A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 4 of 28

dX /dz dx dz dx\ dX /dx dy dx dy~)., , , .

24.] LINE INTEGRAL AND SURFACE INTEGRAL. 27

,,. . dX dx dx , .

adding- and subtracting — — — , this becomes

dx ,dXdx dXdy dX dz\ dfi \dx da dy da dz do)

dx ,dX dx dX dy dX

, .

Let us now suppose that the curves for which a is constant form a series of closed curves surrounding a point on the surface for which a has its minimum value, a0, and let the last curve of the series, for which a = c^ , coincide with the closed curve 5.

Let us also suppose that the curves for which /3 is constant form a series of lines drawn from the point at which a = a0 to the closed curve *, the first, /30, and the last, f3lt being identical.

Integrating (8) by parts, the first term with respect to a and the second with respect to /3, the double integrals destroy each other and the expression becomes

(-f,U/X\ j f x

X — J da. (9)

Since the point (a, fa) is identical with the point (a, /30), the third and fourth terms destroy each other ; and since there is but one value of x at the point where a = a0, the second term is zero, and the expression is reduced to the first term :

Since the curve a = a1 is identical with the closed curve *, we may write the expression in the form

dx 7 ,., .

-J-&, (10)

where the integration is to be performed round the curve s. We may treat in the same way the parts of the surface-integral which depend upon T and Z^ so that we get finally,

where the first integral is extended over the surface $, and the second round the bounding curve s *.

• This theorem was given by Professor Stokes, Smith's Prize Examination, 1854, question 8. It is proved in Thomson and Tait's Natural Philosophy, § 190 (f).

28 PRELIMINARY. [25.

On the effect of the operator V on a vector function.

25.] We have seen that the operation denoted by V is that by which a vector quantity is deduced from its potential. The same operation, however, when applied to a vector function, produces results which enter into the two theorems we have just proved (III and IV). The extension of this operator to vector displace- ments, and most of its further development, is due to Professor Tait*.

Let a be a vector function of />, the vector of a variable point. Let us suppose, as usual, that

p = ix+jy + Jcz, and o- = iX+jY + kZ\

where X, Y, Z are the components of cr in the directions of the axes.

We have to perform on <r the operation

. d .d , d V = t-r- +JJ- + K-J-*

dx dy dz

Performing this operation, and remembering the rules for the multiplication of it j, ft, we find that Vo- consists of two parts, one scalar and the other vector. The scalar part is

,dX dT dZ^ -, TTT

$V(r = — ( -= — f- -r= — h -T-) 5 see Theorem 111, ^dx dy dz'

and the vector part is

dY .dX dZ dY dX.

If the relation between X, Y, Z and f, 77, f is that given by equation (1) of the last theorem, we may write

VV<y = ig+jri + &C See Theorem IV.

It appears therefore that the functions of X, Y, Z which occur in the two theorems are both obtained by the operation V on the vector whose components are X, 7, Z. The theorems themselves may be written

8 V ads=

Iff and fSffdp =s[fs.V<rUvd*\ (IV)

• See Proc. R. S. Edin., April 28, 1862. ' On Green's and other allied Theorems,' Trans. R. 8. Edin., 1869-70, a very valuable paper; and 'On some Quaternion Integrals,' Proc. R. 8. Edin., 1870-71.

26.] HAMILTON'S OPERATOR V. 29

where ds is an element of a volume, ds of a surface, dp of a curve, and Uv a unit- vector in the direction of the normal.

To understand the meaning of these functions of a vector, let us suppose that o-0 is the value of o- at a point P, and let us examine the value of o-— aQ in the neighbourhood of P. If we draw a closed surface round P, then, if the I

surface-integral of <r over this surface is directed \ I / inwards, xSVo- will be positive, and the vector <r— o-0 near the point P will be on the whole p

directed towards P, as in the figure (l). S ik ^\

I propose therefore to call the scalar part of V<r the convergence of a- at the point P. Fig. 1.

To interpret the vector part of Vo-, let us suppose ourselves to be looking in the direction of the vector whose components are f, 77, £ and let us examine the vector a— o-0 near the point P. It will appear "* —

as in the figure (2), this vector being arranged on the whole tangentially in the direction opposite to _ ^

the hands of a watch. ™ 2

I propose (with great diffidence) to call the vector part of Vo- the rotation of o- at the point P.

In Fig. 3 we have an illustration of rotation com- /

bined with convergence. \

Let us now consider the meaning of the equation \

FVo- =0. /

This implies that V o- is a scalar, or that the vector Fig. 3. o- is the space- variation of some scalar function #.

