i ErUtbris w@m& \ OF THE UNIVERSITY OF LIBRARY OF THE SOCIETY r*alue as the direct result of his experiments. - !'--^,^.^Jw / &U*.e* *?i*r&*, &**S&. >-/'; C*<*f&tA^ 6.] DERIVED UNITS. 5 Derived Units. 6.] The unit of Velocity is that velocity in which unit of length is described in unit of time. Its dimensions are [Z7 7 " 1 ]. If we adopt the units of length and time derived from the vibrations of light, then the unit of velocity is the velocity of light. The unit of Acceleration is that acceleration in which the velo- city increases by unity in unit of time. Its dimensions are \_I/T~ 2 ]. The unit of Density is the density of a substance which contains unit of mass in unit of volume. Its dimensions are [3/Jy~ 3 ]. The unit of Momentum is the momentum of unit of mass moving with unit of velocity. Its dimensions are \MLT~ l ~\. The unit of Force is the force which produces unit of momentum in unit of time. Its dimensions are [MLT~ 2 ~]. This is the absolute unit of force, and this definition of it is implied in every equation in Dynamics. Nevertheless, in many text books in which these equations are given, a different unit of force is adopted, namely, the weight of the national unit of mass; and then, in order to satisfy the equations, the national unit of mass is itself abandoned, and an artificial unit is adopted as the dynamical unit, equal to the national unit divided by the numerical value of the intensity of gravity at the place. In this way both the unit of force and the unit of mass are made to depend on the value of the intensity of gravity, which varies from place to place, so that state- ments involving these quantities are not complete without a know- ledge of the intensity of gravity in the places where these statements were found to be true. The abolition, for all scientific purposes, of this method of measur- ing forces is mainly due to the introduction by Gauss of a general system of making observations of magnetic force in countries in which the intensity of gravity is different. All such forces are now measured according to the strictly dynamical method deduced from our definitions, and the numerical results are the same in whatever country the experiments are made. The unit of Work is the work done by the unit of force acting through the unit of length measured in its own direction. Its dimensions are [ML 2 T~ 2 ~\. The Energy of a system, being its capacity of performing work, is measured by the work which the system is capable of performing by the expenditure of its whole energy. 6 PRELIMINARY. [7. The definitions of other quantities, and of the units to which they are referred, will be given when we require them. In transforming the values of physical quantities determined in terms of one unit, so as to express them in terms of any other unit of the same kind, we have only to remember that every expres- sion for the quantity consists of two factors, the unit and the nu- merical part which expresses how often the unit is to be taken. Hence the numerical part of the expression varies inversely as the magnitude of the unit, that is, inversely as the various powers of the fundamental units which are indicated by the dimensions of the derived unit. On Physical Continuity and Discontinuity. 7.] A quantity is said to vary continuously if, when it passes from one value to another, it assumes all the intermediate values. We may obtain the conception of continuity from a consideration of the continuous existence of a particle of matter in time and space. Such a particle cannot pass from one position to another without describing a continuous line in space, and the coordinates of its position must be continuous functions of the time. In the so-called ' equation of continuity,' as given in treatises on Hydrodynamics, the fact expressed is that matter cannot appear in or disappear from an element of volume without passing in or out through the sides of that element. A quantity is said to be a continuous function of its variables if, when the variables alter continuously, the quantity itself alters continuously. Thus, if u is a function of #, and if, while x passes continuously from X Q to # 15 u passes continuously from U Q to u l9 but when x passes from ^ to # 2 , u passes from u{ to u 2 , u being different from !, then u is said to have a discontinuity in its variation with respect to x for the value #=# 1} because it passes abruptly from u to %' while x passes continuously through x. If we consider the differential coefficient of u with respect to x for the value 3?=^ as the limit of the fraction when x. 2 and # are both made to approach x l without limit, then, if X Q and x 2 are always on opposite sides of as l9 the ultimate value of the numerator will be < w lf and that of the denominator will be zero. If u is a quantity physically continuous, the discontinuity 8.] CONTINUITY AND DISCONTINUITY. 7 can exist only with respect to the particular variable x. "We must in this case admit that it has an infinite differential coefficient when X=X L . If u is not physically continuous, it cannot be dif- ferentiated at all. It is possible in physical questions to get rid of the idea of discontinuity without sensibly altering the conditions of the case. If XQ is a very little less than jc l9 and # 2 a very little greater than a? 1} then u will be very nearly equal to % and u 2 to u{. We may now suppose u to vary in any arbitrary but continuous manner from UQ to ?/ 2 between the limits # and # 2 . In many physical questions we may begin with a hypothesis of this kind, and then investigate the result when the values of X Q and # 2 are made to approach that of x and ultimately to reach it. If the result is independent of the arbitrary manner in which we have supposed u to vary between the limits, we may assume that it is true when u is discontinuous. Discontinuity of a Function of more than One Variable. 8.] If we suppose the values of all the variables except x to be constant, the discontinuity of the function will occur for particular values of #, and these will be connected with the values of the other variables by an equation which we may write 4> = $ (,#*,&.) = 0. The discontinuity will occur when $ 0. When $ is positive the function will have the form F 2 (x,y> z, &c.). When < is negative it will have the form F 1 (x,y, #, &c.). There need be no necessary relation between the forms F l and F 2 . To express this discontinuity in a mathematical form, let one of the variables, say #, be expressed as a function of $ and the other variables, and let F 1 and F 2 be expressed as functions of $,y, z, &c. We may now express the general form of the function by any formula which is sensibly equal to F 2 when < is positive, and to F 1 when (/> is negative. Such a formula is the following As long as n is a finite quantity, however great, F will be a continuous function, but if we make n infinite F will be equal to F 2 when < is positive, and equal to F l when < is negative. 8 PRELIMINARY. [9. Discontinuity of the Derivatives of a Continuous Function. The first derivatives of a continuous function may be discon- tinuous. Let the values of the variables for which the discon- tinuity of the derivatives occurs be connected by the equation = (0,#*...) = 0, and let F 1 and F 2 be expressed in terms of < and n 1 other variables, say (y, z . . .). Then, when < is negative, F l is to be taken, and when $ is positive F 2 is to be taken, and, since F is itself continuous, when d> is zero, F, = F 2 * J TJ J Jji Hence, when d> is zero, the derivatives r-^ and ~ may be different, but the derivatives with respect to any of the other 7 TJ J ~fjl variables, such as ^ and - , must be the same. The discon- dy dy / j tinuity is therefore confined to the derivative with respect to $, all the other derivatives being continuous. Periodic and Multiple Functions. 9.] If u is a function of x such that its value is the same for x, x + a, x -\-nctj and all values of x differing by a, u is called a periodic function of #, and a is called its period. If x is considered as a function of u, then, for a given value of M, there must be an infinite series of values of x differing by. multiples of a. In this case x is called a multiple function of u, and a is called its cyclic constant. fj rp The differential coefficient has only a finite number of values du corresponding to a given value of u. On the Relation of Physical Quantities to Directions in Space. 10.] In distinguishing the kinds of physical quantities, it is of great importance to know how they are related to the directions of those coordinate axes which we usually employ in defining the positions of things. The introduction of coordinate axes into geo- metry by Des Cartes was one of the greatest steps in mathematical progress, for it reduced the methods of geometry to calculations performed on numerical quantities. The position of a point is made to depend on the length of three lines which are always drawn in determinate directions, and the line joining two points is in like manner considered as the resultant of three lines. II.] VECTORS, OR DIRECTED QUANTITIES. 9 But for many purposes of physical reasoning 1 , as distinguished from calculation, it is desirable to avoid explicitly introducing- the Cartesian coordinates, and to fix the mind at once on a point of space instead of its three coordinates, and on the magnitude and direction of a force instead of its three components. This mode of contemplating- geometrical and physical quantities is more prim- itive and more natural than the other, although the ideas connected with it did not receive their full development till Hamilton made the next great step in dealing with space, by the invention of his Calculus of Quaternions. As the methods of Des Cartes are still the most familiar to students of science, and as they are really the most useful for purposes of calculation, we shall express all our results in the Cartesian form. I am convinced, however, that the introduction of the ideas, as distinguished from the operations and methods of Quaternions, will be of great use to us in the study of all parts of our subject, and especially in electrodynamics, where we have to deal with a number of physical quantities, the relations of which to each other can be expressed far more simply by a few expressions of Hamilton's, than by the ordinary equations. 11.] One of the most important features of Hamilton's method is the division of quantities into Scalars and Vectors. A Scalar quantity is capable of being completely defined by a single numerical specification. Its numerical value does not in any way depend on the directions we assume for the coordinate axes. A Vector, or Directed quantity, requires for its definition three numerical specifications, and these may most simply be understood as having reference to the directions of the coordinate axes. Scalar quantities do not involve direction. The volume of a geometrical figure, the mass and the energy of a material body, the hydrostatical pressure at a point in a fluid, and the potential at a point in space, are examples of scalar quantities. A vector quantity has direction as well as magnitude, and is such that a reversal of its direction reverses its sign. The dis- placement of a point, represented by a straight line drawn from its original to its final position, may be taken as the typical vector quantity, from which indeed the name of Vector is derived. The velocity of a body, its momentum, the force acting on it, an electric current, the magnetization of a particle of iron, are instances of vector quantities. 10 PKELIMINARY. [l2. There are. physical quantities of another kind which are related to directions in space, but which are not vectors. Stresses and strains in solid bodies are examples of these, and so are some of the properties of bodies considered in the theory of elasticity and in the theory of double refraction. Quantities of this class require for their definition nine numerical specifications. They are ex- pressed in the language of Quaternions by linear and vector functions of a vector. The addition of one vector quantity to another of the same kind is performed according to the rule given in Statics for the com- position of forces. In fact, the proof which Poisson gives of the 'parallelogram of forces' is applicable to the composition of any quantities such that turning them end for end is equivalent to a reversal of their sign. When we wish to denote a vector quantity by a single symbol, and to call attention to the fact that it is a vector, so that we must consider its direction as well as its magnitude, we shall denote it by a German capital letter, as $1, S3, &c. In the calculus of Quaternions, the position of a point in space is defined by the vector drawn from a fixed point, called the origin, to that point. If we have to consider any physical quantity whose value depends on the position of the point, that quantity is treated as a function of the vector drawn from the origin. The function may be itself either scalar or vector. The density of a body, its temperature, its hydrostatic pressure, the potential at a point, are examples of scalar functions. The resultant force at a point, the velocity of a fluid at a point, the velocity of rotation of an element of the fluid, and the couple producing rotation, are examples of vector functions. 