A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 6 of 28

The glow is more easily formed in rare air than in dense air, and more easily when the point is positive than when it is negative.

• See Priestley's History of Electricity, pp. 117 and 591 ; and Cavendish's < Elec- trical Researches,' Phil. Trans., 1771, § 4, or Art. 125 of Reprint of Cavendish.

57-] ELECTRIC SPARK. 57

This and many other differences between positive and negative elec- trification must be studied by those who desire to discover some- thing- about the nature of electricity. They have not, however, been satisfactorily brought to bear upon any existing theory.

The Electric Brush.

56.] The electric brush is a phenomenon which may be pro- duced by electrifying a blunt point or small ball so as to produce an electric field in which the tension diminishes as the distance increases, but in a less rapid manner than when a sharp point is used. It consists of a succession of discharges, ramifying as they diverge from the ball into the air, and terminating either by charging portions of air or by reaching some other conductor. It is accompanied by a sound, the pitch of which depends on the interval between the successive discharges, and there is no current of air as in the case of the glow.

The Electric Spark.

57.] When the tension in the space between two conductors is considerable all the way between them, as in the case of two balls whose distance is not great compared with their radii, the discharge, when it occurs, usually takes the form of a spark, by which nearly the whole electrification is discharged at once.

In this case, when any part of the dielectric has given way, the parts on either side of it in the direction of the electric force are put into a state of greater tension so that they also give way, and so the discharge proceeds right through the dielectric, just as when a little rent is made in the edge of a piece of paper a tension applied to the paper in the direction of the edge causes the paper to be torn through, beginning at the rent, but diverging occasionally where there are weak places in the paper. The electric spark in the same way begins at the point where the electric tension first overcomes the insulation of the dielectric, and proceeds from that point, in an apparently irregular path, so as to take in other weak points, such as particles of dust floating in air.

All these phenomena differ considerably in different gases, and in the same gas at different densities. Some of the forms of electrical discharge through rare gases are exceedingly remarkable. In some cases there is a regular alternation of luminous and dark strata, so that if the electricity, for example, is passing along a tube contain- ing a very small quantity of gas, a number of luminous disks will

58 ELECTROSTATIC PHENOMENA. [58.

be seen arranged transversely at nearly equal intervals along- the axis of the tube and separated by dark strata. If the strength of the current be increased a new disk will start into existence, and it and the old disks will arrange themselves in closer order. In a tube described by Mr. Gassiot* the light of each of the disks is bluish on the negative and reddish on the positive side, and bright red in the central stratum.

These, and many other phenomena of electrical discharge, are exceedingly important, and when they are better understood they will probably throw great light on the nature of electricity as well as on the nature of gases and of the medium pervading space. At present, however, they must be considered as outside the domain of the mathematical theory of electricity.

Electric Phenomena of Tourmaline.

58.] Certain crystals of tourmaline, and of other minerals, possess what may be called Electric Polarity. Suppose a crystal of tour- maline to be at a uniform temperature, and apparently free from electrification on its surface. Let its temperature be now raised, the crystal remaining insulated. One end will be found positively and the other end negatively electrified. Let the surface be de- prived of this apparent electrification by means of a flame or other- wise, then if the crystal be made still hotter,, electrification of the same kind as before will appear, but if the crystal be cooled the end which was positive when the crystal was heated will become negative.

These electrifications are observed at the extremities of the crys- tallographic axis. Some crystals are terminated by a six-sided pyramid at one end and by a three-sided pyramid at the other. In these the end having the six-sided pyramid becomes positive when the crystal is heated.

Sir W. Thomson supposes every portion of these and other hemi- hedral crystals to have a definite electric polarity, the intensity of which depends on the temperature. When the surface is passed through a flame, every part of the surface becomes electrified to such an extent as to exactly neutralize, for all external points, the effect of the internal polarity. The crystal then has no ex- ternal electrical action, nor any tendency to change its mode of electrification. But if it be heated or cooled the interior polariza-

• Intellectual Observer, March, 1866.

59-] pLAN OF THIS TREATISE. 59

tion of each particle of the crystal is altered, and can no longer be balanced by the superficial electrification, so that there is a resultant external action.

Plan of this Treatise.