26.] One of the most remarkable properties of the operator V is that when repeated it becomes

,d* d^ d*\ ~ + + '

an operator occurring in all parts of Physics, which we may refer to as Laplace's Operator.

This operator is itself essentially scalar. When it acts on a scalar function the result is scalar, when it acts on a vector function the result is a vector.

If, with any point P as centre, we draw a small sphere whose radius is r, then if qQ is the value of q at the centre, and q the mean value of q for all points within the sphere,

30 PEELIMINARY. [26.

so that the value at the centre exceeds or falls short of the mean value according as V2 q is positive or negative.

I propose therefore to call V2 q the concentration of q at the point P, because it indicates the excess of the value of q at that point over its mean value in the neighbourhood of the point.

If q is a scalar function, the method of finding its mean value is well known. If it is a vector function, we must find its mean value by the rules for integrating vector functions. The result of course is a vector.

PART I.

ELECTROSTATICS. CHAPTEE I.

DESCRIPTION OF PHENOMENA.

Electrification ly Friction.

27.] EXPERIMENT I*. Let a piece of glass and a piece of resin, neither of which exhibits any electrical properties, be rubbed to- gether and left with the rubbed surfaces in contact. They will still exhibit no electrical properties. Let them be separated. They will now attract each other.

If a second piece of glass be rubbed with a second piece of resin, and if the pieces be then separated and suspended in the neighbourhood of the former pieces of glass and resin, it may be observed —

(1) That the two pieces of glass repel each other.

(2) That each piece of glass attracts each piece of resin.

(3) That the two pieces of resin repel each other.

These phenomena of attraction and repulsion are called Elec- trical phenomena, and the bodies which exhibit them are said to be electrified, or to be charged with electricity.

Bodies may be electrified in many other ways, as well as by friction.

The electrical properties of the two pieces of glass are similar to each other but opposite to those of the two pieces of resin : the glass attracts what the resin repels and repels what the resin attracts.

• See Sir W. Thomson ' On the Mathematical Theory of Electricity,' Cambridge and Dublin Mathematical Journal, March, 1848.

32 ELECTROSTATIC PHENOMENA. [28.

If a body electrified in any manner whatever behaves as the glass does, that is, if it repels the glass and attracts the resin, the body is said to be vitreously electrified, and if it attracts the glass and repels the resin it is said to be resmously electrified. All electrified bodies are found to be either vitreously or resinously electrified.

It is the established practice of men of science to call the vitreous electrification positive, and the resinous electrification negative. The exactly opposite properties of the two kinds of electrification justify us in indicating them by opposite signs, but the applica- tion of the positive sign to one rather than to the other kind must be considered as a matter of arbitrary convention, just as it is a matter of convention in mathematical diagrams to reckon positive distances towards the right hand.

No force, either of attraction or of repulsion, can be observed between an electrified body and a body not electrified. When, in any case, bodies not previously electrified are observed to be acted on by an electrified body, it is because they have become electrified

Electrification ly Induction.

28.] EXPERIMENT II*. Let a hollow vessel of metal be hung up by white silk threads, and let a similar thread be attached to the lid of the vessel so that the vessel may be opened or closed without touching it.

Let the pieces of glass and resin be similarly sus- pended and electrified as before.

Let the vessel be originally unelectrified, then if an electrified piece of glass is hung up within it by its thread without touching the vessel, and the lid closed, the outside of the vessel will be found to be vitreously electrified, and it may be shewn that the electrification outside of the vessel is exactly the Fig. 4. same in whatever part of the interior space the glass

is suspended.

If the glass is now taken out of the vessel without touching it, the electrification of the glass will be the same as before it was put in, and that of the vessel will have disappeared.

This electrification of the vessel, which depends on the glass

• This, and several experiments which follow, are due to Faraday, 'On Static Electrical Inductive Action,' Phil, Mag., 1843, or Exp. Res., vol. ii. p. 279.

2Q-] ELECTRIFICATION. 33

being within it, and which vanishes when the glass is removed, is called electrification by Induction.

Similar effects would be produced if the glass were suspended near the vessel on the outside, but in that case we should find an electrification, vitreous in one part of the outside of the vessel and resinous in another. When the glass is inside the vessel the whole of the outside is vitreously and the whole of the inside resinously electrified.

'Electrification by Conduction.