12.] Physical vector quantities may be divided into two classes, in one of which the quantity is defined with reference to a line, while in the other the quantity is defined with reference to an area. For instance, the resultant of an attractive force in any direction may be measured by finding the work which it would do on a body if the body were moved a short distance in that direction and dividing it by that short distance. Here the attractive force is defined with reference to a line. On the other hand, the flux of heat in any direction at any point of a solid body may be defined as the quantity of heat which crosses a small area drawn perpendicular to that direction divided 13.] INTENSITIES AND FLUXES. 11 by that area and by the time. Here the flux is defined with reference to an area. There are certain cases in which a quantity may be measured with reference to a line as well as with reference to an area. Thus, in treating of the displacements of elastic solids, we may direct our attention either to the original and the actual position of a particle, in which case the displacement of the particle is measured by the line drawn from the first position to the second, or we may consider a small area fixed in space, and determine what quantity of the solid passes across that area during the dis- placement. In the same way the velocity of a fluid may be investigated either with respect to the actual velocity of the individual particles, or with respect to the quantity of the fluid which flows through any fixed area. But in these cases we require to know separately the density of the body as well as the displacement or velocity, in order to apply the first method, and whenever we attempt to form a molecular theory we have to use the second method. In the case of the flow of electricity we do not know anything of its density or its velocity in the conductor, we only know the value of what, on the fluid theory, would correspond to the product of the density and the velocity. Hence in all such cases we must apply the more general method of measurement of the flux across an area. In electrical science, electromotive and magnetic intensity belong to the first class, being defined with reference to lines. When we wish to indicate this fact, we may refer to them as Intensities. On the other hand, electric and magnetic induction, and electric currents, belong to the second class, being defined with reference to areas. When we wish to indicate this fact, we shall refer to them as Fluxes. Each of these forces may be considered as producing, or tending to produce, its corresponding flux. Thus, electromotive intensity produces electric currents in conductors, and tends to produce them in dielectrics. It produces electric induction in dielectrics, and pro- bably in conductors also. In the same sense, magnetic intensity produces magnetic induction. 13.] In some cases the flux is simply proportional to the force and in the same direction, but in other cases we can only affirm 12 PRELIMINARY. [14. that the direction and magnitude of the flux are functions of the direction and magnitude of the force. The case in which the components of the flux are linear functions of those of the force is discussed in the chapter on the Equations of Conduction, Art. 297. There are in general nine coefficients which determine the relation between the force and the flux. In certain cases we have reason to believe that six of these coefficients form three pairs of equal quantities. In such cases the relation be- tween the line of direction of the force and the normal plane of the flux is of the same kind as that between a diameter of an ellipsoid and its conjugate diametral plane. In Quaternion language, the one vector is said to be a linear and vector function of the other, and when there are three pairs of equal coefficients the function is said to be self-conjugate. In the case of magnetic induction in iron, the flux, (the mag- netization of the iron,) is not a linear function of the magnetizing force. In all cases, however, the product of the force and the flux resolved in its direction, give a result of scientific import- ance, and this is always a scalar quantity. 14.] There are two mathematical operations of frequent occur- rence which are appropriate to these two classes of vectors, or directed quantities. In the case of forces, we have to take the integral along a line of the product of an element of the line, and the resolved part of the force along that element. The result of this operation is called the Line-integral of the force. It represents the work done on a body carried along the line. In certain cases in which the line-integral does not depend on the form of the line, but only on the positions of its extremities, the line-integral is called the Potential. In the case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the Surface-integral of the flux. It repre- sents the quantity which passes through the surface. There are certain surfaces across which there is no flux. If two of these surfaces intersect, their line of intersection is a line of flux. In those cases in which the flux is in the same direction as the force, lines of this kind are often called Lines of Force. It would be more correct, however, to speak of them in electrostatics and magnetics as Lines of Induction, and in electrokinematics as Lines of Flow. 1 6.] LINE-INTEGRALS. 13 15.] There is another distinction between different kinds of directed quantities, which, though very important in a physical point of view, is not so necessary to be observed for the sake of the mathematical methods. This is the distinction between longi- tudinal and rotational properties. The direction and magnitude of a quantity may depend upon some action or effect which takes place entirely along a certain line, or it may depend upon something of the nature of rota- tion about that line as an axis. The laws of combination of directed quantities are the same whether they are longitudinal or rotational, so that there is no difference in the mathematical treat- ment of the two classes, but there may be physical circumstances which indicate to which class we must refer a particular pheno- menon. Thus, electrolysis consists of the transfer of certain sub- stances along a line in one direction, and of certain other sub- stances in the opposite direction, which is evidently a longitudinal phenomenon, and there is no evidence of any rotational effect about the direction of the force. Hence we infer that the electric current which causes or accompanies electrolysis is a longitudinal, and not a rotational phenomenon. On the other hand, the north and south poles of a magnet do not differ as oxygen and hydrogen do, which appear at opposite places during electrolysis, so that we have no evidence that mag- netism is a longitudinal phenomenon, while the effect of magnetism in rotating the plane of polarized light distinctly shews that mag- netism is a rotational phenomenon. On Line-integrals. 16.] The operation of integration of the resolved part of a vector quantity along a line is important in physical science generally, and should be clearly understood. Let x, y> z be the coordinates of a point P on a line whose length, measured from a certain point A, is s. These coordinates will be functions of a single variable s. Let R be the numerical value of the vector quantity at P, and let the tangent to the curve at P make with the direction of R the angle e, then R cos e is the resolved part of R along the line, and the integral f* L = / Rcoseds JQ is called the line-integral of R along the line s. 14 PRELIMINARY. [l6. We may write this expression _ C' f^dx dy rydz\ 7 L=l (X-j- +T-f + Z )ds, J v ds ds ds' where X, T, Z are the components of E parallel to #, y^ z respect- ively. This quantity is, in general, different for different lines drawn between A and P. When, however, within a certain region, the quantity x dx + T dy + Z dz = -D*, that is, when it is an exact differential within that region, the value of L becomes and is the same for any two forms of the path between A and P, provided the one form can be changed into the other by continuous motion without passing out of this region. On Potentials. The quantity ^ is a scalar function of the position of the point, and is therefore independent of the directions of reference. It is called the Potential Function, and the vector quantity whose com- ponents are X, Y, Z is said to have a potential ^, if *-, r~<$, ,--<*). When a potential function exists, surfaces for which the potential is constant are called Equipotential surfaces. The direction of E at any point of such a surface coincides with the normal to the surface, dty and if n be a normal at the point P, then E = -- =- dn The method of considering the components of a vector as the first derivatives of a certain function of the coordinates with re- spect to these coordinates was invented by Laplace * in his treat- ment of the theory of attractions. The name of Potential was first given to this function by Green f, who made it the basis of his treatment of electricity. Green's essay was neglected by mathe- maticians till 1846, and before that time most of its important theorems had been rediscovered by Gauss, Chasles, Sturm, and Thomson J. * Me'c. Celeste, liv. iii. t Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Nottingham, 1828. Reprinted in Crette's Jowrnal, and in Mr. Ferrers' edition of Green's Works. t Thomson and Tait, Natural Philosophy, 483. I/.] RELATION BETWEEN FORCE AND POTENTIAL. 15 In the theory of gravitation the potential is taken with the opposite sign to that which is here used, and the resultant force in any direction is then measured by the rate of increase of the potential function in that direction. In electrical and magnetic investigations the potential is defined so that the resultant force in any direction is measured by the decrease of the potential in that direction. This method of using the expression makes it correspond in sign with potential energy, which always decreases when the bodies are moved in the direction of the forces acting on them. 17.] The geometrical nature of the relation between the poten- tial and the vector thus derived from it receives great light from Hamilton's discovery of the form of the operator by which the vector is derived from the potential. The resolved part of the vector in any direction is, as we have seen, the first derivative of the potential with respect to a co- ordinate drawn in that direction, the sign being reversed. Now if i, j, k are three unit vectors at right angles to each other, and if X, Y } Z are the components of the vector J resolved parallel to these vectors, then % = iZ+jT+tZ; (I) and by what we have said above, if ^ is the potential, If we now write V for the operator, . d . d 7 d /_% +/_- +2^-, (3) dx J d dz (4) The symbol of operation V may be interpreted as directing us to measure, in each of three rectangular directions, the rate of increase of ^, and then, considering the quantities thus found as vectors, to compound them into one. This is what we are directed to do by the expression (3). But we may also consider it as directing us first to find out in what direction ^ increases fastest, and then to lay off in that direction a vector representing this rate of increase. M. Lame, in his Traite des Fonctions Inverses, uses the term Differential Parameter to express the magnitude of this greatest rate of increase, but neither the term itself, nor the mode in which . ' / / ^/7~7 16 PRELIMINARY. [l8. Lame* uses it, indicates that the quantity referred to has direction as well as magnitude. On those rare occasions in which I shall have to refer to this relation as a purely geometrical one, I shall call the vector the space-variation of the scalar function #, using the phrase to indicate the direction, as well as the magnitude, of the most rapid decrease of V. 18.] There are cases, however, in which the conditions dZ dY dX dZ dY dX -= ?