mutual distance of their centres, ${\displaystyle \varsigma }$ the quantity of electricity, which under accurately determined and invariable circumstances is carried over from one element to the other, and ${\displaystyle \chi }$ the conductibility between them,
 ${\displaystyle \chi =\varsigma \cdot s}$.
We will now endeavour to determine more precisely the quantity of electricity denoted by ${\displaystyle \varsigma }$. According to § 4 the quantity of electricity, which is transferred in an exceedingly short time from one element to the other, is, the distance being invariable, in general proportional to the difference between the electroscopic forces, the duration, and the size of each of the two elements. If therefore we designate the electroscopic forces of the two elements ${\displaystyle E}$ and ${\displaystyle E'}$ by ${\displaystyle u}$ and ${\displaystyle u'}$, and the space they occupy by ${\displaystyle m}$ and ${\displaystyle m'}$, we obtain for the quantity of electricity carried over from ${\displaystyle E'}$ to ${\displaystyle E}$ in the element of time ${\displaystyle d\,t}$ the following expression:
 ${\displaystyle \alpha mm'(u'-u)\,d\,t}$,
where ${\displaystyle \alpha }$ represents a coefficient depending in some way on the distance ${\displaystyle s}$. This quantity changes every moment if ${\displaystyle u'-u}$ is variable; but if we suppose that the forces ${\displaystyle u'}$ and ${\displaystyle u}$ remain constant at all times, it merely depends on the magnitude of the instant of time ${\displaystyle d\,t}$, we can consequently extend it to the unity of time; if we place the present constant difference of the forces ${\displaystyle u'-u}$ equal to the unity of force, it then becomes
 ${\displaystyle \alpha mm'}$.
This quantity of electricity is for the two elements ${\displaystyle E}$ and ${\displaystyle E'}$ whose position is invariable, constant under the same circumstances, on which account it may be employed in the determination of the power of conduction just mentioned. For if we understand by ${\displaystyle \varsigma }$ the quantity of electricity transferred from ${\displaystyle E'}$ to ${\displaystyle E}$ in the unity of time, with a constant difference of the electroscopic forces equal to the unity of force, we have
 ${\displaystyle \varsigma =\alpha mm'}$,
and then
 ${\displaystyle \chi =\alpha mm's}$.
If we take from this last equation the value of a ${\displaystyle mm'}$ and substitute it in the expression
 ${\displaystyle \alpha mm'(u'-u)\,d\,t}$,
we obtain for the variable quantity of electricity which passes over in the instant of time ${\displaystyle d\,t}$ from ${\displaystyle E'}$ to ${\displaystyle E}$, the following:
 ${\displaystyle {\frac {\chi (u'-u)\,d\,t}{s}}}$, (♂)

which expression is not accompanied by the above-mentioned disproportion between the members of the differential equation, as will soon be perceived.

7. The course hitherto pursued was based upon the supposition that the action exerted by one element on the other is proportional to the product of the space occupied by the two elements, an assumption which, as was already observed in § 4, can no longer be allowed in cases where it is a question of the mutual action of elements situated indefinitely near each other, because it either establishes a relation between the magnitudes of the elements and their mutual distances, or prescribes to these elements a certain form. The previously found expression (♂) for the variable quantity of electricity passing from one element to the other, possesses therefore no slight advantage in being entirely independent of this supposition; for whatever may have to be placed in any determinate case instead of the product ${\displaystyle mm'}$, the expression (♂) constantly remains the same, this peculiarity being solely referrible to the power of conduction ${\displaystyle \chi }$. If, for instance, ${\displaystyle F}$ designate, as was stated in § 4, the function, corresponding to such a case, of the dimensions and of the mean distance of both elements, the expression
 ${\displaystyle \alpha mm'(u'-u)\,d\,t}$
not merely changes apparently into
 ${\displaystyle F(u'-u)\,d,t}$,
but also the equation
 ${\displaystyle \chi =\alpha mm's}$
into the other,
 ${\displaystyle \chi =F\cdot s}$, (☉)
so that if we take the value of ${\displaystyle F}$ from this equation and place it in the above expression, we always obtain
 ${\displaystyle {\frac {\chi (u'-u)\,d\,t}{s}}}$.

Moreover, the circumstance of the expression (♂) still remaining valid for corpuscles, whose dimensions are no longer indefinitely small, is of some importance when the same electroscopic force only exists merely at all points of each such part. It is hence evident how intimately our considerations are allied to the spirit of the differential calculus; for uniformity in all points with reference to the property which enters into the calculation is precisely the distinctive characteristic required by the differential calculus from that which it is to receive as an element.

If we institute a more profound comparison between the process originating with Laplace and that here advanced, we shall arrive at some interesting points of comparison. If for instance we consider that for infinitely small masses at infinitely short distances all particular relations must necessarily have the same weight as for finite masses at finite distances, it is not directly evident how the method of the immortal Laplace—to whom we are indebted for so many valuable explanations respecting the nature of molecular actions,—according to which the elements must be constantly treated as if they were placed at finite distances from each other, could nevertheless still afford correct results; but we shall find on closer examination that it acts in fact otherwise than it expresses. Indeed, since Laplace, when determining the changes of an element by all surrounding it, makes the higher powers of the distance disappear compared with the lower, he therewith assumes, quite in the spirit of the differential calculus, the difference of action itself to be infinitely small, but terms it finite, and treats it also as such; whence it is immediately apparent that he in fact treats that which is infinitely small at an infinitely short distance as finite. Disregarding however the great certainty and distinctness which accompany our manner of representation, there might still be something more to say, and perhaps with some justice, against Laplace's mode of treatment in favour of ours, in this respect, that the former takes not the least account of the possible nature of the given elements of bodies, but merely has to do with imaginary elements of space, by which the physical nature of the bodies is almost entirely lost sight of. We may, to render our assertion intelligible by an example, undoubtedly imagine bodies in nature which consist only of homogeneous elements, but whose position to each other, taken in one direction, might be different than when in another direction; such bodies, as our mode of representation immediately shows, might conduct the electricity in one direction in a different manner than in another, notwithstanding that they might appear uniform and equally dense. In such a case, did it occur, we should have to take refuge, according to Laplace, in considerations which have remained entirely foreign to the general process. On the other hand, the mode in which bodies conduct affords us the means by which we are enabled to judge of their internal structure, which, from our almost total ignorance on the subject, cannot be immediately shown. Lastly, we may add, that this, our hitherto developed view of molecular actions, unites in itself the two advanced by Laplace and by Fourier in his theory of heat, and reconciles them with each other.