59.] In the following treatise I propose first to explain the ordinary theory of electrical action, which considers it as depending only on the electrified bodies and on their relative position, with- out taking account of any phenomena which may take place in the intervening media. In this way we shall establish the law of the inverse square, the theory of the potential, and the equations of Laplace and Poisson. We shall next consider the charges and potentials of a system of electrified conductors as connected by a system of equations, the coefficients of which may be supposed to be determined by experiment in those cases in which our present mathematical methods are not applicable, and from these we shall determine the mechanical forces acting between the different elec- trified bodies.

We shall then investigate certain general theorems by which Green, Gauss, and Thomson have indicated the conditions of so- lution of problems in the distribution of electricity. One result of these theorems is, that if Poisson's equation is satisfied by any function, and if at the surface of every conductor the function has the value of the potential of that conductor, then the func- tion expresses the actual potential of the system at every point. We also deduce a method of finding problems capable of exact solution.

In Thomson's theorem, the total energy of the system is ex- pressed in the form of the integral of a certain quantity extended over the whole space between the electrified bodies, and also in the form of an integral extended over the electrified surfaces only. The equality of these two expressions may be thus inter- preted physically. We may conceive the physical relation between the electrified bodies, either as the result of the state of the intervening medium, or as the result of a direct action between the electrified bodies at a distance. If we adopt the latter con- ception, we may determine the law of the action, but we can go no further in speculating on its cause. If, on the other hand, we adopt the conception of action through a medium, we are led to enquire into the nature of that action in each part of the medium.

60 ELECTROSTATIC PHENOMENA. [59.

It appears from the theorem, that if we are to look for the seat of the electric energy in the different parts of the dielectric me- dium, the amount of energy in any small part must depend on the square of the resultant electromotive intensity at that place multiplied by a coefficient called the specific inductive capacity of the medium.

It is better, however, in considering the theory of dielectrics from the most general point of view, to distinguish between the electromotive intensity at any point and the electric polarization of the medium at that point, since these directed quantities, though related to one another, are not, in some solid substances, in the same direction. The most general expression for the electric energy of the medium per unit of volume is half the product of the electromotive intensity and the electric polarization multiplied by the cosine of the angle between their directions. In all fluid dielectrics the electromotive intensity and the electric polarization are in the same direction and in a constant ratio.

If we calculate on this hypothesis the total energy residing in the medium, we shall find it equal to the energy due to the electrification of the conductors on the hypothesis of direct action at a distance. Hence the two hypotheses are mathematically equivalent.

If we now proceed to investigate the mechanical state of the medium on the hypothesis that the mechanical action observed between electrified bodies is exerted through and by means of the medium, as in the familiar instances of the action of one body on another by means of the tension of a rope or the pressure of a rod, we find that the medium must be in a state of mechanical stress.

The nature of this stress is, as Faraday pointed out *, a tension along the lines of force combined with an equal pressure in all directions at right angles to these lines. The magnitude of these stresses is proportional to the energy of the electrification per unit of volume, or, in other words, to the square of the resultant electro- motive intensity multiplied by the specific inductive capacity of the medium.

This distribution of stress is the only one consistent with the observed mechanical action on the electrified bodies, and also with the observed equilibrium of the fluid dielectric which surrounds them. I have therefore thought it a warrantable step in scientific

• Exp. Res., series xi. 1297.

60.] STEESS IN DIELECTRICS. 61

procedure to assume the actual existence of this state of stress, and to follow the assumption into its consequences. Finding the phrase electric tension used in several vague senses, I have attempted to confine it to what I conceive to have been in the mind of some of those who have used it, namely, the state of stress in the dielectric medium which causes motion of the electrified bodies, and leads, when continually augmented, to disruptive .discharge. Electric tension, in this sense, is a tension of exactly the same kind, and measured in the same way, as the tension of a rope, and the dielectric medium, which can support a certain tension and no more, may be said to have a certain strength in exactly the same sense as the rope is said to have a certain strength. Thus, for example, Thomson has found that air at the ordinary pressure and temperature can support an electric tension of 9600 grains weight per square foot before a spark passes.

60.] From the hypothesis that electric action is not a direct action between bodies at a distance, but is exerted by means of the medium between the bodies, we have deduced that this medium must be in a state of stress. We have also ascertained the cha- racter of the stress, and compared it with the stresses which may occur in solid bodies. Along the lines of force there is tension, and perpendicular to them there is pressure, the numerical mag- nitude of these forces being equal, and each proportional to the square of the resultant intensity at the point. Having established these results, we are prepared to take another step, and to form an idea of the nature of the electric polarization of the dielectric medium.