- = 0, ^ T- = J an d -7 7- = 0, dy dz dz d% dx dy which are those of Xdx+Ydy+Zdz being a complete differential, are satisfied throughout a certain region of space, and yet the line- integral from A to P may be different for two lines, each of which lies wholly within that region. This may be the case if the region is in the form of a ring, and if the two lines from A to P pass through opposite segments of the ring. In this case, the one path cannot be transformed into the other by continuous motion without passing out of the region. We are here led to considerations belonging to the Geometry of Position, a subject which, though its importance was pointed out by Leibnitz and illustrated by Gauss, has been little studied. The most complete treatment of this subject has been given by J. B. Listing*. Let there be p points in space, and let I lines of any form be drawn joining these points so that no two lines intersect each other, and no point is left isolated. We shall call a figure com- posed of lines in this way a Diagram. Of these lines, p 1 are sufficient to join the p points so as to form a connected system. Every new line completes a loop or closed path, or, as we shall call it, a Cycle. The number of independent cycles in the diagram is therefore K = I p + 1 . Any closed path drawn along the lines of the diagram is com- posed of these independent cycles, each being taken any number of times and in either direction. The existence of cycles is called Cyclosis, and the number of cycles in a diagram is called its Cyclomatic number. Cyclosis in Surfaces and Regions. Surfaces are either complete or bounded. Complete surfaces are either infinite or closed. Bounded surfaces are limited by one or * Der Census Raumlicher Complete, Gott. Abli., Bd. x. S. 97 (1861). 1 9.] CYCLIC REGIONS. 17 more closed lines, which may in the limiting cases become double finite lines or points. A finite region of space is bounded by one or more closed surfaces. Of these one is the external surface, the others are included in it and exclude each other, and are called internal surfaces. If the region has one bounding surface, we may suppose that surface to contract inwards without breaking its continuity or cutting itself. If the region is one of simple continuity, such as a sphere, this process may be continued till it is reduced to a point ; but if the region is like a ring, the result will be a closed curve; and if the region has multiple connexions, the result will be a diagram of lines, and the cyclomatic number of the diagram will be that of the region. The space outside the region has the same cyclomatic number as the region itself. Hence, if the region is bounded by internal as well as external surfaces, its cyclomatic number is the sum of those due to all the surfaces. When a region encloses within itself other regions, it is called a Periphractic region. The number of internal bounding surfaces of a region is called its periphractic number. A closed surface is also periphractic, its periphractic number being unity. The cyclomatic number of a closed surface is twice that of either of the regions which it bounds. To find the cyclomatic number of a bounded surface, suppose all the boundaries to contract inwards, without breaking continuity, till they meet. The surface will then be reduced to a point in the case of an acyclic surface, or to a linear diagram in the case of cyclic surfaces. The cyclomatic number of the diagram is that of the surface. 19.] THEOREM I. If throughout any acyclic region Xdx + Ydy + Z-dz = -DV, the value of the line-integral from a point A to a point P taken along any path within the region will be the same. We shall first shew that the line-integral taken round any closed path within the region is zero. Suppose the equipotential surfaces drawn. They are all either closed surfaces or are bounded entirely by the surface of the re- gion, so that a closed line within the region, if it cut^ any of the surfaces at one part of its path, must cut the same surface in the opposite direction at some other part of its path, and the VOL. i. c 18 PRELIMINARY. [20. corresponding portions of the line-integral being equal and opposite, the total value is zero. Hence if AQP and AQ'P are two paths from A to P, the line- integral for AQ'P is the sum of that for AQP and the closed path AQ'P Q A. But the line-integral of the closed path is zero, there- fore those of the two paths are equal. Hence if the potential is given at any one point of such a region, that at any other point is determinate. 20.] THEOREM II. In a cyclic region in which the equation Xdx + Ydy+Zdz = -Dy is everywhere satisfied, the line-integral from A to P, along a line drawn within the region, will not in general be determinate unless the channel of communication between A and P be specified. Let K be the cyclomatic number of the region, then K sections of the region may be made by surfaces which we may call Dia- phragms, so as to close up K of the channels of communication, and reduce the region to an acyclic condition without destroying its continuity. The line-integral from A to any point P taken along a line which does not cut any of these diaphragms will be, by the last theorem, determinate in value. Now let A and P be taken indefinitely near to each other, but on opposite sides of a diaphragm, and let K be the line-integral from A to P. Let A' and P' be two other points on opposite sides of the same diaphragm and indefinitely near to each other, and let K' be the line-integral from A' to P'. Then K'= K. For if we draw AA' and PP', nearly coincident, but on opposite sides of the diaphragm, the line-integrals along these lines will be equal. Suppose each equal to L, then K', the line-integral of A'P', is equal to that of A'A + AP + PP'=-L + K+L=K= that ofAP. Hence the line-integral round a closed curve which passes through one diaphragm of the system in a given direction is a constant quantity K. This quantity is called the Cyclic constant corre- sponding to the given cycle. Let any closed curve be drawn within the region, and let it cut the diaphragm of the first cycle p times in the positive direction and jt/ times in the negative direction, and let p p'= %. Then the line-integral of the closed curve will be n t K : . 21.] SURFACE-INTEGRALS. 1 9 Similarly the line-integral of any closed curve will be where n K represents the excess of the number of positive passages of the curve through the diaphragm of the cycle K over the number of negative passages. If two curves are such that one of them may be transformed into the other by continuous motion without at any time passing through any part of space for which the condition of having a potential is not fulfilled, these two curves are called Reconcileable curves. Curves for which this transformation cannot be effected are called Irreconcileable curves *. The condition that Xdx + Ydf+Zd* is a complete differential of some function ^ for all points within a certain region, occurs in several physical investigations in which the directed quantity and the potential have different physical interpretations, In pure kinematics we may suppose X, Y, Z to be the com- ponents of the displacement of a point of a continuous body whose original coordinates are %, y> z; the condition then expresses that these displacements constitute a non-rotational strain f. If X, Y, Z represent the components of the velocity of a fluid at the point #,y, z, then the condition expresses that the motion of the fluid is irrotational. If X, Y, Z represent the components of the force at the point a?, y, z, then the condition expresses that the work done on a particle passing from one point to another is the difference of the potentials at these points, and the value of this difference is the same for all reconcileable paths between the two points. On Surface-Integrals. 21.] Let dS be the element of a surface, and e the angle which a normal to the surface drawn towards the positive side of the surface makes with the direction of the vector quantity R, then / IE cos e dS is called the surface-integral of It over the surface 8. THEOREM III. The surface-integral of the flux inwards through a closed surface may le expressed as the volume-integral of its con- vergence taken within the surface. (See Art. 25.) Let X, Y, Z be the components of R, and let I, m, n be the * See Sir W. Thomson ' On Vortex Motion,' Trans. R. S. Edin. t 1867-8. t See Thomson and Tait's Natural Philosophy, 190 (i). C 2, 20 PRELIMINARY. [21. direction-cosines of the normal to S measured inwards. Then the surface-integral of E over S is ffjt cos - dS =JJxidS +f/Ym dS+ffzndS; (1) the values of X, Y, Z being those at a point in the surface, and the integrations being extended over the whole surface. If the surface is a closed one, then, when y and z are given, the coordinate x must have an even number of values, since a line parallel to x must enter and leave the enclosed space an equal number of times provided it meets the surface at all. At each entrance IdS = dydz> and at each exit 7 , -, , Ida dydz. Let a point travelling from a? = oo to # = + oo first enter the space when so = se l9 then leave it when x = % 2 > an ^ so on > and let the values of X at these points be X 1) X 2 , &c., then f/XldS = //{fr-JQ + (Z,-!*) + &c. + (X^-X^)} Aydz. (2) If X is a quantity which is continuous, and has no infinite values between as l and x z) then where the integration is extended from the first to the second intersection, that is, along the first segment of x which is within the closed surface. Taking into account all the segments which lie within the closed surface, we find the double integration being confined to the closed surface, but the triple integration being extended to the whole enclosed space. Hence, if X, J", Z are continuous and finite within a closed surface S, the total surface-integral of R over that surface will be tf I JJ -n f j RcosedS =_.///( + + \dxdydz, (5) JJJ^dxdydz' the triple integration being extended over the whole space within S. Let us next suppose that X, Y, Z are not continuous within the closed surface, but that at a certain surface F(x, y, z) = the values of X, Y, Z alter abruptly from X, Y, Z on the negative side of the surface to JT, J', Z' on the positive side. 22.] SOLENOIDAL DISTRIBUTION. 21 If this discontinuity occurs, say, between x l and # 2 , the value (6) where in the expression under the integral sign only the finite values of the derivative of X are to be considered. In this case therefore the total surface-integral of E over the closed surface will be expressed by fJR cos , d8 = -///(g + f + f ) &** +//(X'-X) Ay dz ; (7) or, if V, m', n' are the direction-cosines of the normal to the surface of discontinuity, and dS' an element of that surface, ff J J dy dz \ (8) where the integration of the last term is to be extended over the surface of discontinuity. If at every point where X, J", Z are continuous ^,^1,^-0 (9) dx + dy + dz ~ ( ' and at every surface where they are discontinuous l'X' + mT+n'Z'= I'X+m'Y+n'Z (10) then the surface-integral over every closed surface is zero, and the distribution of the vector quantity is said to be Solenoidal. We shall refer to equation (9) as the General solenoidal con- dition, and to equation (10) as the Superficial solenoidal condition. 22.] Let us now consider the case in which at every point within the surface S the equation dX dY dZ is satisfied. We have as a consequence of this the surface-integral over the closed surface equal to zero. Now let the closed surface S consist of three parts S lt S , and S 2 . Let $! be a surface of any form bounded by a closed line L^. Let S be formed by drawing lines from every point of L^ always 22 PKELIMINARY. [22. coinciding with the direction of E. If I, m, n are the direction- cosines of the normal at any point of the surface S , we have R cose = Xl+Im + Zn = 0. (12) Hence this part of the surface contributes nothing towards the value of the surface-integral. Let S 2 be another surface of any form bounded by the closed curve I/ 2 in which it meets the surface S . Let Q 1} Q , Q 2 be the surface-integrals of the surfaces S lt $ , S 2) and let Q be the surface-integral of the closed surface S. Then Q= Qi+Qo+ 2 =0; (13) and we know that Q = ; (14) therefore Q 2 = 5i ; (15) or, in other words, the surface-integral over the surface S 2 is equal and opposite to that over S l whatever be the form and position of $ 2 , provided that the intermediate surface $ is one for which R is always tangential. If we suppose L a closed curve of small area, S will be a tubular surface having the property that the surface-integral over every complete section of the tube is the same. Since the whole space can be divided into tubes of this kind provided $% dY dZ -7- + T- + T- = dx dy dz a distribution of a vector quantity consistent with this equation is called a Solenoidal Distribution. On Tubes and Lines of Flow. If the space is so divided into tubes that the surface-integral for every tube is unity, the tubes are called Unit tubes, and the surface-integral over any finite surface 8 bounded by a closed curve L is equal to the number of such tubes which pass through S in the positive direction, or, what is the same thing, the number which pass through the closed curve L. Hence the surface-integral of S depends only on the form of its boundary L t and not on the form of the surface within its boundary. On Periphractic Regions. If, throughout the whole region bounded externally by the single closed surface $, the solenoidal condition dX dY dZ_ dx dy dz 22.] PERIPHRACTIC REGIONS. 