8. We need now no longer hesitate about allowing the electrical action of an element not to extend beyond the adjacent surrounding elements, so that the action entirely disappears at every finite distance, however small. The extremely limited circle of action with the almost infinite velocity with which electricity passes through many bodies might indeed appear suspicious; but we did not overlook on its admission, that our comparison in such cases is only effected by an imaginary relative standard, which is deceitful, and does therefore not justify us to vary a law so simple and independent until the conclusions drawn from it are in contradiction to nature, which in our subject, however, does not seem to be the case.

The sphere of action thus fixed by us, has, although it is infinitely small, precisely the same circumference as that introduced by Laplace, and called finite, where he lets the higher powers of the distance vanish compared with the lower, the reason of which may be found already in what has been stated above. The supposition of a finite distance of action in our sense would correspond to the case where Laplace still retains higher powers of distance together with the lower.

9. The bodies on which we observe electric phænomena are in most cases surrounded by the atmosphere; it is therefore requisite, in order to investigate profoundly the entire process, not to disregard the changes which may be produced by the adjacent air. According to the experiments left us by Coulomb on the diffusion of electricity in the surrounding atmosphere, the loss in force thus occasioned is (during a very short constant time), at least when the intensities are not very considerable, on the one hand proportional to the energy of the electricity, and on the other is dependent on a coefficient varying according to the cotemporaneous nature of the air, but otherwise invariable for the same air. The knowledge of this enables us to bring the influence of the atmosphere on galvanic phænomena into calculation wherever it might be requisite. It must however not be overlooked here, that Coulomb's experiments were made on electricity which had entered into equilibrium and was no longer in the process of excitation, with respect to which both observation and calculation have convinced us that it is confined to the surface of bodies, or merely penetrates to a very slight depth into their interior; for from thence may be drawn the conclusion, of some importance with respect to our subject, that all the electricity present in those experiments may have been directly concerned in the transference to the atmosphere. If we now connect with this observation the law just announced, according to which two elements, situated at any finite distance from each other, no longer exert any direct action on each other, we are justified in concluding, that where the electricity is uniformly diffused throughout the entire mass of a finite body, or at least so that proportionately but a small quantity resides in the vicinity of the surface, which case does not in general occur when it has entered into motion, the loss which is occasioned by the circumambient air can be but extremely small in comparison to that which takes place when the entire force is situated immediately at the surface, which invariably happens when it has entered into equilibrium; and thence, therefore, it happens that the atmosphere exerts no perceptible influence on galvanic phænomena in the closed circuit when this is composed of good conductors, so that the changes produced by the presence of the atmosphere in phænomena of contact-electricity may be neglected in such cases. This conclusion, moreover, receives new support from the circumstance, that in the same cases the contact-electricity only remains during an exceedingly short time in the conductors, and even on that account would only give up a very slight portion to the air, even if it were in immediate contact with it.

Although, from what has been stated, it is placed beyond all doubt that the action of the atmosphere has no perceptible influence on the magnitude of effect of the usual galvanic circuits, it by no means is intended to admit the reverse of the conclusion, viz. that the galvanic conductor exerts no perceptible influence on the electric state of the atmosphere; for mathematical investigation teaches us that the electroscopic action of a body on another has no direct connexion with the quantity of electricity which is carried over from the one to the other.

10. We arrive at last at that position founded on experiment, and which is of the highest importance for the whole of natural philosophy, since it forms the basis of all the phænomena to which we apply the name of galvanic: it may be expressed thus: Different bodies, which touch each other, constantly preserve at the place of contact the same difference between their electroscopic forces by virtue of a contrariety proceeding from their nature, which we are accustomed to designate by the expression electric tension, or difference of bodies. Thus enounced, the position stands, without losing any of its simplicity, in all the generality which belongs to it; for we are nearly always referred to it by every single phænomenon. Moreover, the above expression is adopted in all its generality, either expressly or tacitly, by all philosophers in the explanation of the electroscopic phænomena of the voltaic pile. According to our previously developed ideas respecting the mode in which elements act on one another, we must seek for the source of this phænomenon in the elements directly in contact, and consequently we must allow the abrupt transition to take place from one body to the other in an infinitely small extent of space.

11. This being established, we will now proceed to the subject, and in the first place consider the motion of the electricity in a homogeneous cylindric or prismatic body, in which all points throughout the whole extent of each section, perpendicular to its axis, possess contemporaneously equal electroscopic force, so that the motion of the electricity can only take place in the direction of its axis. If we imagine this body divided by a number of such sections into disks of infinitely small thickness, and so that in the whole circumference of each disk the electroscopic force does not vary sensibly for each pair of such disks, the expression ♂ given in § 6 can be applied to determine the quantity of electricity passing from one disk to the other; but by the limitation of the distance of action to only infinitely small distances mentioned in the preceding paragraph, its nature is so modified that it disappears as soon as the divisor ceases to be infinitely small.