An elementary portion of a body may be said to be polarized when it acquires equal and opposite properties on two opposite sides. The idea of internal polarity may be studied to the greatest advantage as exemplified in permanent magnets, and it will be explained at greater length when we come to treat of magnetism.

The electric polarization of an elementary portion of a dielectric is a forced state into which the medium is thrown by the action of electromotive force, and which disappears when that force is removed. We may conceive it to consist in what we may call an electrical displacement, produced by the electromotive intensity. When the electromotive force acts on a conducting medium it produces a current through it, but if the medium is a non-con- ductor or dielectric, the current cannot flow through the medium, but the electricity is displaced within the medium in the direction

62 ELECTEOSTATIC PHENOMENA. [60.

of the electromotive intensity, the extent of this displacement depending- on the magnitude of the electromotive intensity, so that if the electromotive intensity increases or diminishes, the electric displacement increases and diminishes in the same ratio.

The amount of the displacement is measured by the quantity of electricity which crosses unit of area, while the displacement increases from zero to its actual amount. This, therefore, is the measure of the electric polarization.

The analogy between the action of electromotive force in pro- ducing electric displacement and of ordinary mechanical force in producing the displacement of an elastic body is so obvious that I have ventured to call the ratio of the electromotive intensity to the corresponding electric displacement the coefficient of electric elasticity of the medium. This coefficient is different in different media, and varies inversely as the specific inductive capacity of each medium.

The variations of electric displacement evidently constitute electric currents. These currents, however, can only exist during the variation of the displacement, and therefore, since the displace- ment cannot exceed a certain value without causing disruptive discharge, they cannot be continued indefinitely in the same direc- tion, like the currents through conductors.

In tourmaline, and other pyro-electric crystals, it is probable that a state of electric polarization exists, which depends upon tem- perature, and does not require an external electromotive force to produce it. If the interior of a body were in a state of permanent electric polarization, the outside would gradually become charged in such a manner as to neutralize the action of the internal polarization for all points outside the body. This external super- ficial charge could not be detected by any of the ordinary tests, and could not be removed by any of the ordinary methods for discharging superficial electrification. The internal polarization of the substance would therefore never be discovered unless by some means, such as change of temperature, the amount of the internal polarization could be increased or diminished. The external elec- trification would then be no longer capable of neutralizing the external effect of the internal polarization, and an apparent elec- trification would be observed, as in the case of tourmaline.

If a charge e is uniformly distributed over the surface of a sphere, the resultant force at any point of the medium surrounding the sphere is numerically equal to the charge e divided by the square of

60.] ELECTRIC DISPLACEMENT. 63

the distance from the centre of the sphere. This resultant force, according- to our theory, is accompanied by a displacement of elec- tricity in a direction outwards from the sphere.

If we now draw a concentric spherical surface of radius r, the whole displacement, U, through this surface will be proportional to the resultant force multiplied by the area of the spherical surface. -"^ ' ' But the resultant force is directly as the charge e and inversely as IL^ the square of the radius, while the area of the surface is directly *"' as the square of the radius.

Hence the whole displacement, E, is proportional to the charge ey and is independent of the radius.

To determine the ratio between the charge £, and the quantity of electricity, E, displaced outwards through any one of the spherical surfaces, let us consider the work done upon the medium in the region between two concentric spherical surfaces, while the displacement is increased from E to E-}- §E. If Tt and Fz denote the potentials at the inner and the outer of these surfaces respect- ively, the electromotive force by which the additional displacement is produced is 7^— F2, so that the work spent in augmenting the displacement is (V^—V^E.

If we now make the inner surface coincide with that of the electrified sphere, and make the radius of the other infinite, V± becomes T9 the potential of the sphere, and F"2 becomes zero, so that the whole work done in the surrounding medium is VbE.

But by the ordinary theory, the work done in augmenting the charge is Fbe, and if this is spent, as we suppose, in augmenting the displacement, b^E—be, and since E and e vanish together, E = e, or —

The displacement outwards through any spherical surface concentric with the sphere is equal to the charge on the sphere.