23 is satisfied, then the surface-integral taken over any closed surface drawn within this region will be zero, and the surface-integral taken over a bounded surface within the region will depend only on the form of the closed curve which forms its boundary. It is not, however, generally true that the same results follow if the region within which the solenoidal condition is satisfied is bounded otherwise than by a single surface. For if it is bounded by more than one continuous surface, one of these is the external surface and the others are internal surfaces, and the region S is a periphractic region, having within it other regions which it completely encloses. If within one of these enclosed regions, say, that bounded by the closed surface S lt the solenoidal condition is not satisfied, let be the surface-integral for the surface enclosing this region, and let Q 2 , Q 3 , &c. be the corresponding quantities for the other en- closed regions S 2 , S 3 , &c. Then, if a closed surface $' is drawn within the region S, the value of its surface-integral will be zero only when this surface S' does not include any of the enclosed regions S lt S 2i &c. If it includes any of these, the surface-integral is the sum of the surface- integrals of the different enclosed regions which lie within it. For the same reason, the surface-integral taken over a surface bounded by a closed curve is the same for such surfaces only, bounded by the closed curve, as are reconcileable with the given surface by continuous motion of the surface within the region 8. When we have to deal with a periphractic region, the first thing to be done is to reduce it to an aperiphractic region by drawing lines L lt L 2 > & c * joking the internal surfaces S lt S 2 , &c. to the external surface S. Each of these lines, provided it joins surfaces which were not already in continuous connexion, reduces the periphractic number by unity, so that the whole number of lines to be drawn to remove the periphraxy is equal to the periphractic number, or the number of internal surfaces. In drawing these lines we must remember that any line joining surfaces which are already connected does not diminish the periphraxy, but introduces cyclosis. When these lines have been drawn we may assert that if the solenoidal condition is satisfied in the region 8, any closed surface drawn entirely within 8, and not cutting any of the lines, has its surface-integral zero. If it cuts any line, say L^ , once or any odd 24 PEELIMINAEY. [23. number of times, it encloses the surface S 1 and the surface-integral IBI. The most familiar example of a periphractic region within which the solenoidal condition is satisfied is the region surrounding a mass attracting or repelling inversely as the square of the distance. In this case we have X=m-^> Y = m^-, Z=m^j r 3 r 3 r s where m is the mass, supposed to be at the origin of coordinates. At any point where r is finite dX dY dZ ^ + ^ + ^ =;0> but at the origin these quantities become infinite. For any closed surface not including the origin, the surface-integral is zero. If a closed surface includes the origin, its surface-integral is 4 itm. If, for any reason, we wish to treat the region round m as if it were not periphractic, we must draw a line from m to an infinite distance, and in taking surface-integrals we must remember to add 4 irm whenever this line crosses from the negative to the positive side of the surface. On Right-handed and Left-handed Relations in Space. 23.] In this treatise the motions of translation along any axis and of rotation about that axis will be assumed to be of the same sign when their directions correspond to those of the translation and rotation of an ordinary or right-handed screw *. For instance, if the actual rotation of the earth from west to east is taken positive, the direction of the earth's axis from south to north will be taken positive, and if a man walks forward in the positive direction, the positive rotation is in the order, head, right- hand, feet, left-hand. * The combined action of the muscles of the arm when we turn the upper side of the right-hand outwards, and at the same time thrust the hand forwards, will impress the right-handed screw motion on the memory more firmly than any verbal definition. A common corkscrew may be used as a material symbol of the same relation. Professor W. H. Miller has suggested to me that as the tendrils of the vine are right-handed screws and those of the hop left-handed, the two systems of relations in space might be called those of the vine and the hop respectively. The system of the vine, which we adopt, is that of Linnaeus, and of screw-makers in all civilized countries except Japan. De Candolle was the first who called the hop-tendril right-banded, and in this he is followed by Listing, and by most writers on the circular polarization of light. Screws like the hop-tendril are made for the couplings of railway-carriages, and for the fittings of wheels on the left side of or- dinary carriages, but they are always called left-handed screws by those who use them. 24.] LINE-INTEGRAL AND SURFACE-INTEGRAL. 25 If we place ourselves on the positive side of a surface, the positive direction along its bounding curve will be opposite to the motion of the hands of a watch with its face towards us. This is the right-handed system which is adopted in Thomson and Tait's Natural Philosophy, 243, and in Tait's Quaternions. The opposite, or left-handed system, is adopted in Hamilton's Quaternions (Lectures, p. 76, and Elements, p. 108, and p. 117 note). The operation of passing from the one system to the other is called, by Listing, Perversion. The reflexion of an object in a mirror is a perverted image of the object. When we use the Cartesian axes of #, y, z, we shall draw them so that the ordinary conventions about the cyclic order of the symbols lead to a right-handed system of directions in space. Thus, if x is drawn eastward and y northward, z must be drawn upward. The areas of surfaces will be taken positive when the order of integration coincides with the cyclic order of the symbols. Thus, the area of a closed curve in the plane of xy may be written either jxdy or jydx; the order of integration being x, y in the first expression, and y, x in the second. This relation between the two products dx dy and dy dx may be compared with the rule for the product of two perpendicular vectors in the method of Quaternions, the sign of which depends on the order of multiplication ; and with the reversal of the sign of a determinant when the adjoining rows or columns are ex- changed. For similar reasons a volume-integral is to be taken positive when the order of integration is in the cyclic order of the variables x, y, z, and negative when the cyclic order is reversed. We now proceed to prove a theorem which is useful as esta- blishing a connexion between the surface-integral taken over a finite surface and a line-integral taken round its boundary, 24.] THEOREM IV. A line-integral taken round a closed curve may be expressed in terms of a surface-integral taJcen over a surface bounded by the curve. Let X, Y, Z be the components of a vector quantity 21 whose line- integral is to be taken round a closed curve s. Let S be any continuous finite surface bounded entirely by the 26 PRELIMINARY. [24. closed curve , 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 the quantities e and / being always understood to be taken with their proper signs. Variation of the Force with the Distance. 40.] Having established the law of force at a fixed distance, we may measure the force between bodies charged in a constant manner and placed at different distances. It is found by direct measurement that the force, whether of attraction or repulsion, varies inversely as the square of the distance, so that if f is the repulsion between two units at unit distance, the repulsion at dis- tance r will be/>~ 2 , and the general expression for the repulsion between e units and e' units at distance r will be Definition of the Electrostatic Unit of 'Electricity . 41.] We have hitherto used a wholly arbitrary standard for our unit of electricity, namely, the electrification of a certain piece of glass as it happened to be electrified at the commencement of our experiments. We are now able to select a unit on a definite 44 ELECTROSTATIC PHENOMENA. [42. principle, and in order that this unit may belong 1 to a general system we define it so thatj^may be unity, or in other words The electrostatic unit of electricity is that quantity of positive elec- tricity which) when placed at unit of distance from an equal quantity ', repels it with unit of force. This unit is called the Electrostatic unit to distinguish it from the Electromagnetic unit, to be afterwards defined. We may now write the general law of electrical action in the simple form jF_^' r -2. or ^ The repulsion between two small bodies charged respectively with e and e f units of electricity is numerically eqiial to the product of the charges divided ~by the square of the distance. Dimensions of the Electrostatic Unit of Quantity. 42.] If [Q] is the concrete electrostatic unit of quantity itself, and e, e' the numerical values of particular quantities; if [Z] is the unit of length, and r the numerical value of the distance ; and if [F] is the unit of force, and F the numerical value of the force, then the equation becomes whence [Q] = \LF*\ This unit is called the Electrostatic Unit of electricity. Other units may be employed for practical purposes, and in other depart- ments of electrical science, but in the equations of electrostatics quantities of electricity are understood to be estimated in electro- static units, just as in physical astronomy we employ a unit of mass which is founded on the phenomena of gravitation, and which differs from the units of mass in common use. Proof of the Law of Electrical Force. 43.] The experiments of Coulomb with the torsion-balance may be considered to have established the law of force with a certain approximation to accuracy. Experiments of this kind, however, are rendered difficult, and in some degree uncertain, by several disturbing causes, which must be carefully traced and corrected for. In the first place, the two electrified bodies must be of sensible dimensions relative to the distance between them, in order to be capable of carrying charges sufficient to produce measurable forces. 44-] LAW OF ELECTRIC FORCE. 45 The action of each body will then produce an effect on the dis- tribution of electricity on the other, so that the charge cannot be considered as evenly distributed over the surface, or collected at the centre of gravity ; but its effect must be calculated by an intricate investigation. This, however, has been done as regards two spheres by Poisson in an extremely able manner, and the investigation has been greatly simplified by Sir W. Thomson in his Theory of Electrical Images. See Arts. 172-175. Another difficulty arises from the action of the electricity induced on the sides of the case containing the instrument. By making the inner surface of the instrument of metal, this effect can be rendered definite and measurable. An independent difficulty arises from the imperfect insulation of the bodies, on account of which the charge continually de- creases. Coulomb investigated the law of dissipation, and made corrections for it in his experiments. The methods of insulating charged conductors, and of measuring electrical effects, have been greatly improved since the time of Coulomb, particularly by Sir W. Thomson ; but the perfect ac- curacy of Coulomb's law of force is established, not by any direct experiments and measurements (which may be used as illustrations of the law), but by a mathematical consideration of the pheno- menon described as Experiment VII, namely, that an electrified conductor .5, if made to touch the inside of a hollow closed con- ductor C and then withdrawn without touching (?, is perfectly dis- charged, in whatever manner the outside of C may be electrified. By means of delicate electroscopes it is easy to shew that no electricity remains on B after the operation, and by the mathe- matical theory given at Art. 74, this can only be the case if the force varies inversely as the square of the distance, for if the law were of any different form B would be electrified. The Electric Field. 44.] The Electric Field is the portion of space in the neigh- bourhood of electrified bodies, considered with reference to electric phenomena. It may be occupied by air or other bodies, or it may be a so-called vacuum, from which we have withdrawn every sub- stance which we can act upon with the means at our disposal. If an electrified body be placed at any part of the electric field it will, in general, produce a sensible disturbance in the electri- fication of the other bodies. 