If we now choose one of the infinite number of sections invariably for the origin of the abscissæ, and imagine anywhere a second, whose distance from the first we denote by ${\displaystyle x}$, then ${\displaystyle d\,x}$ represents the thickness of the disk there situated, which we will designate by ${\displaystyle M}$. If we conceive this thickness of the disk to be of like magnitude at all places, and term ${\displaystyle u}$ the electroscopic force present at the time ${\displaystyle t}$ in the disk ${\displaystyle M}$, whose abscissa is ${\displaystyle x}$, so that therefore ${\displaystyle u}$ in general will be a function of ${\displaystyle t}$ and ${\displaystyle x}$; if we further suppose${\displaystyle u'}$ and ${\displaystyle u_{j}}$ to be the values of ${\displaystyle u}$ when ${\displaystyle x+d\,x}$ and ${\displaystyle x-dx}$ are substituted respectively for ${\displaystyle x}$, then ${\displaystyle u'}$ and ${\displaystyle u_{'}}$ evidently express the electroscopic forces of the disks situated next the two sides of the disk ${\displaystyle M}$, of which we will denote the one belonging to the abscissa ${\displaystyle x+d\,x}$ by ${\displaystyle M'}$, and that belonging to the abscissa ${\displaystyle x+d\,x}$ by ${\displaystyle M_{'}}$; and it is clearly evident that the distance of the centre of each of the disks ${\displaystyle M'}$ and ${\displaystyle M_{'}}$ from the centre of the disk ${\displaystyle M}$ is ${\displaystyle d\,x}$. Consequently, by virtue of the expression (♂) given in § 6, if ${\displaystyle \chi }$ represents the conducting power of the disk ${\displaystyle M'}$ to ${\displaystyle M}$,
 ${\displaystyle {\frac {\chi (u'-u)d\,t}{d\,x}}}$
is the quantity of electricity which is transferred during the interval of time ${\displaystyle d\,t}$ from the disk ${\displaystyle M'}$ to the disk ${\displaystyle M}$, or from the latter to the former, according as ${\displaystyle u'-u}$ is positive or negative. In the same manner, when we admit the same power of conduction between ${\displaystyle M_{'}}$ and ${\displaystyle M}$,
 ${\displaystyle {\frac {\chi (u_{\prime }-u)\,dt}{dx}}}$
is the quantity of electricity passing over from ${\displaystyle M_{\prime }}$ to ${\displaystyle M}$ when the expression is positive, and from ${\displaystyle M}$ to ${\displaystyle M_{\prime }}$ when it is negative. The total change of the quantity of electricity which the disk ${\displaystyle M}$ undergoes from the motion of the electricity in the interior of the body in the particle of time ${\displaystyle dt}$, is consequently
 ${\displaystyle {\frac {\chi (u'+u_{\prime }-2u)\,dt}{dx}},}$

and an increase in the quantity of electricity is denoted when this value is positive, and when negative a diminution of the same.

But according to Taylor's theorem
 ${\displaystyle u'=u+{\frac {du}{dx}}\centerdot dx+{\frac {d^{2}u}{dx^{2}}}\centerdot {\frac {dx^{2}}{2}}+\ldots \ldots ,}$
and in the same way
 ${\displaystyle u_{\prime }=u-{\frac {du}{dx}}\centerdot dx+{\frac {d^{2}u}{dx^{2}}}\centerdot {\frac {dx^{2}}{2}}-\ldots ;}$
consequently
 ${\displaystyle u'+u_{\prime }=2u+{\frac {d^{2}u}{dx^{2}}}dx^{2}.}$
According to this the expression just found for the total change of the quantity of electricity present in the disk ${\displaystyle M}$ is converted during the time ${\displaystyle dt}$ into
 ${\displaystyle \chi \cdot {\frac {d^{2}u}{dx^{2}}}dx\,dt,}$
where ${\displaystyle \chi }$ represents the power of conduction which prevails from one disk to the adjacent one, which we suppose to be invariable throughout the length of the homogeneous body. It must here be observed, that this value ${\displaystyle \chi }$ is, on account of the infinitely small distance of action, proportional to the section of the cylindric or prismatic body; if therefore we denote the magnitude of this section by ${\displaystyle \omega }$, and separate this factor from the value ${\displaystyle \chi }$, always calling the remaining portion ${\displaystyle \chi }$, the former expression changes into the following:
 ${\displaystyle \chi \omega {\frac {d^{2}u}{dx^{2}}}dx\,dt,}$

in which ${\displaystyle \chi }$ now represents the conductibility of the body independent of the magnitude of the section, which we will term the absolute conductibilty of the body in opposition to the former, which may be called the relative. Henceforward wherever the word conductibility occurs without any closer definition, the absolute conductibility is always to be understood.

Hitherto we have left out of consideration the change which the disk suffers from the adjacent atmosphere. This influence may easily be determined. If, for instance, we designate by ${\displaystyle c}$ the circumference of the disk belonging to the abscissa ${\displaystyle x}$, then ${\displaystyle c\,dx}$ is the portion of its surface which is exposed to the air; consequently, according to the experiments of Coulomb, mentioned in § 9,
 ${\displaystyle bcu\,dx\,dt}$

is the change of the quantity of electricity which is occasioned in the disk ${\displaystyle M}$ by the passing off of the electricity into the atmosphere during the moment of time ${\displaystyle dt}$, where ${\displaystyle b}$ represents a coefficient dependent on the cotemporaneous nature of the atmosphere, but constant for the same atmosphere. It expresses a decrease when ${\displaystyle u}$ is positive, and an increase when ${\displaystyle u}$ is negative. But in accordance with our original supposition, this action cannot occasion an inequality of the electroscopic force in the same section of the body; or at least, this inequality must be so slight that no perceptible alteration is produced in the other quantities; a circumstance which may nearly always be supposed in the galvanic circuit.

Accordingly, the entire change which the quantity of electricity in the disk ${\displaystyle M}$ undergoes in the moment of time ${\displaystyle dt}$ is
 ${\displaystyle \chi \omega {\frac {d^{2}u}{dx^{2}}}dx\cdot dt-bcu\,dx\,dt,}$

in which the portion is comprised which arises from the motion of the electricity in the interior of the body as well as that which is caused by the circumambient atmosphere.

But the entire change of the electroscopic force ${\displaystyle u}$ in the disk ${\displaystyle M}$ effected in the moment of time ${\displaystyle dt}$ is
 ${\displaystyle {\frac {du}{dt}}dt,}$
consequently the total change in the quantity of electricity in the disk ${\displaystyle M}$ during the time ${\displaystyle dt}$ is
 ${\displaystyle \omega {\frac {du}{dt}}dx\,dt,}$

where, however, it is supposed that under all circumstances similar changes in the electroscopic force correspond to similar, changes in the quantity of electricity. If observation showed that different bodies of the same surface underwent a different change in their electroscopic force by the same quantity of electricity, then there would still remain to be added a coefficient ${\displaystyle \gamma }$ corresponding to this property of the various bodies. Experience has not yet decided respecting this supposition borrowed from the relation of heat to bodies.

If we assume the two expressions just found for the entire change in the quantity of electricity in the disk ${\displaystyle M}$ during the moment of time ${\displaystyle dt}$ to be equal, and divide all the members of the equation by ${\displaystyle \omega \,dx\,dt}$ we obtain
 ${\displaystyle \gamma {\frac {du}{dt}}=\chi {\frac {d^{2}u}{dx^{2}}}-{\frac {bc}{\omega }}u,}$ (a)

from which the electroscopic force ${\displaystyle u}$ has to be determined as a function of ${\displaystyle x}$ and ${\displaystyle t}$.