29.] EXPERIMENT III. Let the metal vessel be electrified by induction, as in the last experiment, let a second metallic body be suspended by white silk threads near it, and let a metal wire, similarly suspended, be brought so as to touch simultaneously the electrified vessel and the second body.

The second body will now be found to be vitreously electrified, and the vitreous electrification of the vessel will have diminished.

The electrical condition has been transferred from the vessel to the second body by means of the wire. The wire is called a con- ductor of electricity, and the second body is said to be electrified ly conduction.

Conductors and Insulators.

EXPERIMENT IV. If a glass rod, a stick of resin or gutta-percha, or a white silk thread, had been used instead of the metal wire, no transfer of electricity would have taken place. Hence these latter substances are called Non-conductors of electricity. Non-conduc- tors are used in electrical experiments to support electrified bodies without carrying off their electricity. They are then called In- sulators.

The metals are good conductors ; air, glass, resins, gutta-percha, vulcanite, paraffin, &c. are good insulators; but, as we shall see afterwards, all substances resist the passage of electricity, and all substances allow it to pass, though in exceedingly different degrees. This subject will be considered when we come to treat of the motion of electricity. For the present we shall consider only two classes of bodies, good conductors, and good insulators.

In Experiment II an electrified body produced electrification in the metal vessel while separated from it by air, a non-conducting medium. Such a medium, considered as transmitting these electrical effects without conduction, has been called by Faraday a Dielectric

VOL. I. D

34 ELECTROSTATIC PHENOMENA. [30.

medium, and the action which takes place through it is called Induction.

In Experiment III the electrified vessel produced electrification in the second metallic body through the medium of the wire. Let us suppose the wire removed, and the electrified piece of glass taken out of the vessel without touching it, and removed to a sufficient distance. The second body will still exhibit vitreous electrifica- tion, but the vessel, when the glass is removed, will have resinous electrification. If we now bring the wire into contact with both bodies, conduction will take place along the wire, and all electri- fication will disappear from both bodies, shewing that the elec- trification of the two bodies was equal and opposite.

30.] EXPERIMENT V. In Experiment II it was shewn that if a piece of glass, electrified by rubbing it with resin, is hung up in an insulated metal vessel, the electrification observed outside does not depend on the position of the glass. If we now introduce the piece of resin with which the glass was rubbed into the same vessel, without touching it or the vessel, it will be found that there is no electrification outside the vessel. From this we conclude that the electrification of the resin is exactly equal and opposite to that of the glass. By putting in any number of bodies, electrified in any way, it may be shewn that the electrification of the outside of the vessel is that due to the algebraic sum of all the electrifica- tions, those being reckoned negative which are resinous. We have thus a practical method of adding the electrical effects of several bodies without altering the electrification of each.

31.] EXPERIMENT VI. Let a second insulated metallic vessel, J5, be provided, and let the electrified piece of glass be put into the first vessel A, and the electrified piece of resin into the second vessel _Z?. Let the two vessels be then put in communication by the metal wire, as in Experiment III. All signs of electrification will dis- appear.

Next, let the wire be removed, and let the pieces of glass and of resin be taken out of the vessels without touching them. It will be found that A is electrified resinously and B vitreously.

If now the glass and the vessel A be introduced together into a larger insulated vessel C, it will be found that there is no elec- trification outside C. This shews that the electrification of A is exactly equal and opposite to that of the piece of glass, and that of B may be shewn in the same way to be equal and opposite to that of the piece of resin.

33-] SUMMATION OF ELECTRIC EFFECTS. 35

We have thus obtained a method of charging a vessel with a quantity of electricity exactly equal and opposite to that of an electrified body without altering the electrification of the latter, and we may in this way charge any number of vessels with exactly equal quantities of electricity of either kind, which we may take for provisional units.

32.] EXPERIMENT VII, Let the vessel B, charged with a quan- tity of positive electricity, which we shall call, for the present, unity, be introduced into the larger insulated vessel C without touching it. It will produce a positive electrification on the out- side of C. Now let B be made to touch the inside of C. No change of the external electrification will be observed. If B is now taken out of C without touching it, and removed to a sufficient distance, it will be found that B is completely discharged, and that C has become charged with a unit of positive electricity.

We have thus a method of transferring the charge of B to C.

Let B be now recharged with a unit of electricity, introduced into C already charged, made to touch the inside of C, and re- moved. It will be found that B is again completely discharged, so that the charge of C is doubled.

If this process is repeated, it will be found that however highly C is previously charged, and in whatever way B is charged, when B is first entirely enclosed in C, then made to touch (7, and finally removed without touching C, the charge of B is completely trans- ferred to C, and B is entirely free from electrification.