46 ELECTROSTATIC PHENOMENA. [45. But if the body is very small, and its charge also very small, the electrification of the other bodies will not be sensibly disturbed, and we may consider the position of the body as determined by its centre of mass. The force acting- on the body will then be proportional to its charge, and will be reversed when the charge is reversed. Let e be the charge of the body, and F the force acting on the body in a certain direction, then when e is very small F is propor- tional to e, or F=Re, where R depends on the distribution of electricity on the other bodies in the field. If the charge e could be made equal to unity without disturbing the electrification of other bodies we should have F E. We shall call R the Resultant Electromotive Intensity at the given point of the field. When we wish to express the fact that this quantity is a vector we shall denote it by the German letter (. Electromotive Force and Potential. 45.] If the small body carrying the small charge e be moved from one given point, A, to another H, along a given path, it will experience at each point of its course a force Re, where R varies from point to point of the course. Let the whole work done on the body by the electrical force be Ee> then E is called the Total Electromotive Force along the path A B. If the path forms a complete circuit, and if the total electromotive force round the circuit does not vanish, the electricity cannot be in equi- librium but a current will be produced. Hence in Electrostatics the electromotive force round any closed circuit must be zero, so that if A and B are two points on the circuit, the electromotive force from A to B is the same along either of the two paths into which the circuit is broken, and since either of these can be altered independently of the other, the electromotive force from A to B is the same for all paths from A to B. If B is taken as a point of reference for all other points, then the electromotive force from A to B is called the Potential of A. It depends only on the position of A. In mathematical investi- gations, B is generally taken at an infinite distance from the electrified bodies. A body charged positively tends to move from places of greater positive potential to places of smaller positive, or of negative, 46.] ELECTRIC POTENTIAL. 47 potential, and a body charged negatively tends to move in the opposite direction. In a conductor the electrification is free to move relatively to the conductor. If therefore two parts of a conductor have different potentials, positive electricity will move from the part having greater potential to the part having less potential as long as that difference continues. A conductor therefore cannot be in electrical equilibrium unless every point in it has the same potential. This potential is called the Potential of the Conductor. Equipotential Surfaces. 46.] If a surface described or supposed to be described in the electric field is such that the electric potential is the same at every point of the surface it is called an Equipotential surface. An electrified particle constrained to rest upon such a surface will have no tendency to move from one part of the surface to another, because the potential is the same at every point. An equipotential surface is therefore a surface of equilibrium or a level surface. The resultant force at any point of the surface is in the direction of the normal to the surface, and the magnitude of the force is such that the work done on an electrical unit in passing from the surface Fto the surface V is V V. No two equipotential surfaces having different potentials can meet one another, because the same point cannot have more than one potential, but one equipotential surface may meet itself, and this takes place at all points and along all lines of equilibrium. The surface of a conductor in electrical equilibrium is necessarily an equipotential surface. If the electrification of the conductor is positive over the whole surface, then the potential will diminish as we move away from the surface on every side, and the conductor will be surrounded by a series of surfaces of lower potential. But if (owing to the action of external electrified bodies) some regions of the conductor are charged positively and others ne- gatively, the complete equipotential surface will consist of the surface of the conductor itself together with a system of other surfaces, meeting the surface of the conductor in the lines which divide the positive from the negative regions. These lines will be lines of equilibrium, and an electrified particle placed on one of these lines will experience no force in any direction. When the surface of a conductor is charged positively in some 4:8 ELECTROSTATIC PHENOMENA. [47. parts and negatively in others, there must be some other electrified body in the field besides itself. For if we allow a positively electrified particle, starting* from a positively charged part of the surface, to move always in the direction of the resultant force upon it, the potential at the point will continually diminish till the point reaches either a negatively charged surface at a potential less than that of the first conductor, or moves off to an infinite distance. Since the potential at an infinite distance is zero, the latter case can only occur when the potential of the conductor is positive. In the same way a negatively electrified particle, moving off from a negatively charged part of the surface^ must either reach a positively charged surface, or pass off to infinity, and the latter case can only happen when the potential of the conductor is negative. Therefore, if both positive and negative charge exist on a conductor, there must be some other body in the field whose potential has the same sign as that of the conductor but a greater numerical value, and if a conductor of any form is alone in the field the charge of every part is of the same sign as the potential of the conductor. The interior surface of a hollow conducting vessel containing no charged bodies is entirely free from charge. For if any part of the surface were charged positively, a positively electrified particle moving in the direction of the force upon it, must reach a nega- tively charged surface at a lower potential. But the whole in- terior surface has the same potential. Hence it can have no charge. A conductor placed inside the vessel and communicating with it, may be considered as bounded by the interior surface. Hence such a conductor has no charge. Lines of Force. 47.] The line described by a point moving always in the direc- tion of the resultant intensity is called a Line of force. It cuts the equipotential surfaces at right angles. The properties of lines of force will be more fully explained afterwards, because Faraday has expressed many of the laws of electrical action in terms of his conception of lines of force drawn in the electric field, and in- dicating both the direction and the intensity at every point. 50.] ELECTRIC TENSION. 49 Electric Tension. 48.] Since the surface of a conductor is an equipotential surface, the resultant force is normal to the surface, and it will be shewn in Art. 78 that it is proportional to the superficial density of the electrification. Hence the electricity on any small area of the surface will be acted on by a force tending from the conductor and proportional to the product of the resultant force and the density, that is, proportional to the square of the resultant force. This force, which acts outwards as a tension on every part of the conductor, will be called electric Tension. It is measured like ordinary mechanical tension, by the force exerted on unit of area. The word Tension has been used by electricians in several vague senses, and it has been attempted to adopt it in mathematical language as a synonym for Potential ; but on examining the cases in which the word has been used, I think it will be more con- sistent with usage and with mechanical analogy to understand by tension a pulling force of so many pounds weight per square inch exerted on the surface of a conductor or elsewhere. We shall find that the conception of Faraday, that this electric tension exists not only at the electrified surface but all along the lines of force, leads to a theory of electric action as a phenomenon of stress in a medium. Electromotive Force. 49.] When two conductors at different potentials are connected by a thin conducting wire, the tendency of electricity to flow along the wire is measured by the difference of the potentials of the two bodies. The difference of potentials between two con- ductors or two points is therefore called the Electromotive force between them. Electromotive force cannot in all cases be expressed in the form of a difference of potentials. These cases, however, are not treated of in Electrostatics. We shall consider them when we come to heterogeneous circuits, chemical actions, motions of mag- nets, inequalities of temperature, &c. Capacity of a Conductor. 50.] If one conductor is insulated while all the surrounding con- ductors are kept at the zero potential by being put in commu- nication with the earth, and if the conductor, when charged with VOL. I. E 50 ELECTROSTATIC PHENOMENA. [51. a quantity E of electricity, has a potential F", the ratio of 2? to F" is called the Capacity of the conductor. If the conductor is com- pletely enclosed within a conducting* vessel without touching- it, then the charge on the inner conductor will be equal and op- posite to the charge on the inner surface of the outer conductor, and will be equal to the capacity of the inner conductor multiplied by the difference of the potentials of the two conductors. Electric Accumulators. A system consisting of two conductors whose opposed surfaces are separated from each other by a thin stratum of an insulating medium is called an electric Accumulator. The two conductors are called the Electrodes and the insulating medium is called the Dielectric. The capacity of the accumulator is directly propor- tional to the area of the opposed surfaces and inversely proportional to the thickness of the stratum between them. A Leyden jar is an accumulator in which glass is the insulating medium. Accumu- lators are sometimes called Condensers, but I prefer to restrict the term ' condenser ' to an instrument which is used not to hold electricity but to increase its superficial density. PROPERTIES OP BODIES IN RELATION TO STATICAL ELECTRICITY. Resistance to the Passage of Electricity through a Body. 51.] When a charge of electricity is communicated to any part of a mass of metal the electricity is rapidly transferred from places of high to places of low potential till the potential of the whole mass becomes the same. In the case of pieces of metal used in ordinary experiments this process is completed in a time too short to be observed, but in the case of very long and thin wires, such as those used in telegraphs, the potential does not become uniform till after a sensible time, on account of the resistance of the wire to the passage of electricity through it. The resistance to the passage of electricity is exceedingly dif- ferent in different substances, as may be seen from the tables at Arts. 362, 366, and 369, which will be explained in treating of Electric Currents. All the metals are good conductors, though the resistance of lead is 12 times that of copper or silver, that of iron 6 times, and that of mercury 60 times that of copper. The resistance of all metals increases as their temperature rises. 51.] ELECTRIC RESISTANCE. 51 Many liquids conduct electricity by electrolysis. This mode of conduction will be considered in Part II. For the present, we may regard all liquids containing water and all damp bodies as con- ductors, far inferior to the metals, but incapable of insulating a charge of electricity for a sufficient time to be observed. The re- sistance of electrolytes diminishes as the temperature rises. On the other hand, the gases at the atmospheric pressure, whether dry or moist, are insulators so nearly perfect when the electric tension is small that we have as yet obtained no evidence of electricity passing through them by ordinary conduction. The gradual loss of charge by electrified bodies may in every case be traced to imperfect insulation in the supports, the electricity either passing through the substance of the support or creeping over its surface. Hence, when two charged bodies are hung up near each other, they will preserve their charges longer if they are electrified in opposite ways, than if they are electrified in the same way. For though the electromotive force tending to make the electricity pass through the air between them is much greater when they are oppositely electrified, no per- ceptible loss occurs in this way. The actual loss takes place through the supports, and the electromotive force through the supports is greatest when the bodies are electrified in the same way. The result appears anomalous only when we expect the loss to occur by the passage of electricity through the air between the bodies. The passage of electricity through gases takes place, in general, by dis- ruptive discharge, and does not begin till the electromotive force has reached a certain value. The value of the electromotive force which can exist in a dielectric without a discharge taking place is called the Electric Strength of the dielectric. The electric strength of air diminishes as the pressure is reduced from the atmo- spheric pressure to that of about three millimetres of mercury. When the pressure is still further reduced, the electric strength rapidly increases ; and when the exhaustion is carried to the highest degree hitherto attained, the electromotive force required to produce a spark of a quarter of an inch is greater than that which will give a spark of eight inches in air at the ordinary pressure. A vacuum, that is to say, that which remains in a vessel after we have removed everything which we can remove from it, is there- fore an insulator of very great electric strength. The electric strength of hydrogen is much less than that of air. Certain kinds of glass when cold are marvellously perfect in- sulators, and Sir W. Thomson has preserved charges of electricity E 2 52 ELECTROSTATIC PHENOMENA. [52. for years in bulbs hermetically sealed. The same glass, however, becomes a conductor at a temperature below that of boiling water. Gutta-percha, caoutchouc, vulcanite, paraffin, and resins are good insulators, the resistance of gutta-percha at 75 F. being about 6 x 1 19 times that of copper. Ice, crystals, and solidified electrolytes, are also insulators. Certain liquids, such as naphtha, turpentine, and some oils, are insulators, but inferior to the best solid insulators. DIELECTRICS. Specific Inductive Capacity. 