12. We have in the preceding paragraph found for the change in the quantity of electricity occurring between the disks ${\displaystyle M'}$ and ${\displaystyle M}$ during the time ${\displaystyle dt}$
 ${\displaystyle {\frac {\chi (u'-u)\,dt}{dx}},}$
and have seen that the direction of the passage is opposed to the course of the abscissæ when the expression is positive; on the contrary, it proceeds in the direction of the abscissæ when it is negative. In the same way the magnitude of the transition between the disks ${\displaystyle M_{\prime }}$ and ${\displaystyle M}$, when we retain the same relation to its direction, is
 ${\displaystyle {\frac {\chi (u_{\prime }-u)\,dt}{dx}}.}$
If we substitute in these two expressions for ${\displaystyle u_{\prime }}$ and ${\displaystyle u'}$ the transformations given in the same paragraph, and at the same time ${\displaystyle \chi \omega }$ for ${\displaystyle \chi }$, i. e. the absolute power of conduction for the relative, we obtain in both cases
 ${\displaystyle \chi \omega {\frac {du}{dx}}\,dt,}$
whence it results that the same quantity of electricity which enters from the one side into the disk ${\displaystyle M}$ during the element of time ${\displaystyle dt}$, is again in the same time expelled from it towards the other side. If we imagine this transmission of the electricity, occurring at the time ${\displaystyle t}$ in the disk belonging to the abscissa ${\displaystyle x}$, of invariable energy reduced to the unity of time, call it the electric current, and designate the magnitude of this current by ${\displaystyle S}$, then
 ${\displaystyle S=\chi \omega {\frac {du}{dx}};}$ (b)

and in this equation positive values for ${\displaystyle S}$ show that the current takes place opposed to the direction of the abscissæ; negative, that it occurs in the direction of the abscissæ.

13. In the two preceding paragraphs we have constantly had in view a homogeneous prismatic body, and have inquired into the diffusion of the electricity in such a body, on the supposition that throughout the whole extent of each section, perpendicular to its length or axis, the same electroscopic force exists at any time whatsoever. We will now take into consideration the case where two prismatic bodies ${\displaystyle A}$ and ${\displaystyle B}$, of the same kind, but formed of different substances, are adjacent, and touch each other in a common surface. If we establish for both ${\displaystyle A}$ and ${\displaystyle B}$ the same origin of abscissæ, and designate the electroscopic force of ${\displaystyle A}$ by ${\displaystyle u}$, that of ${\displaystyle B}$ by ${\displaystyle u'}$, then both ${\displaystyle u}$ and ${\displaystyle u'}$ are determined by the equation (a) in paragraph 11, if ${\displaystyle \chi }$ only retain the value each time corresponding to the peculiar substance of each body: but ${\displaystyle u}$ represents a function of ${\displaystyle t}$ and ${\displaystyle x}$, which holds only so long as the abscissa ${\displaystyle x}$ corresponds to points in the body ${\displaystyle A}$; ${\displaystyle u}$ on the other hand denotes a function of ${\displaystyle t}$ and ${\displaystyle x}$, which holds only when the abscissa ${\displaystyle x}$ corresponds to the body B. But there are still some other conditions at this common surface, which we will now explain. If we denote for this purpose the separate values of ${\displaystyle u}$ and ${\displaystyle u'}$, which they first assume at the common surface, by enclosing the general ones between crotchets, we find according to the law advanced in § 10 the following equation between these separate values:
 ${\displaystyle (u)-(u')=a,}$
where ${\displaystyle a}$ represents a constant magnitude otherwise dependent on the nature of the two bodies. Besides this condition, which relates to the electroscopic force, there is still a second, which has reference to the electric current. It consists in this, that the electric current in the common surface must in the first place possess equal magnitude and like direction in both bodies, or, if we retain the common factor ${\displaystyle \omega }$,
 ${\displaystyle \chi \omega \left({\frac {du}{dx}}\right)=\chi '\omega \left({\frac {du'}{dx}}\right),}$

where ${\displaystyle \chi }$ represents the actual power of conduction of the body ${\displaystyle A}$, ${\displaystyle x'}$ that of the body ${\displaystyle B}$, and ${\displaystyle \left({\frac {du}{dx}}\right)}$, ${\displaystyle \left({\frac {du'}{dx}}\right)}$ the particular values of ${\displaystyle {\frac {du}{dx}}}$, ${\displaystyle {\frac {du'}{dx}}}$ immediately belonging to them at the common surface, and in which it was assumed that the origin of the abscissæ was not taken on this common surface. The necessity of this last equation may easily be conceived; for were it otherwise, the two currents would not be of equal energy in the common surface, but there would be more conveyed from the one body to this surface than would be abstracted from it by the other; and if this difference were a finite portion of the entire current, the electroscopic force would increase at that very place, and indeed, considering the surprising fertility of the electric current, would arrive in the shortest time to an exceedingly high degree, as observation has long since demonstrated. Nor can a smaller quantity of electricity be imparted from the one body to the common surface than it is deprived of by the other, as this circumstance would be evinced by an infinitely high degree of negative electricity.

It is not absolutely requisite for the validity of the preceding determinations, that the two bodies in contact have the same base. The section in the one prismatic body may be different in size and form to that in the other, if this does not render the electroscopic force sensibly different at the various points of the same section, which, considering the great energy with which the electricity tends to equilibrium, will not be the case when the bodies are good conductors, whose length far surpasses their other dimensions. In this case everything remains as before, only that the section of the body ${\displaystyle B}$ must everywhere be distinguished from that of ${\displaystyle A}$; consequently the second conditional equation for the place where the two bodies are in contact changes into the following:—
 ${\displaystyle \chi \omega \left({\frac {du}{dx}}\right)=\chi '\omega '\left({\frac {du'}{dx}}\right),}$

where ${\displaystyle \omega }$ still represents the section of ${\displaystyle A}$, but ${\displaystyle \omega '}$ that of the body ${\displaystyle B}$, which at present differs from the former.