This experiment indicates a method of charging a body with any number of units of electricity. We shall find, when we come to the mathematical theory of electricity, that the result of this experiment affords an accurate test of the truth of the theory.

33.] Before we proceed to the investigation of the law of electrical force, let us enumerate the facts we have already estab- lished.

By placing any electrified system inside an insulated hollow con- ducting vessel, and examining the resultant effect on the outside of the vessel, we ascertain the character of the total electrification •of the system placed inside, without any communication of elec- tricity between the different bodies of the system.

The electrification of the outside of the vessel may be tested with great delicacy by putting it in communication with an elec- troscope.

We may suppose the electroscope to consist of a strip of gold

36 ELECTROSTATIC PHENOMENA. [34.

leaf hanging4 between two bodies charged, one positively, and the other negatively. If the gold leaf becomes electrified it will incline towards the body whose electrification is opposite to its own. By increasing the electrification of the two bodies and the delicacy of the suspension, an exceedingly small electrification of the gold leaf may be detected.

When we come to describe electrometers and multipliers we shall find that there are still more delicate methods of detecting electrification and of testing the accuracy of our theories, but at present we shall suppose the testing to be made by connecting the hollow vessel with a gold leaf electroscope.

This method was used by Faraday in his very admirable de- monstration of the laws of electrical phenomena *.

34.] I. The total electrification of a body, or system of bodies, remains always the same, except in so far as it receives electrifi- cation from or gives electrification to other bodies.

In all electrical experiments the electrification of bodies is found to change, but it is always found that this change is due to want of perfect insulation, and that as the means of insulation are im- proved, the loss of electrification becomes less. We may therefore assert that the electrification of a body placed in a perfectly in- sulating medium would remain perfectly constant.

II. When one body electrifies another by conduction, the total electrification of the two bodies remains the same, that is, the one loses as much positive or gains as much negative electrification as the other gains of positive or loses of negative electrification.

For if the two bodies are enclosed in the hollow vessel, no change of the total electrification is observed.

III. When electrification is produced by friction, or by any other known method, equal quantities of positive and negative elec- trification are produced.

For the electrification of the whole system may be tested in the hollow vessel, or the process of electrification may be carried on within the vessel itself, and however intense the electrification of the parts of the system may be, the electrification of the whole, as indicated by the gold leaf electroscope, is invariably zero.

The electrification of a body is therefore a physical quantity capable of measurement, and two or more electrifications can be combined experimentally with a result of the same kind as when

• 'On Static Electrical Inductive Action,' Phil. Mag., 1843, or Exp. Res., vol. ii. p. 249.

35-] ELECTRICITY AS A QUANTITY. 37

two quantities are added algebraically. We therefore are entitled to use language fitted to deal with electrification as a quantity as well as a quality, and to speak of any electrified body as ' charged with a certain quantity of positive or negative electricity.'

35.] While admitting electricity, as we have now done, to the rank of a physical quantity, we must not too hastily assume that it is, or is not, a substance, or that it is, or is not, a form of energy, or that it belongs to any known category of physical quantities. All that we have hitherto proved is that it cannot be created or annihilated, so that if the total quantity of elec- tricity within a closed surface is increased or diminished, the in- crease or diminution must have passed in or out through the closed surface.

This is true of matter, and is expressed by the equation known as the Equation of Continuity in Hydrodynamics.

It is not true of heat, for heat may be increased or diminished within a closed surface, without passing in or out through the surface, by the transformation of some other form of energy into heat, or of heat into some other form of energy.

It is not true even of energy in general if we admit the imme- diate action of bodies at a distance. For a body outside the closed surface may make an exchange of energy with a body within the surface. But if all apparent action at a distance is the result of the action between the parts of an intervening medium, it is conceivable that in all cases of the increase or diminution of the energy within a closed surface we may be able, when the nature of this action of the parts of the medium is clearly under- stood, to trace the passage of the energy in or out through that surface.

There is, however, another reason which warrants us in asserting that electricity, as a physical quantity, synonymous with the total electrification of a body, is not, like heat, a form of energy. An electrified system has a certain amount of energy, and this energy can be calculated by multiplying the quantity of electricity in each of its parts by another physical quantity, called the Potential of that part, and taking half the sum of the products. The quan- tities ' Electricity ' and ' Potential,' when multiplied together, produce the quantity ' Energy.' It is impossible, therefore, that electricity and energy should be quantities of the same category, for electricity is only one of the factors of energy, the other factor being ' Potential.'