To fix our ideas of electric displacement, let us consider an accu- mulator formed of two conducting plates A and 5, separated by a stratum of a dielectric C. Let W be a conducting wire joining A and .Z?, and let us suppose that by the action of an electromotive force a quantity Q of positive electricity is transferred along the wire from B to A. The positive electrification of A and the negative electrification of B will produce a certain electromotive force acting from A towards B in the dielectric stratum, and this will produce an electric displacement from A towards B within the dielectric. The amount of this displacement, as measured by the quantity of electricity forced across an imaginary section of the

64: ELECTROSTATIC PHENOMENA. [6 1.

dielectric dividing it into two strata, will be, according to our theory, exactly Q. See Arts. 75, 76, 111.

It appears, therefore, that at the same time that a quantity Q of electricity is being transferred along the wire by the electro- motive force from B towards A, so as to cross every section of the wire, the same quantity of electricity crosses every section of the dielectric from A towards £ by reason of the electric dis- placement.

The displacements of electricity during the discharge of the accu- mulator will be the reverse of these. In the wire the discharge will be Q from A to .Z?, and in the dielectric the displacement will subside, and a quantity of electricity Q will cross every section from B towards A.

Every case of charge or discharge may therefore be considered as a motion in a closed circuit, such that at every section of the circuit the same quantity of electricity crosses in the same time, and this is the case, not only in the voltaic circuit where it has always been recognised, but in those cases in which elec- tricity has been generally supposed to be accumulated in certain places.

61.] We are thus led to a very remarkable consequence of the theory which we are examining, namely, that the motions of elec- tricity are like those of an incompressible fluid, so that the total quantity within an imaginary fixed closed surface remains always the same. This result appears at first sight in direct contradiction to the fact that we can charge a conductor and then introduce it into the closed space, and so alter the quantity of electricity within that space. But we must remember that the ordinary theory takes no account of the electric displacement in the sub- stance of dielectrics which we have been investigating, but confines its attention to the electrification at the bounding surfaces of the conductors and dielectrics. In the case of the charged conductor let us suppose the charge to be positive, then if the surrounding dielectric extends on all sides beyond the closed surface there will be electric polarization, accompanied with displacement from within outwards all over the closed surface, and the surface-integral of the displacement taken over the surface will be equal to the charge on the conductor within.

Thus when the charged conductor is introduced into the closed space there is immediately a displacement of a quantity of elec- tricity equal to the charge through the surface from within out-

62.] THEORY PROPOSED. 65

wards, and the whole quantity within the surface remains the same.

The theory of electric polarization will be discussed at greater length in Chapter V, and a mechanical illustration of it will be given in Art. 334, but its importance cannot be fully understood till we arrive at the study of electromagnetic phenomena.

62.] The peculiar features of the theory are : —

That the energy of electrification resides in the dielectric medium, whether that medium be solid, liquid, or gaseous, dense or rare, or even what is called a vacuum, provided it be still capable of transmitting electrical action.

That the energy in any part of the medium is stored up in the form of a state of constraint called electric polarization, the amount of which depends on the resultant electromotive intensity at the place.

That electromotive force acting on a dielectric produces what we have called electric displacement, the relation between the in- tensity and the displacement being in the most general case of a kind to be afterwards investigated in treating of conduction, but in the most important cases the displacement is in the same direc-' tion as the force, and is numerically equal to the intensity mul- tiplied by — K, where K is the specific inductive capacity of the

dielectric. /

That the energy per unit of volume of the dielectric arising from ^^~ the electric polarization is half the product of the electromotive ..: * intensity and the electric displacement, multiplied, if necessary, by the cosine of the angle between their directions.

That in fluid dielectrics the electric polarization is accompanied by a tension in the direction of the lines of induction, combined with an equal pressure in all directions at right angles to the lines of induction, the tension or pressure per unit of area being numerically equal to the energy per unit of volume at the same place.

That the surface of any elementary portion into which we may conceive the volume of the dielectric divided must be conceived to be charged so that the surface-density at any point of the surface is equal in magnitude to the displacement through that point of the surface reckoned inwards. If the displacement is in the positive direction, the surface of the element will be charged negatively on the positive side of the element, and positively on

VOL. i. p

66 ELECTROSTATIC PHENOMENA, [62.

the negative side. These superficial charges will in general destroy one another when consecutive elements are considered, except where the dielectric has an internal charge, or at the surface of the dielectric.