There may even exist in the prolongation of the body ${\displaystyle A}$ two prismatic bodies, ${\displaystyle B}$ and ${\displaystyle C}$, separated from each other, which are both situated immediately on the one surface of ${\displaystyle A}$. If in this case ${\displaystyle \chi '\omega 'u'}$ signifies for the body ${\displaystyle B}$, and ${\displaystyle \chi ''\omega ''}$, ${\displaystyle u''}$ for the body ${\displaystyle C}$ what ${\displaystyle \chi \omega u}$ does for ${\displaystyle A}$, we obtain instead of the one conditional equation the two following:—
 {\displaystyle {\begin{aligned}&(u)-(u')=a,\\&(u)-(u'')=a',\end{aligned}}}
where ${\displaystyle a}$ represents the electric tension between the bodies ${\displaystyle A}$ and ${\displaystyle B}$, and ${\displaystyle a'}$ that between ${\displaystyle A}$ and ${\displaystyle C}$. In the same manner we now obtain instead of the second conditional equation the following:—
 ${\displaystyle \chi \omega \left({\frac {du}{dx}}\right)=\chi '\omega '\left({\frac {du'}{dx}}\right)+\chi ''\omega ''\left({\frac {du''}{dx}}\right).}$

It is immediately apparent how these equations must change when a greater number of bodies are combined. We shall not enter further into these complications, as what has been stated suffices to throw sufficient light upon the changes which have in such a case to be performed on the equations.

14. To avoid misconception, I will, at the close of these general observations, once more accurately define the circle of application within which our formulæ have universal validity. Our whole inquiry is confined to the case where all the parts of the same section possess equal electroscopic force, and the magnitude of the section varies only from one body to the other. The nature of the subject, however, frequently gives rise to circumstances which render one or the other of these conditions superfluous, or at least diminishes their importance. Since the knowledge of such circumstances is not without use, I will here illustrate the most prominent by an example.

A circuit of copper, zinc, and an aqueous fluid, will wholly come under the above formula when the copper and zinc are prismatic and of equal section; when, further, the fluid is likewise prismatic and of the same or of smaller section, and its terminal surfaces everywhere in contact with the metals. Nay, when only these last conditions are fulfilled with respect to the fluid, the metals may possess equal sections or not, and touch one another with their full sections, or only at some points, and even their form may deviate considerably from the prismatic form, and nevertheless the circuit must constantly obey the laws deduced from our formulæ; for the motion of the electricity produced with such ease in the metals, is obstructed to such a considerable extent by the non-conductive nature of the fluid, that it gains sufficient time to diffuse itself thoroughly with equal energy over the metals, and thus re-establishes in the fluid the conditions upon which our calculation is founded. But it is a very different matter when the prismatic fluid is only touched in disproportionately small portions of its surfaces by the metals, as the electricity arriving there can only advance slowly and with considerable loss of energy to the untouched surfaces of the fluid, whence currents of various kinds and directions result. The existence of such currents has been sufficiently demonstrated by Pohl's manifoldly varied experiments, and nothing more now stands in the way of their determination by analysis, after the additions which it has received from the successful investigations respecting the theory of heat, than the complexities of the expressions. Since their determination exceeds the hmits of this small work, which has for its object to investigate the current only in one dimension, we will defer them to a more fit occasion.

We will now proceed to the application of the formulæ advanced, and divide, for the sake of a more easy and general survey, the whole into two sections, of which the one will treat of the electroscopic phænomena, and the other of the phænomena of the electric current.

B. Electroscopic Phænomena.

15. In our preceding general determinations we have constantly confined our attention to prismatic bodies, whose axes, upon which the abscissæ have been taken, formed a straight line. But all these considerations still retain their entire value, if we imagine the conductor constantly curved in any way whatsoever, and take the abscissæ on the present curved axis of the conductor. The above formulæ acquire their entire applicability from this observation, since galvanic circuits, from their very nature, can but seldom be extended in a straight line. Having anticipated this point, we mil immediately proceed to the most simple case, where the prismatic conductor is formed in its entire length of the same material, and is curved backwards on itself, and conceive the seat of the electric tension to be where its two ends touch. Although no case in nature resembles this imaginary one, it will nevertheless be of great service in the treatment of the other cases which do really occur in nature.

The electroscopic force, at any place of such a prismatic body, may be deduced from the differential equation (a) found in § 11. For this purpose we have only to integrate it, and to determine, in accordance with the other conditions of the problem, the arbitrary functions or constants entering into the integral. This matter is, however, generally very much facilitated, with respect to our subject, by omitting one or even two members, according to the nature of the subject, from the equation (a). Thus nearly all galvanic actions are such that the phænomena are permanent and invariable immediately at their origin. In this case, therefore, the electroscopic force is independent of time, consequently the equation (a) passes into
 ${\displaystyle 0=\chi {\frac {d^{2}u}{dx^{2}}}-{\frac {bc}{\omega }}u.}$

Moreover, the surrounding atmosphere has (as we have already noticed in § 9.) in most cases no influence on the electric nature of the galvanic circuit; then ${\displaystyle b=0}$, by which the last equation is converted into
 ${\displaystyle 0={\frac {d^{2}u}{dx^{2}}}.}$
But the integral of this last equation is
 ${\displaystyle u=fx+c,}$ (c)

where ${\displaystyle f}$ and ${\displaystyle c}$ represent any constants remaining to be determined. The equation (c) consequently expresses the law of electrical diffusion, in a homogeneous prismatic conductor, in all cases where the abduction by the air is insensible, and the action no longer varies with time. As these circumstances in reality most frequently accompany the galvanic circuit, we shall on that account dwell longest upon them.

We are enabled to determine one of the constants by the tension occurring at the extremities of the conductor, which has to be regarded as invariable and given in each case. If, for instance, we imagine the origin of the abscissæ anywhere in the axis of the body, and designate the abscissa belonging to one of its ends by ${\displaystyle x_{1}}$, then the electroscopic force there situated is, according to the equation (c),
 ${\displaystyle fx_{1}+c\colon }$
in the same way we obtain for the electroscopic force of the other extremity, when we represent its abscissa by ${\displaystyle x_{2}}$,
 ${\displaystyle fx_{2}+c.}$
If we now call the given tension or difference of the electroscopic force ${\displaystyle a}$, we have
 ${\displaystyle a=\pm (x_{1}-x_{2}).}$
But ${\displaystyle x_{1}-x_{2}}$ evidently represents the entire, positive or negative, length of the prismatic conductor; if we designate this by ${\displaystyle l}$, we obtain accordingly
 ${\displaystyle a=\pm fl,}$
whence the constant ${\displaystyle f}$ may be determined. If we now introduce the value of the constant thus found into the equation (c), it is converted into
 ${\displaystyle u=\pm {\frac {a}{l}}x+c,}$
so that only the constant ${\displaystyle c}$ remains to be determined. We may consider the ambiguity of the sign ${\displaystyle \pm }$ to be owing to the tension ${\displaystyle a}$, by ascribing to it a positive value when the extremity of the conductor, belonging to the greater abscissa, possesses the greatest electroscopic force, and when the contrary a negative. Under this supposition is then generally
 ${\displaystyle u={\frac {a}{l}}x+c.}$ (d)

The constant ${\displaystyle c}$ remains in general wholly undetermined, which admits of our allowing the diffusion of the electricity in the conductor to vary arbitrarily, by external influences, in such manner that it occupies the entire conductor everywhere uniformly.