38 ELECTROSTATIC PHENOMENA. [36.

Energy, which is the product of these factors, may also be con- sidered as the product of several other pairs of factors, such as A Force x A distance through which the force is to act.

A Mass x Gravitation acting through a certain height.

A Mass X Half the square of its velocity.

A Pressure x A volume of fluid introduced into a vessel at

that pressure. A Chemical Affinity x A chemical change, measured by the number

of electro-chemical equivalents which enter

into combination.

If we ever should obtain distinct mechanical ideas of the nature of electric potential, we may combine these with the idea of energy to determine the physical category in which ' Electricity ' is to be placed.

36.] In most theories on the subject, Electricity is treated as a substance, but inasmuch as there are two kinds of electrification which, being combined, annul each other, and since we cannot conceive of two substances annulling each other, a distinction has been drawn between Free Electricity and Combined Electricity.

Theory of Two Fluids.

In what is called the Theory of Two Fluids, all bodies, in their unelectrified state, are supposed to be charged with equal quan- tities of positive and negative electricity. These quantities are supposed to be so great that no process of electrification has ever yet deprived a body of all the electricity of either kind. The pro- cess of electrification, according to this theory, consists in taking a certain quantity P of positive electricity from the body A and communicating it to .5, or in taking a quantity N of negative electricity from B and communicating it to A, or in some com- bination of these processes.

The result will be that A will have P + N units of negative electricity over and above its remaining positive electricity, which is supposed to be in a state of combination with an equal quantity of negative electricity. This quantity P -f N is called the Free elec- tricity, the rest is called the Combined, Latent, or Fixed electricity.

In most expositions of this theory the two electricities are called

• Fluids,' because they are capable of being transferred from one body to another, and are, within conducting bodies, extremely mobile. The other properties of fluids, such as their inertia,

36.] THEOEY OF TWO FLUIDS. 39

weight, and elasticity, are not attributed to them by those who have used the theory for merely mathematical purposes; but the use of the word Fluid has been apt to mislead the vulgar, including many men of science who are not natural philosophers, and who have seized on the word Fluid as the only term in the statement of the theory which seemed intelligible to them.

We shall see that the mathematical treatment of the subject has been greatly developed by writers who express themselves in terms of the ' Two Fluids ' theory. Their results, however, have been deduced entirely from data which can be proved by experiment, and which must therefore be true, whether we adopt the theory of two fluids or not. The experimental verification of the mathe- matical results therefore is no evidence for or against the peculiar doctrines of this theory.

The introduction of two fluids permits us to consider the negative electrification of A and the positive electrification of B as the effect of any one of three different processes which would lead to the same result. We have already supposed it produced by the transfer of P units of positive electricity from A to B, together with the transfer of N units of negative electricity from B to A. But if P + N units of positive electricity had been transferred from A to B, or if P + N units of negative electricity had been transferred from B to A, the resulting ' free electricity' on A and on B would have been the same as before, but the quantity of ' combined electricity' in A would have been less in the second case and greater in the third than it was in the first.

It would appear therefore, according to this theory, that it is possible to alter not only the amount of free electricity in a body, but the amount of combined electricity. But no. phenomena have ever been observed in electrified bodies which can be traced to the varying amount of their combined electricities. Hence either the combined electricities have no observable properties, or the amount of the combined electricities is incapable of variation. The first of these alternatives presents no difficulty to the mere mathema- tician, who attributes no properties to the fluids except those of attraction and repulsion, for he conceives the two fluids simply to annul one another, like -f e and — *?, and their combination to be a true mathematical zero. But to those who cannot use the word Fluid without thinking of a substance it is difficult to conceive how the combination of the two fluids can have no properties at all, so that the addition of more or less of the combination to a body shall

40 ELECTROSTATIC PHENOMENA. [37.

not in any way affect it, either by increasing- its mass or its weight, or altering some of its other properties. Hence it has been supposed by some, that in every process of electrification exactly equal quan- tities of the two fluids are transferred in opposite directions, so that the total quantity of the two fluids in any body taken to- gether remains always the same. By this new law they ' contrive to save appearances,' forgetting that there would have been no need of the law except to reconcile the ' two fluids ' theory with facts, and to prevent it from predicting non-existent phenomena.

Theory of One Fluid.