52.] All bodies whose insulating power is such that when they are placed between two conductors at different potentials the elec- tromotive force acting on them does not immediately distribute their electricity so as to reduce the potential to a constant value, are called by Faraday Dielectrics. It appears from the hitherto unpublished researches of Cavendish that he had, before 1773, measured the capacity of plates of glass, rosin, beeswax, and shellac, and had determined the ratio in which their capacity exceeded that of plates of air of the same dimensions. Faraday, to whom these researches were unknown, discovered that the capacity of an accumulator depends on the nature of the insulating medium between the two conductors, as well as on the dimensions and relative position of the conductors themselves. By substituting other insulating media for air as the dielectric of the accumulator, without altering it in any other respect, he found that when air and other gases were employed as the insulating medium the capacity of the accumulator remained sensibly the same, but that when shellac, sulphur, glass, &c. were substituted for air, the capacity was increased in a ratio which was different for each substance. By a more delicate method of measurement Boltzmann succeeded in observing the variation of the inductive capacity of gases at different pressures. This property of dielectrics, which Faraday called Specific In- ductive Capacity, is also called the Dielectric Constant of the sub- stance. It is defined as the ratio of the capacity of an accumulator when its dielectric is the given substance, to its capacity when the dielectric is a vacuum. If the dielectric is not a good insulator, it is difficult to measure 53-] ELECTRIC ABSORPTION. 53 its inductive capacity, because the accumulator will not hold a charge for a sufficient time to allow it to be measured ; but it is certain that inductive capacity is a property not confined to good insulators, and it is probable that it exists in all bodies. Absorption of Electricity. 53.] It is found that when an accumulator is formed of certain dielectrics, the following phenomena occur. When the accumulator has been for some time electrified and is then suddenly discharged and again insulated, it becomes recharged in the same sense as at first, but to a smaller degree, so that it may be discharged again several times in succession, these discharges always diminishing. This phenomenon is called that of the Re- sidual Discharge. The instantaneous discharge appears always to be proportional to the difference of potentials at the instant of discharge, and the ratio of these quantities is the true capacity of the accumulator; but if the contact of the discharger is prolonged so as to include some of the residual discharge, the apparent capacity of the accu- mulator, calculated from such a discharge, will be too great. The accumulator if charged and left insulated appears to lose its charge by conduction, but it is found that the proportionate rate of loss is much greater at first than it is afterwards, so that the measure of conductivity, if deduced from what takes place at first, would be too great. Thus, when the insulation of a submarine cable is tested, the insulation appears to improve as the electrifi- cation continues. Thermal phenomena of a kind at first sight analogous take place in the case of the conduction of heat when the opposite sides of a body are kept at different temperatures. In the case of heat we know that they depend on the heat taken in and given out by the body itself. Hence, in the case of the electrical phenomena, it has been supposed that electricity is absorbed and emitted by the parts of the body. We shall see, however, in Art. 329, that the phenomena can be explained without the hypothesis of absorp- tion of electricity, by supposing the dielectric in some degree heterogeneous. That the phenomenon called Electric Absorption is not an actual absorption of electricity by the substance may be shewn by charging the substance in any manner with electricity while it is surrounded by a closed metallic insulated vessel. If, when the 54: ELECTEOSTATIC PHENOMENA. [54. substance is charged and insulated, the vessel be instantaneously discharged and then left insulated, no charge is ever communicated to the vessel by the gradual dissipation of the electrification of the charged substance within it. 54.] This fact is expressed by the statement of Faraday that it is impossible to charge matter with an absolute and independent charge of one kind of electricity *. In fact it appears from the result of every experiment which has been tried that in whatever way electrical actions may take place among a system of bodies surrounded by a metallic vessel, the charge on the outside of that vessel is not altered. Now if any portion of electricity could be forced into a body so as to be absorbed in it, or to become latent, or in any way to exist in it, without being connected with an equal portion of the opposite electricity by lines of induction, or if, after having being absorbed, it could gradually emerge and return to its ordi- nary mode of action, we should find some change of electrification in the surrounding vessel. As this is never found to be the case, Faraday concluded that it is impossible to communicate an absolute charge to matter, and that no portion of matter can by any change of state evolve or render latent one kind of electricity or the other. He therefore regarded induction as * the essential function both in the first development and the consequent phenomena of electricity.' His 'induction' is (1298) a polarized state of the particles of the dielectric, each particle being positive on one side and negative on the other, the positive and the negative electrification of each particle being always exactly equal. Disruptive Discharge f. 55.] If the electromotive intensity at any point of a dielectric is gradually increased, a limit is at length reached at which there is a sudden electrical discharge through the dielectric, generally accompanied with light and sound, and with a temporary or per- manent rupture of the dielectric. The intensity of the electromotive force when this takes place is a measure of what we may call the electric^strength of the di- electric. It depends on the nature of the dielectric, and is greater in dense air than in rare air, and greater in glass than in air, but * Exp. Res., vol. i. series xi. *[[ ii. 'On the Absolute Charge of Matter,' and (1244). t See Faraday, Exp. Res., vol. i., series xii. and xiii. 55-] ELECTRIC GLOW. 55 in every case, if the electromotive force be made great enough, the dielectric gives way and its insulating power is destroyed, so that a current of electricity takes place through it. It is for this reason that distributions of electricity for which the electromotive intensity becomes anywhere infinite cannot exist. The Electric Glow. Thus, when a conductor having a sharp point is electrified, the theory, based on the hypothesis that it retains its charge, leads to the conclusion that as we approach the point the superficial density of the electricity increases without limit, so that at the point itself the surface-density, and therefore the resultant electrical force, would be infinite. If the air, or other surrounding dielectric, had an invincible insulating power, this result would actually occur ; but the fact is, that as soon as the resultant force in the neigh- bourhood of the point has reached a certain limit, the insulating power of the air gives way, so that the air close to the point becomes a conductor. At a certain distance from the point the resultant force is not sufficient to break through the insulation of the air, so that the electric current is checked, and the electricity accumulates in the air round the point. The point is thus surrounded by particles of air charged with electricity of the same kind with its own. The effect of this charged air round the point is to relieve the air at the point itself from part of the enormous electromotive force which it would have ex- perienced if the conductor alone had been electrified. In fact the surface of the electrified body is no longer pointed, because the point is enveloped by a rounded mass of charged air, the surface of which, rather than that of the solid conductor, may be regarded as the outer electrified surface. If this portion of charged air could be kept still, the electrified body would retain its charge, if not on itself at least in its neighbourhood, but the charged particles of air being free to move under the action of electrical force, tend to move away from the electrified body because it is charged with the same kind of elec- tricity. The charged particles of air therefore tend to move off in the direction of the lines of force and to approach those sur- rounding bodies which are oppositely electrified. When they are gone, other uncharged particles take their place round the point, and since these cannot shield those next the point itself from the excessive electric tension, a new discharge takes place, after which 56 ELECTROSTATIC PHENOMENA. [55. the newly charged particles move off, and so on as long as the body remains electrified. In this way the following phenomena are produced : At and close to the point there is a steady glow, arising from the con- stant discharges which are taking place between the point and the air very near it. The charged particles of air tend to move off in the same general direction, and thus produce a current of air from the point, con- sisting of the charged particles, and probably of others carried along by them. By artificially aiding this current we may increase the glow, and by checking the formation of the current we may pre- vent the continuance of the glow *. The electric wind in the neighbourhood of the point is sometimes very rapid, but it soon loses its velocity, and the air with its charged particles is carried about with the general motions of the atmo- sphere, and constitutes an invisible electric cloud. When the charged particles come near to any conducting surface, such as a wall, they induce on that surface a charge opposite to their own, and are then attracted towards the wall, but since the electro- motive force is small they may remain for a long time near the wall without being drawn up to the surface and discharged. They thus form an electrified atmosphere clinging to conductors, the presence of which may sometimes be detected by the electrometer. The electrical forces, however, acting between large masses of charged air and other bodies are exceedingly feeble compared with the ordinary forces which produce winds, and which depend on inequalities of density due to differences of temperature, so that it is very improbable that any observable part of the motion of ordinary thunder clouds arises from electrical causes. The passage of electricity from one place to another by the motion of charged particles is called Electrical Convection or Con- vective Discharge. The electrical glow is therefore produced by the constant passage of electricity through a small portion of air in which the tension is very high, so as to charge the surrounding particles of air which are continually swept off by the electric wind, which is an essential part of the phenomenon. 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-] p LAN 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 e y 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 T t and F z 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^ F 2 , 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 T 9 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^Ebe, 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 ] 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 */ r 2 ' 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 {(-aO a + (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. 