Among the various considerations respecting this constant, there is one of especial importance to the galvanic circuit, I mean that which supposes the circuit to be connected at some one place with a perfect conductor, so that the electroscopic force has to be regarded as constantly destroyed at this place. If we call the abscissa belonging to this place ${\displaystyle \lambda }$, then according to the equation (d)
 ${\displaystyle 0={\frac {a}{l}}\lambda +c.}$
By determining from this the constant ${\displaystyle c}$, and placing its value in the same equation (d), we obtain
 ${\displaystyle u={\frac {a}{l}}(x-\lambda ),}$

from which the electroscopic force of a galvanic circuit of the length ${\displaystyle l}$, and of the tension ${\displaystyle a}$, which is touched at any given place whose abscissa is ${\displaystyle \lambda }$, may be found for every other place.

If any constant and perfect adduction, from outwards to the galvanic circuit, were to be given instead of the permanent abduction outwards, so that the electroscopic force pertaining to the abscissa ${\displaystyle \lambda }$ were compelled to assume constantly a given energy, which we will designate by ${\displaystyle \alpha }$, we should obtain for the determination of the constant ${\displaystyle c}$ the equation
 ${\displaystyle \alpha ={\frac {a}{l}}\lambda +c,}$
and for the determination of the electroscopic force of the circuit at any other place the following:
 ${\displaystyle u={\frac {a}{l}}(x-\lambda )+\alpha .}$

We have seen how the constant ${\displaystyle c}$ may be determined when the electroscopic force is indicated at any place of the circuit by external circumstances; but now the question arises, what value are we to ascribe to the constant when the circuit is left entirely to itself, and this value can consequently no longer be deduced from outward circumstances? The answer to this question is found in the consideration, that each time both electricities proceed contemporaneously, and in like quantity from a previously indifferent state. It may, therefore, be asserted, that a simple circuit of the present kind, which is formed in a perfectly neutral and isolated condition, would assume on each side of the place of contact an equal but opposite electric condition, whence it is self-evident that their centre would be indifferent. For the same reason, however, it is also apparent that when the circuit at the moment of its origin is compelled by any circumstance to deviate from this, its normal state, it would certainly assume the abnormal one until again caused to change.

The properties of a simple galvanic circuit, such as we have hitherto considered them to be, accordingly consist essentially in the following, as is directly evident from the equation (d):

a. The electroscopic force of such a circuit varies throughout the whole length of the conductor continually, and on like extents constantly to the same amount; but where the two extremities are in contact, it changes suddenly, and, indeed, from one extremity to the other, to the extent of an entire tension.

b. When any place of the circuit is disposed by any circumstance to change its electric state, all the other places of the circuit change theirs at the same time, and to the same amount.

16. We will now imagine a galvanic circuit, composed of two parts, ${\displaystyle P}$ and ${\displaystyle P'}$, at whose two points of contact a different electric tension occurs, which case comprises in it the thermal circuit. If we call ${\displaystyle u}$ the electroscopic force of the part ${\displaystyle P}$, and ${\displaystyle u'}$ that of the part ${\displaystyle P'}$, then, according to the preceding paragraph, as here, the case there noticed is repeated twice, in consequence of the equation (c),
 ${\displaystyle u=fx+c}$
for the part ${\displaystyle P}$, and
 ${\displaystyle u'=f'x+c'}$
for the part ${\displaystyle P'}$, where ${\displaystyle f}$, ${\displaystyle c}$, ${\displaystyle f'}$, ${\displaystyle c'}$ are any constant magnitudes to be deduced from the peculiar circumstances of our problem, and each equation is only valid so long as the abscissæ refer to that part to which the equations belong. If we now place the origin of the abscissæ at one of the places of contact of the part ${\displaystyle P}$, and suppose the direction of the abscissæ in this part to proceed inwards; moreover, designate by ${\displaystyle l}$ the length of the part ${\displaystyle P}$, and by ${\displaystyle l'}$ that of ${\displaystyle P'}$; and, lastly, represent by ${\displaystyle u'_{2}}$ and ${\displaystyle u_{1}}$ the values of ${\displaystyle u}$ and ${\displaystyle u'}$ at the place of contact where ${\displaystyle x=0}$, and by ${\displaystyle u_{2}}$ and ${\displaystyle u'_{1}}$ the values of ${\displaystyle u}$ and ${\displaystyle u'}$ at the place of contact where ${\displaystyle x=l}$, we then obtain
 {\displaystyle {\begin{aligned}u'_{2}&=f'(l+l')+c'&u_{1}&=c\\u_{2}&=fl+c&u'_{1}&=f'l+c'.\end{aligned}}}
If we now designate by ${\displaystyle a}$ the tension which occurs at the place of contact where ${\displaystyle x=0}$, and by ${\displaystyle a'}$ that which occurs at the place of contact where ${\displaystyle x=l}$; and if we once for all assume, for the sake of uniformity, that the tension at each individual place of contact always expresses the value which is obtained when we deduct the electroscopic force of one extremity from the electroscopic force of that extremity belonging to the place in question, upon which the abscissa falls before the abrupt change takes place—(it is not difficult to perceive that this general rule contains that advanced in the preceding paragraph, and which, in fact, expresses nothing more than that the tensions of such places of contact, by the springing over of which, in the direction of the abscissæ, we arrive from the greater to the smaller electroscopic force, are to be regarded as positive, in the contrary case as negative, where, however, it must not be overlooked that every positive force has to be taken as greater than every negative, and the negative as greater than the actually smaller), we obtain
 ${\displaystyle a=f'(l+l')+c'-c,}$
and
 ${\displaystyle a'=fl-f'l+c-c',}$
whence directly results
 ${\displaystyle a+a'=fl+f'l'.}$