That whatever electricity may be, and whatever we may under- stand by the movement of electricity, the phenomenon which we have called electric displacement is a movement of electricity in the same sense as the transference of a definite quantity of electricity through a wire is a movement of electricity, the only difference being that in the dielectric there is a force which we have called electric elasticity which acts against the electric displacement, and forces the electricity back when the electromotive force is removed; whereas in the conducting wire the electric elasticity is continually giving way, so that a current of true conduction is set up, and the resistance depends, not on the total quantity of electricity dis- placed from its position of equilibrium, but on the quantity which crosses a section of the conductor in a given time.

That in every case the motion of electricity is subject to the same condition as that of an incompressible fluid, namely, that at every instant as much must flow out of any given closed surface as flows into it.

It follows from this that every electric current must form a closed circuit. The importance of this result will be seen when we investigate the laws of electro-magnetism.

Since, as we have seen, the theory of direct action at a distance is mathematically identical with that of action by means of a medium, the actual phenomena may be explained by the one theory as well as by the other, provided suitable hypotheses be introduced when any difficulty occurs. Thus, Mossotti has .deduced the mathematical theory of dielectrics from the ordinary theory of attraction merely by giving an electric instead of a magnetic interpretation to the symbols in the investigation by which Poisson has deduced the theory of magnetic induction from the theory of magnetic fluids. He assumes the existence within the dielectric of small conducting elements, capable of having their opposite surfaces oppositely electrified by induction, but not capable of losing or gaining electricity on the whole, owing to their being insulated from each other by a non-conducting medium. This theory of dielectrics is consistent with the laws of electricity, and may be actually true. If it is true, the specific inductive capacity of a dielectric may be greater, but cannot be less, than that of a

62.] METHOD OF THIS WORK. 67

vacuum. No instance has yet been found of a dielectric having an inductive capacity less than that of a vacuum, but if such should be discovered, Mossotti's physical theory must be abandoned, although his formulae would all remain exact, and would only require us to alter the sign of a coefficient.

In many parts of physical science, equations of the same form are found applicable to phenomena which are certainly of quite different natures, as, for instance, electric induction through di- electrics, conduction through conductors, and magnetic induction. In all these cases the relation between the force and the effect produced is expressed by a set of equations of the same kind, so that when a problem in one of these subjects is solved, the problem and its solution may be translated into the language of the other subjects and the results in their new form will still be true.

CHAPTEE II.

ELEMENTARY MATHEMATICAL THEORY OF STATICAL ELECTRICITY.

Definition of Electricity as a Mathematical Quantity.

63.] We have seen that the properties of charged bodies are such that the charge of one body may be equal to that of an- other, or to the sum of the charges of two bodies, and that when two bodies are equally and oppositely charged they have no elec- trical effect on external bodies when placed together within a closed insulated conducting vessel. We may express all these results in a concise and consistent manner by describing an electrified body as charged with a certain quantity of electricity, which we may denote by e. When the charge is positive, that is, according to the usual convention, vitreous, e will be a positive quantity. When the charge is negative or resinous, e will be negative, and the quantity — e may be interpreted either as a negative quantity of vitreous electricity or as a positive quantity of resinous electricity.

The effect of adding together two equal and opposite charges of electricity, +£ and — e, is to produce a state of no charge expressed by zero. We may therefore regard a body not charged as virtually charged with equal and opposite charges of indefinite magnitude, and a charged body as virtually charged with unequal quantities of positive and negative electricity, the algebraic sum of these charges constituting the observed electrification. It is manifest, however, that this way of regarding an electrified body is entirely artificial, and may be compared to the conception of the velocity of a body as compounded of two or more different velocities, no one of which is the actual velocity of the body.

ON ELECTRIC DENSITY.

Distribution in Three Dimensions.

64.] Definition. The electric volume-density at a given point in space is the limiting ratio of the quantity of electricity within

64.] ELECTRIC DENSITY. 69

a sphere whose centre is the given point to the volume of the sphere, when its radius is diminished without limit.

We shall denote this ratio by the symbol p, which may be posi- tive or negative.

Distribution over a Surface.

It is a result alike of theory and of experiment, that, in certain cases, the charge of a body is entirely on the surface. The density at a point on the surface, if defined according to the method given above, would be infinite. We therefore adopt a different method for the measurement of surface-density.