68.] In order to simplify the mathematical process, it is con- venient to consider the action of an electrified body, not on another body of any form, but on an indefinitely small body, charged with an indefinitely small amount of electricity, and placed at any point of the space to which the electrical action extends. By making the charge of this body indefinitely small we render insensible its disturbing action on the charge of the first body. 72 ELECTROSTATICS. [69. Let e be the charge of the small body, and let the force acting on it when placed at the point (#,y, z) be Re, and let the direction- cosines of the force be I, m, n, then we may call R the resultant electrical_Intensity at the point (#, ^, z). If X, Y, Z denote the components of R, then X-Rl, Y=Rm, Z=Rn. In speaking of the resultant electrical intensity at a point, we do not necessarily imply that any force is actually exerted there, but only that if an electrified body were placed there it would be acted on by a force Re, where e is the charge of the body *. Definition. The Resultant electric Intensity at any point is the force which would be exerted on a small body charged with the unit of positive electricity, if it were placed there without disturbing the actual distribution of electricity. This force not only tends to move a body charged with electricity, but to move the electricity within the body, so that the positive electricity tends to move in the direction of R and the negative electricity in the opposite direction. Hence the quantity R is also called the E lee t r omo t i ve Intensity at the point (a?, y, z). When we wish to express the fact that the resultant intensity is a vector, we shall denote it by the German letter (. If the body is a dielectric, then, according to the theory adopted in this treatise, the electricity is displaced within it, so that the quantity of electricity which is forced in the direction of ( across unit of area fixed perpendicular to ( is > = #(; where $) is the displacement, ( the resultant intensity, and K the specific inductive capacity of the dielectric. If the body is a conductor, the state of constraint is continually giving way, so that a current of conduction is produced and main- tained as long as ( acts on the medium. Line-Integral of Electric Intensity, or Electromotive Force along an Arc of a Curve. 69.] The Electromotive force along a given arc AP of a curve is numerically measured by the work which would be done by the * The Electric and Magnetic Intensity correspond, in electricity and mag- netism, to the intensity of gravity, commonly denoted by g, in the theory of heavy bodies. 70.] ELECTKOMOTIVE FORCE. 73 electric force on a unit of positive electricity carried along the curve from J, the beginning, to P, the end of the arc. If s is the length of the arc, measured from A, and if the re- sultant intensity 72 at any point of the curve makes an angle e with the tangent drawn in the positive direction, then the work done on unit of electricity in moving along the element of the curve /fo will be Rcostds, and the total electromotive force E will be E = / ~R cos e ds, J the integration being extended from the beginning to the end of the arc. If we make use of the components of the intensity, the expres- sion becomes ds If X, Y, and Z are such that Xd% + Ydy + Zdz is the complete differential of F, a function of #, y, z, then E = f P (Xdx + Ydy + Zdz) = - f*dr = V A - V P ; J A J A where the integration is performed in any way from the point A to the point P, whether along the given curve or along any other line between A and P. In this case V is a scalar function of the position of a point in space, that is, when we know the coordinates of the point, the value of V is determinate, and this value is independent of the position and direction of the axes of reference. See Art. 16. On Functions of the Position of a Point. In what follows, when we describe a quantity as a function of the position of a point, we mean that for every position of the point the function has a determinate value. We do not imply that this value can always be expressed by the same formula for all points of space, for it may be expressed by one formula on one side of a given surface and by another formula on the other side. On Potential Functions. 70.] The quantity Xdx+ Ydy + Zdz is an exact differential whenever the force arises from attractions or repulsions whose in- tensity is a function of the distances from any number of points. 74 ELECTEOSTATICS. [71. For if r-L be the distance of one of the points from the point (#, y, z\ and if R l be the repulsion, then with similar expressions for Y l and ^, so that and since ^ is a function of r x only, R 1 dr^ is an exact differential of some function of r^ , say V^ . Similarly for any other force R^ acting from a centre at dis- tance r 2 , X 2 dx + Y 2 dy + Zzdz = R 2 dr 2 = <*JJ. But X = Xj + Xg-f&c. and T and # are compounded in the same way, therefore Xdx+ Ydy+Zdz = -d^-d^+be. - -dV. The integral of this quantity, under the condition that it vanishes at an infinite distance, is called the Potential Function. The use of this function in the theory of attractions was intro- duced by Laplace in the calculation of the attraction of the earth. Green, in his essay ' On the Application of Mathematical Analysis to Electricity/ gave it the name of the Potential Function. Gauss, working independently of Green, also used the word Potential. Clausius and others have applied the term Potential to the work which would be done if two bodies or systems were removed to an infinite distance from one another. We shall follow the use of the word in recent English works, and avoid ambiguity by adopting the following definition due to Sir W. Thomson. Definition of Potential. The Potential at a Point is the work which would be done on a unit of positive electricity by the elec- tric forces if it were placed at that point without disturbing the electric distribution, and carried from that point to an infinite distance : or, what comes to the same thing, the work which must be done by an external agent in order to bring the unit of positive electricity from an infinite distance (or from any place where the potential is zero) to the given point. 71.] Expressions for the Resultant Intensity and its components in terms of the Potential. Since the total electromotive force along any arc AB is 72.] POTENTIAL^ 75 if we put ds for the arc AB we shall have for the force resolved in the direction of ds, .72 cos e = =- : ds whence, by assuming ds parallel to each of the axes in succession, we get X-- . r-- Z--' dx dy dz ' E ldV ~~ Ida dV dV T 7 r We shall denote the intensity itself, whose magnitude, or tensor, is E and whose components are X, Y, Z, by the German letter ($, as in Arts. 17 and 68. The Potential at all Points within a Conductor is the same. 72.] A conductor is a body which allows the electricity within it to move from one part of the body to any other when acted on by electromotive force. When the electricity is in equilibrium there can be no electromotive force -acting within the conductor. Hence E = throughout the whole space occupied by the con- ductor. From this it follows that dV dV dV ^=> *=' ^= ; and therefore for every point of the conductor r=c, where C is a constant quantity. Since the potential at all points within the substance of the conductor is C, the quantity C is called the Potential of the con- ductor. C may be denned as the work which must be done by external agency in order to bring a unit of electricity from an infinite distance to the conductor, the distribution of electricity being supposed not to be disturbed by the presence of the unit. It will be shewn at Art. 246 that in general when two bodies of different kinds are in contact, an electromotive force acts from one to the other through the surface of contact, so that when they are in equilibrium the potential of the latter is higher than that of the former. For the present, therefore, we shall suppose all our conductors made of the same metal, and at the same temperature. If the potentials of the conductors A and B be V A and V B re- spectively, then the electromotive force along a wire joining A and B will be F-F 76 ELECTROSTATICS. [73. in the direction AS, that is, positive electricity will tend to pass from the conductor of higher potential to the other. Potential, in electrical science, has the same relation to Elec- tricity that Pressure, in Hydrostatics, has to Fluid, or that Tem- perature, in Thermodynamics, has to Heat. Electricity, Fluids, and Heat all tend to pass from one place to another, if the Poten- tial, Pressure, or Temperature is greater in the first place than in the second. A fluid is certainly a substance, heat is as certainly not a substance, so that though we may find assistance from ana- logies of this kind in forming clear ideas of formal relations of electrical quantities, we must be careful not to let the one or the other analogy suggest to us that electricity is either a substance like water, or a state of agitation like heat. Potential due to any Electrical System. 73.] Let there be a single electrified point charged with a quantity e of electricity, and let r be the distance of the point #', y', / from it, then A> ra> 7= / Edr = ~dr = - J r J r r 2 r Let there be any number of electrified points whose coordinates are (# 15 y l5 ^), (# 2) y^ z 2 ), &c. and their charges e lt e 2 , &c., and let their distances from the point (#', y'> z) be r lt r 2 , &c., then the potential of the system at (#', y\ /) will be Let the electric density at any point (UK, y> z) within an elec- trified body be p, then the potential due to the body is 7 = (j( p -dxdydz', where r = {(cc-xj + (y-yj + (z-zj}^ the integration being extended throughout the body. On the Proof of the Law of the Inverse Square. 74 .] The fact that the force between electrified bodies is inversely as the square of the distance may be considered to be established by Coulomb's direct experiments with the torsion-balance. The results, however, which we derive from such experiments must be regarded as affected by an error depending on the probable error of each experiment, and unless the skill of the operator be very great, PROOF OF THE LAW OF FORCE. 77 the probable error of an experiment with the torsion-balance is considerable. A far more accurate verification of the law of force may be deduced from an experiment similar to that described at Art 32 (Exp. VII). Cavendish, in his hitherto unpublished work on electricity, makes the evidence of the law of force depend on an experiment of this kind. He fixed a globe on an insulating 1 support, and fastened two hemispheres by glass rods to two wooden frames hinged to an axis so that the hemispheres, when the frames were brought together, formed an insulated spherical shell concentric with the globe. The globe could then be made to communicate with the hemispheres by means of a short wire, to which a silk string was fastened so that the wire could be removed without discharging the apparatus. The globe being in communication with the hemispheres, he charged the hemispheres by means of a Leyden jar, the potential of which had been previously measured by an electrometer, and immediately drew out the communicating wire by means of the silk string, removed and discharged the hemispheres, and tested the electrical condition of the globe by means of a pith ball electro- meter. No indication of any charge of the globe could be detected by the pith ball electrometer, which at that time (1773) was considered the most delicate electroscope. Cavendish next communicated to the globe a known fraction of the charge formerly communicated to the hemispheres, and tested the globe again with his electrometer. He thus found that the charge of the globe in the original experiment must have been less than ^ of the charge of the whole apparatus, for if it had been greater it would have been detected by the electrometer. He then calculated the ratio of the charge of the globe to that of the hemispheres on the hypothesis that the repulsion is inversely as a power of the distance differing slightly from 2, and found that if this difference was -^ there would have been a charge on the globe equal to -^ of that of the whole apparatus, and therefore capable of being detected by the electrometer. 74 b.