But now at each of the places of contact when ${\displaystyle \chi }$ and ${\displaystyle \omega }$ represent the power of conduction and the section of the part ${\displaystyle P}$, and ${\displaystyle \chi '}$ and ${\displaystyle \omega '}$ the same for ${\displaystyle P'}$, in accordance with the considerations developed in § 13, there arises the conditional equation
 ${\displaystyle \chi \omega \left({\frac {du}{dx}}\right)=\chi '\omega '\cdot \left({\frac {du'}{dx}}\right),}$
where ${\displaystyle \left({\frac {du}{dx}}\right)}$ and ${\displaystyle \left({\frac {du'}{dx}}\right)}$ represent the values of ${\displaystyle {\frac {du}{dx}}}$ and ${\displaystyle {\frac {du'}{dx}}}$ at the place of contact. From the equations at the commencement of this paragraph for the determination of the electroscopic force in each single part of the circuit, we, however, obtain the value of ${\displaystyle x}$ to be allowed to each,
 ${\displaystyle {\frac {du}{dx}}=f\qquad {\mbox{and}}\qquad {\frac {du'}{dx}}=f',}$
which converts the conditional equation in question into
 ${\displaystyle \chi \omega f=\chi '\omega 'f'.}$
From this, and the equation ${\displaystyle a+a'=fl+f'l'}$ just deduced from the tensions, we now find the values of ${\displaystyle f}$ and ${\displaystyle f'}$ thus;
 {\displaystyle {\begin{aligned}f&={\frac {(a+a')\chi '\omega '}{\chi '\omega 'l+\chi \omega l'}},\\f'&={\frac {(a+a')\chi \omega }{\chi '\omega 'l+\chi \omega l'}},\end{aligned}}}
and with the help of these values we find
 ${\displaystyle c'=c-a'+{\frac {(a+a')(\chi '\omega 'l-\chi \omega l')}{\chi '\omega 'l+\chi \omega l'}}.}$
Hence the electroscopic force of the circuit in the part ${\displaystyle P}$ is expressed by the equation
 ${\displaystyle u={\frac {(a+a')\chi '\omega 'x}{\chi '\omega 'l+\chi \omega l'}}+c,}$
and that in the part ${\displaystyle P'}$ by the equation
 ${\displaystyle u'={\frac {(a+a')\chi \omega x-\chi \omega l+\chi '\omega 'l}{\chi '\omega 'l+\chi \omega l'}}-a+c.}$
If we substitute ${\displaystyle \lambda }$ and ${\displaystyle \lambda '}$ for ${\displaystyle {\frac {l}{\chi \omega }}}$ and ${\displaystyle {\frac {l'}{\chi '\omega '}}}$, the following more simple form may be given to these equations:—
 {\displaystyle \left.{\begin{aligned}u&={\frac {a+a'}{\lambda +\lambda '}}\centerdot {\frac {x}{\chi \omega }}+c\\u'&={\frac {a+a'}{\lambda +\lambda '}}\left({\frac {x-l}{\chi '\omega '}}+{\frac {l}{\chi \omega }}\right)-a'+c\end{aligned}}\right\}} (L).

From the form of these equations it will be immediately perceived, that when the conductibility, or the magnitude of the section, is the same in both parts, the expressions for ${\displaystyle u}$ and ${\displaystyle u'}$ undergo no other change than that the letter representing the conductibility or the section entirely disappears.

17. We will now proceed to the consideration of a galvanic circuit, composed of three distinct parts ${\displaystyle P}$, ${\displaystyle P'}$, and ${\displaystyle P''}$, which case comprises the hydro-circuit.

If we represent by ${\displaystyle u}$, ${\displaystyle u'}$, ${\displaystyle u''}$ respectively the electroscopic forces of the parts ${\displaystyle P}$, ${\displaystyle P'}$, and ${\displaystyle P''}$, then, according to § 15, the case there mentioned being here thrice repeated, we have, in accordance with the equation (c) there found, with respect to the part ${\displaystyle P}$,
 ${\displaystyle u=fx+c,}$
with respect to the part ${\displaystyle P'}$,
 ${\displaystyle u'=f'x+c',}$
and with respect to the part ${\displaystyle P''}$,
 ${\displaystyle u''=f''x+c'',}$
where ${\displaystyle f}$, ${\displaystyle f'}$, ${\displaystyle f''}$, ${\displaystyle c}$, ${\displaystyle c'}$, ${\displaystyle c''}$ may represent any constant magnitudes remaining to be determined from the nature of the problem, and each equation has only so long any meaning as the abscissæ refer to that part to which the equations appertain. If we suppose the origin of the abscissæ at that extremity of the part ${\displaystyle P}$, which is connected with the part ${\displaystyle P''}$, and choose the direction of the abscissæ so that they lead from the part ${\displaystyle P}$ to that of ${\displaystyle P'}$, and from thence into ${\displaystyle P''}$; if we further respectively designate by ${\displaystyle l}$, ${\displaystyle l'}$, and ${\displaystyle l''}$ the lengths of the parts ${\displaystyle P}$, ${\displaystyle P'}$, ${\displaystyle P''}$; and lastly, let ${\displaystyle u''_{2}}$ and ${\displaystyle u_{1}}$ represent the values of ${\displaystyle u''}$ and ${\displaystyle u}$ at the place of contact where ${\displaystyle x=0}$, and ${\displaystyle u_{2}}$ and ${\displaystyle u'_{1}}$ the values of ${\displaystyle u}$ and ${\displaystyle u'}$ at the place of contact where ${\displaystyle x=l}$, and ${\displaystyle u'_{2}}$ and ${\displaystyle u''_{1}}$ the values of ${\displaystyle u'}$ and ${\displaystyle u''}$ at the place of contact where ${\displaystyle x=l+l'}$, then we obtain
 {\displaystyle {\begin{aligned}u''_{2}&=f''(l+l'+l'')+c''&u_{1}&=c\\u_{2}&=fl+c&u'_{1}&=f'l+c'\\u'_{2}&=f'(l+l')+c'&u''_{1}&=f''(l+l')+c''.\end{aligned}}}
If we call ${\displaystyle a}$ the tension which occurs at the place of contact where ${\displaystyle x=0}$, ${\displaystyle a'}$ that at the place of contact where ${\displaystyle x=l}$, and ${\displaystyle a''}$ that at the place of contact where ${\displaystyle x=l+l'}$, we obtain, if we pay due attention to the general rule stated in the preceding paragraph,
 {\displaystyle {\begin{aligned}a&=f''(l+l'+l'')+c''-c\\a'&=fl-f'l+c-c'\\a''&=f'(l+l')-f''(l+l')+c'-c'',\end{aligned}}}
and hence
 ${\displaystyle a+a'+a''=fl+f'l'+f''l''.}$