37.] In the theory of One Fluid everything is the same as in the theory of Two Fluids except that, instead of supposing the two substances equal and opposite in all respects, one of them, gene- rally the negative one, has been endowed with the properties and name of Ordinary Matter, while the other retains the name of The Electric Fluid. The particles of the fluid are supposed to repel one another according to the law of the inverse square of the distance, and to attract those of matter according to the same law. Those of matter are supposed to repel each other and attract those of electricity.

If the quantity of the electric fluid in a body is such that a particle of the electric fluid outside the body is as much repelled by the electric fluid in the body as it is attracted by the matter of the body, the body is said to be Saturated. If the quantity of fluid in the body is greater than that required for saturation, the excess is called the Redundant fluid, and the body is said to be Overcharged. If it is less, the body is said to be Undercharged, and the quantity of fluid which would be required to saturate it is sometimes called the Deficient fluid. The number of units of electricity required to saturate one gramme of ordinary matter must be very great, because a gramme of gold may be beaten out to an area of a square metre, and when in this form may have a negative charge of at least 60,000 units of electricity. In order to saturate the gold leaf, this quantity of electric fluid must be communicated to it, so that the whole quantity required to saturate it must be greater than this. The attraction between the matter and the fluid in two saturated bodies is supposed to be a very little greater than the repulsion between the two portions of matter and that between the two portions of fluid. This residual force is sup- posed to account for the attraction of gravitation.

38.] THEORY OF ONE FLUID. 41

This theory does not, like the Two-Fluid theory, explain too much. It requires us, however, to suppose the mass of the electric fluid so small that no attainable positive or negative electrification has yet perceptibly increased or diminished either the mass or the weight of a body, and it has not yet been able to assign sufficient reasons why the vitreous rather than the resinous electrification should be supposed due to an excess of electricity.

One objection has sometimes been urged against this theory by men who ought to have reasoned better. It has been said that the doctrine that the particles of matter uncombined with elec- tricity repel one another, is in direct antagonism with the well- established fact that every particle of matter attracts every other particle throughout the universe. If the theory of One Fluid were true we should have the heavenly bodies repelling one another.

But it is manifest that the heavenly bodies, according to this theory, if they consisted of matter uncombined with electricity, would be in the highest state of negative electrification, and would repel each other. We have no reason to believe that they are in such a highly electrified state, or could be maintained in that state. The earth and all the bodies whose attraction has been observed are rather in an unelectrified state, that is, they contain the normal charge of electricity, and the only action between them is the residual force lately mentioned. The artificial manner, how- ever, in which this residual force is introduced is a much more valid objection to the theory.

In the present treatise I propose, at different stages of the in- vestigation, to test the different theories in the light of additional classes of phenomena. For my own part, I look for additional light on the nature of electricity from a study of what takes place in the space intervening between the electrified bodies. Such is the essential character of the mode of investigation pursued by Faraday in his Experimental Researches, and as we go on I intend to exhibit the results, as developed by Faraday, W. Thomson, &c., in a con- nected and mathematical form, so that we may perceive what phenomena are explained equally well by all the theories, and what phenomena indicate the peculiar difficulties of each theory.

Measurement of the Force between Electrified Bodies. 38.] Forces may be measured in various ways. For instance, one of the bodies may be suspended from one arm of a delicate balance, and weights suspended from the other arm, till the body,

42 ELECTROSTATIC PHENOMENA. [39.

when unelectrified, is in equilibrium. The other body may then be placed at a known distance beneath the first, so that the attraction or repulsion of the bodies when electrified may increase or diminish the apparent weight of the first. The weight which must be added to or taken from the other arm, when expressed in dynamical measure, will measure the force between the bodies. This arrangement was used by Sir W. Snow Harris, and is that adopted in Sir W. Thomson's absolute electrometers. See Art. 217.

It is sometimes more convenient to use a torsion-balance, in which a horizontal arm is suspended by a fine wire or fibre, so as to be capable of vibrating about the vertical wire as an axis, and the body is attached to one end of the arm and acted on by the force in the tangential direction, so as to turn the arm round the vertical axis, and so twist the suspension wire through a certain angle. The torsional rigidity of the wire is found by observing the time of oscillation of the arm, the moment of inertia of the arm being otherwise known, and from the angle of torsion and the torsional rigidity the force of attraction or repulsion can be deduced. The torsion-balance was devised by Michell for the de- termination of the force of gravitation between small bodies, and was used by Cavendish for this purpose. Coulomb, working in- dependently of these philosophers, reinvented it, thoroughly studied its action, and successfully applied it to discover the laws of electric and magnetic forces ; and the torsion-balance has ever since been used in all researches where small forces have to be measured. See Art. 215.