Definition. The electric density at a given point on a surface is the limiting ratio of the quantity of electricity within a sphere whose centre is the given point to the area of the surface contained within the sphere, when its radius is diminished without limit.

We shall denote the surface-density by the symbol <r.

Those writers who supposed electricity to be a material fluid or a collection of particles, were obliged in this case to suppose the electricity distributed on the surface in the form of a stratum of a certain thickness 0, its density being /o0, or that value of p which would result from the particles having the closest contact of which they are capable. It is manifest that on this theory

PQ e = o-.

When a- is negative, according to this theory, a certain stratum of thickness 0 is left entirely devoid of positive electricity, and filled entirely with negative electricity, or, on the theory of one fluid, with matter.

There is, however, no experimental evidence either of the elec- tric stratum having any thickness, or of electricity being a fluid or a collection of particles. We therefore prefer to do without the symbol for the thickness of the stratum, and to use a special symbol for surface-density.

Distribution on a Line.

It is sometimes convenient to suppose electricity distributed on a line, that is, a long narrow body of which we neglect the thickness. In this case we may define the line-density at any point to be the limiting ratio of the charge on an element of the line to the length of that element when the element is diminished without limit.

70 ELECTROSTATICS. [65.

If A denotes the line-density, then the whole quantity of elec- tricity on a curve is e = / A ds, where ds is the element of the curve.

Similarly, if o- is the surface-density, the whole quantity of elec- tricity on the surface is

where dS is the element of surface.

If p is the volume-density at any point of space, then the whole electricity within a certain volume is

e = / / / p dx dy dz.

where dx dy dz is the element of volume. The limits of integration in each case are those of the curve, the surface, or the portion of space considered.

It is manifest that £, X, a- and p are quantities differing in kind, each being one dimension in space lower than the preceding, so that if I be a line, the quantities 0, l, I2vy and 1B p will be all of the same kind, and if [Z] be the unit of length, and [A.], [o-], [p] the units of the different kinds of density, [e], [-Z/A], [X2<r], and [^/3/>] will each denote one unit of electricity,

Definition of the Unit of Electricity.

65.] Let A and B be two points the distance between which is the unit of length. Let two bodies, whose dimensions are small compared with the distance AS, be charged with equal quantities of positive electricity and placed at A and B respectively, and let the charges be such that the force with which they repel each other is the unit of force, measured as in Art. 6. Then the charge of either body is said to be the unit of electricity.

If the charge of the body at B were a unit of negative electricity, then, since the action between the bodies would be reversed, we should have an attraction equal to the unit of force. If the charge of A were also negative, and equal to unity, the force would be repulsive, and equal to unity.

Since the action between any two portions of electricity is not affected by the presence of other portions, the repulsion between e units of electricity at A and e' units at B is ee\ the distance AB being unity. See Arfe 39.

Law of Force between Charged Bodies. 66.] Coulomb shewed by experiment that the force between.

68.] LAW OF ELECTRIC FORCE. 71

charged bodies whose dimensions are small compared with the distance between them, varies inversely as the square of the dis- tance. Hence the repulsion between two such bodies charged with quantities e and / and placed at a distance r is

*/ r2 '

We shall prove in Art. 74 that this law is the only one con- sistent with the observed fact that a conductor, placed in the inside of a closed hollow conductor and in contact with it, is deprived of all electrical charge. Our conviction of the accuracy of the law of the inverse square of the distance may be considered to rest on experiments of this kind, rather than on the direct measure- ments of Coulomb.

Resultant Force letiveen Two Bodies.

67.] In order to calculate the resultant force between two bodies we might divide each of them into its elements of volume, and consider the repulsion between the electricity in each of the elements of the first body and the electricity in. each of the elements of the second body. We should thus get a system of forces equal in number to the product of the numbers of the elements into which we have divided each body, and we should have to combine the effects of these forces by the rules of Statics. Thus, to find the component in the direction of x we should have to find the value of the sextuple integral

r r rr r r ? />>-#') dx ay dz ax'atfM

JJJJJJ {(«-aOa + (y-yj + (-/)2} '

where #, y, z are the coordinates of a point in the first body at which the electrical density is p, and #', if, z, and p' are the corresponding quantities for the second body, and the integration is extended first over the one body and then over the other.

Resultant Intensity at a Point.