But from the considerations developed in § 13, when ${\displaystyle \chi }$ and ${\displaystyle \omega }$ represent the power of conduction and the section for the part ${\displaystyle P}$, ${\displaystyle \chi '}$ and ${\displaystyle \omega '}$ the same for the part ${\displaystyle P'}$, and ${\displaystyle \chi ''}$ and ${\displaystyle \omega ''}$ for the part ${\displaystyle P''}$, at the individual places of contact, the following conditional equations are obtained:
 ${\displaystyle \chi \omega \left({\frac {du}{dx}}\right)=\chi '\omega '\left({\frac {du'}{dx}}\right)=\chi ''\omega ''\left({\frac {du''}{dx}}\right),}$
where ${\displaystyle \left({\frac {du}{dx}}\right)}$, ${\displaystyle \left({\frac {du'}{dx}}\right)}$, ${\displaystyle \left({\frac {du''}{dx}}\right)}$ represent the particular values of ${\displaystyle {\frac {du}{dx}}}$, ${\displaystyle {\frac {du'}{dx}}}$, ${\displaystyle {\frac {du''}{dx}}}$, belonging to the places of contact. From the equations stated at the commencement of the present paragraph for the determination of the electroscopic force in the single parts of the circuit, we obtain for every admissible value of ${\displaystyle x}$,
 ${\displaystyle {\frac {du}{dx}}=f,\qquad {\frac {du'}{dx}}=f',\qquad {\frac {du''}{dx}}=f'',}$
by which the preceding conditional equations are converted into
 ${\displaystyle \chi \omega f=\chi '\omega 'f'=\chi ''\omega ''f''.}$
From these, and the equation between ${\displaystyle f}$, ${\displaystyle f'}$, and ${\displaystyle f''}$ above deduced from the tensions, we now find, when ${\displaystyle \lambda }$, ${\displaystyle \lambda '}$, ${\displaystyle \lambda ''}$ are respectively substituted for ${\displaystyle {\frac {l}{\chi \omega }}}$, ${\displaystyle {\frac {l'}{\chi '\omega '}}}$, ${\displaystyle {\frac {l''}{\chi ''\omega ''}}}$,
 {\displaystyle {\begin{aligned}f&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot {\frac {1}{\chi \omega }},\\f'&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot {\frac {1}{\chi '\omega '}},\\f''&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot {\frac {1}{\chi ''\omega ''}}.\end{aligned}}}
and by the aid of these values we find further,
 {\displaystyle {\begin{aligned}c'&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\left({\frac {l}{\chi \omega }}-{\frac {l}{\chi '\omega '}}\right)-a'+c,\\c''&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot \left({\frac {l}{\chi '\omega '}}-{\frac {l+l'}{\chi ''\omega ''}}+{\frac {l}{\chi \omega }}\right)-(a'+a'')+c.\end{aligned}}}
By substituting these values, we obtain for the determination of the electroscopic force of the circuit in the parts ${\displaystyle P}$, ${\displaystyle P'}$, ${\displaystyle P''}$ respectively, the following equations:
 {\displaystyle \left.{\begin{aligned}u&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot {\frac {x}{\chi \omega }}+c\\u'&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot \left({\frac {x-l}{\chi '\omega '}}+{\frac {l}{\chi \omega }}\right)-a'+c\\u''&={\frac {a+a'+a''}{\lambda +\lambda '+\lambda ''}}\centerdot \left({\frac {x-(l+l')}{\chi ''\omega ''}}+{\frac {l'}{\chi '\omega '}}+{\frac {l}{\chi \omega }}\right)-(a'+a'')+c\end{aligned}}\right\}} (L′).

and it is easy to see, that these equations, with the omission of the letter ${\displaystyle \chi }$ or ${\displaystyle \omega }$ (both where they are explicit, as well as in the expressions for ${\displaystyle \lambda }$, ${\displaystyle \lambda '}$, ${\displaystyle \lambda ''}$), are the true ones for the case ${\displaystyle \chi =\chi '}$ , or ${\displaystyle \omega =\omega '=\omega ''}$.

18. These few cases suffice to demonstrate the law of progression of the formulæ ascertained for the electroscopic force, and to comprise them all in a single general expression. To do this with the requisite brevity, for the sake of a more easy and general survey, we will call the quotients, formed by dividing the length of any homogeneous part of the circuit by its power of conduction and its section, the reduced length of this part; and when the entire circuit comes under consideration, or a portion of it, composed of several homogeneous parts, we understand by its reduced length the sum of the reduced lengths of all its parts. Having premised this, all the previously found expressions for the electroscopic force, which arc given by the equations (L) and (L′), may be comprised in the following general statement, which is true when the circuit consists of any number of parts whatever.

The electroscopic force of any place of a galvanic circuit, composed of any number of parts, is found by dividing the sum of all its tensions by its reduced length, multiplying this quotient by the reduced length of the part of the circuit comprised by the abscissa, and subtracting from this product the sum of all the tensions abruptly passed over by the abscissa; lastly, by varying the value thus obtained by a constant magnitude to be determined elsewhere.

If, therefore, we designate by ${\displaystyle A}$ the sum of all the tensions of the circuit, by ${\displaystyle L}$ its entire reduced length, by ${\displaystyle y}$ the reduced length of the part which the abscissa passes through, and by ${\displaystyle O}$ the sum of all the tensions to the points to which the abscissa corresponds, lastly, by ${\displaystyle u}$, the electroscopic force of any place in any part of the circuit, then
 ${\displaystyle u={\frac {A}{L}}y-O+c,}$

where ${\displaystyle c}$