~\ The experiment has recently been repeated at the Cavendish Laboratory in a somewhat different manner. The hemispheres were fixed on an insulating stand, and the globe 78 ELECTROSTATICS. [74 6. fixed in its proper position within them by means of an ebonite ring. By this arrangement the insulating support of the globe was never exposed to the action of any sensible electric force, and therefore never became charged, so that the disturbing effect of electricity creeping along the surface of the insulators was entirely removed. Instead of removing the hemispheres before testing the potential of the globe, they were left in their position, but discharged to earth. The effect of a given charge of the globe on the electro- meter was not so great as if the hemispheres had been removed, but this disadvantage was more than compensated by the perfect security afforded by the conducting vessel against all external electric disturbances. The short wire which made the connexion between the shell and the globe was fastened to a small metal disk which acted as a lid to a small hole in the shell, so that when the wire and the lid were lifted up by a silk string, the electrode of the electrometer could be made to dip into the hole and rest on the globe within. The electrometer was Thomson's Quadrant Electrometer described in Art. 219. The case of the electrometer and one of the electrodes were always connected to earth, and the testing electrode was con- nected to earth till the electricity of the shell had been discharged. To estimate the original charge of the shell, a small brass ball was placed on an insulating support at a considerable distance from the shell. The operations were conducted as follows : The shell was charged by communication with a Leyden jar. The small ball was connected to earth so as to give it a negative charge by induction, and was then left insulated. The communicating wire between the globe and the shell was removed by a silk string. The shell was then discharged, and kept connected to earth. The testing electrode was disconnected from earth, and made to touch the globe, passing through the hole in the shell. Not the slightest effect on the electrometer could be observed. To test the sensitiveness of the apparatus the shell was discon- nected from earth and the small ball was discharged to earth. The electrometer then showed a positive deflection, D. The negative charge of the brass ball was about -fa of the ori- ginal charge of the shell, and the positive charge induced by the ball when the shell was put to earth was about . J of that of the balL 74 .] PEOOF OF THE LAW OF FORCE. 79 Hence when the ball was put to earth the potential of the shell, as indicated by the electrometer, was about T | F of its original potential. But if the repulsion had been as r q ~ 2 , the potential of the globe would have been 0-1478 q of that of the shell by equation 22, p. 81. Hence if + d be the greatest deflexion of the electrometer which could escape observation, and D the deflexion observed in the second part of the experiment, q cannot exceed lil. ~ 72 D Now even in a rough experiment D was more than 300 d, so that q cannot exceed 1 - 21600* Theory of the Experiment. 74 c.~\ To find the potential at any point due to a uniform spherical shell, the repulsion between two units of matter being any given function of the distance. Let (/) be the repulsion between two units at distance /-, and let/(r) be such that (r)rfr. (1) Let the radius of the shell be a, and its surface density o-, then, if a denotes the whole mass of the shell, a = 477 a 2 a-. (2) Let b denote the distance of the given point from the centre of the shell, and let r denote its distance from any given point of the shell. If we refer the point on the shell to spherical coordinates, the pole being the centre of the shell, and the axis the line drawn to the given point, then r 2 = 2 + 5 2 -2dO, (4) and the potential due to this element at the given point is aa 2 s { n of-&ded; (5) and this has to be integrated with respect to from = to $ = 2 TT, which gives (6) r which has to be integrated from = to = it. 80 ELECTROSTATICS. [74 C. Differentiating (3) we find rdr = absmOdO. (?) Substituting the value of dd in (6) we obtain 2w from which we may determine q in terms of the results of the experiment. 740.] Laplace gave the first demonstration that no function of the distance except the inverse square satisfies the condition that a uniform spherical shell exerts no force on a particle within it *. If we suppose that /3 in equation (15) is always zero, we may apply the method of Laplace to determine the form of f(r). We have by (15), Differentiating twice with respect to , and dividing by #, we find f"(a + b] =f"(a b). If this equation is generally true /" ( r ) = (r)dr = & = C Q + ^-, j r r r We may observe, however, that though the assumption of Cavendish, that the force varies as some power of the distance, may appear less general than that of Laplace, who supposes it to be any function of the distance, it is the only one consistent with the fact that similar figures can be electrified so as to have similar electrical properties. For if the force were any function of the distance except a power of the distance, the ratio of the force at two different distances would not be a function of the ratio of the distances, but would depend on the absolute value of the distances, and would therefore involve the ratios of these distances to an absolutely fixed length. Indeed Cavendish himself points out that on his own hypothesis as to the constitution of the electric fluid, it is impossible for the distribution of electricity to be accurately similar in two conductors geometrically similar, unless the charges are proportional to the volumes. For he supposes the particles of the electric fluid to be closely pressed together near the surface of the body, and this is equivalent to supposing that the law of repulsion is no longer the inverse square, but that as soon as the particles come into contact, their repulsion begins to increase at a much greater rate with any further diminution of their distance. Surface-Integral of Electric Induction, and Electric Displacement through a surface. 75.] Let R be the resultant intensity at any point of the surface, and e the angle which R makes with the normal drawn towards the positive side of the surface, then R cos e is the component of the intensity normal to the surface, and if dS is the element of the surface, the electric displacement through dS will be, by Art. 68, KRcoscdS. 4?r since we do not at present consider any dielectric except air, K 1. We may, however, avoid introducing at this stag-e the theory of electric displacement, by calling R cos e dS the Induction through the element dS. This quantity is well known in mathematical 76.] ELECTRIC INDUCTION. 83 physics, but the name of induction is borrowed from Faraday. The surface-integral of induction is R cos e dS t and it appears by Art. 21, that if X, Y, Z are the components of R, and if these quantities are continuous within a region bounded by a closed surface S, the induction reckoned from within outwards is f dY dZ the integration being extended through the whole space within the surface. Induction through a Closed Surface due to a Single Centre of Force. 76.] Let a quantity e of electricity be supposed to be placed at a point 0, and let r be the distance of any point P from 0, the force at that point is R er~ 2 in the direction OP. Let a line be drawn from in any direction to an infinite dis- tance. If is without the closed surface this line will either not cut the surface at all, or it will issue from the surface as many times as it enters. If is within the surface the line must first issue from the surface, and then it may enter and issue any number of times alternately, ending by issuing from it. Let e be the angle between OP and the normal to the surface drawn outwards where OP cuts it, then where the line issues from the surface, cos e will be positive, and where it enters, cos will be negative. Now let a sphere be described with centre and radius unity, and let the line OP describe a conical surface of small angular aperture about as vertex. This cone will cut off a small element d<* from the surface of the sphere, and small elements dS 19 dS 2 , &c. from the closed surface at the different places where the line OP intersects it. Then, since any one of these elements dS intersects the cone at a distance r from the vertex and at an obliquity e, dS = r 2 sec e ^w ; and, since R = er~ 2 , we shall have R cos c dS = e d<* ; the positive sign being taken when r issues from the surface, and the negative where it enters it. If the point is without the closed surface, the positive values G 2 84 ELECTROSTATICS. [77. are equal in number to the negative ones, so that for any direction of r, ^.R cos e dS = 0, and therefore / / R cos e dS = 0, the integration being extended over the whole closed surface. If the point is within the closed surface the radius vector OP first issues from the closed surface, giving a positive value of e da>, and then has an equal number of entrances and issues, so that in this case 2 R cos e dS = e da. Extending the integration over the whole closed surface, we shall include the whole of the spherical surface, the area of which is 4 TT, so that rr rr I I R cos e dS = e I I d = kite. Hence we conclude that the total induction outwards through a closed surface due to a centre of force e placed at a point is zero when is without the surface, and ne when is within the surface. Since in air the displacement is equal to the induction divided by 4 TT, the displacement through a closed surface, reckoned out- wards, is equal to the electricity within the surface. Corollary. It also follows that if the surface is not closed but is bounded by a given closed curve, the total induction through it is co e, where G> is the solid angle subtended by the closed curve at 0. This quantity, therefore, depends only on the closed curve, and the form of the surface of which it is the boundary may be changed in any way, provided it does not pass from one side to the other of the centre of force. On the Equations of Laplace and Poisson. 77.] Since the value of the total induction of a single centre of force through a closed surface depends only on whether the centre is within the surface or not, and does not depend on its position in any other way, if there are a number of such centres e lt e 2 , &c. within the surface, and /, /, &c. without the surface, we shall have r r I I Rcost dS = 47T0; where e denotes the algebraical sum of the quantities of electricity at all the centres of force within the closed surface, that is, the total electricity within the surface, resinous electricity being reck- oned negative. j3a.] EQUATIONS OF LAPLACE AND POISSON. 85 If the electricity is so distributed within the surface that the density is nowhere infinite, we shall have by Art. 64, 4 TT e = 4 TT / / / p dx dy dz, and by Art. 75, If we take as the closed surface that of the element of volume dx dy dz, we shall have, by equating these expressions, dX dY dZ -T -- \r -j -- h -7- = 47rp; dx dy dz and if a potential V exists, we find by Art. 7 1 , d 2 7 This equation, in the case in which the density is zero, is called Laplace's Equation. In its more general form it was first given by Poisson. It enables us, when we know the potential at every point, to determine the distribution of electricity. We shall denote, as in Art. 26, the quantity . T5 + T2 + TT bv - oar dy 2 dz 2 and we may express Poisson's equation in words by saying that the electric density multiplied by 4w is the concentration of the potential. Where there is no electrification, the potential has no concentration, and this is the interpretation of Laplace's equation. By Art. 72, V is constant within a conductor. Hence within a conductor the volume-density is zero, and the whole charge must be on the surface. If we suppose that in the superficial and linear distributions of electricity the volume-density p remains finite, and that the elec- tricity exists in the form of a thin stratum or a narrow fibre, then, by increasing p and diminishing the depth of the stratum or the section of the fibre, we may approach the limit of true superficial or linear distribution, and the equation being true throughout the process will remain true at the limit, if interpreted in accordance with the actual circumstances. Variation of the Potential at a Charged Surface. 78 a.] The potential function, F, must be physically continuous in the sense defined in Art. 7, except at the bounding surface of 86 ELECTBOSTATICS. [78 a. two different media, in which case, as we shall see in Art. 246, there may be a difference of potential between the substances, so that when the electricity is in equilibrium, the potential at a point in one substance is higher than the potential at the contiguous point in the other substance by a constant quantity, C, depending on the natures of the two substances and on their temperatures. But the first derivatives of V with respect to #, y, or z may be discontinuous, and, by Art. 8, the points at which this discontinuity occurs must lie in a surface, the equation of which may be expressed in the form = (# } y^ z ) = o. (l) This surface separates the region in which $ is negative from the region in which is positive. Let VL denote the potential at any given point in the negative region, and V 2 that at any given point in the positive region, then at any point in the surface at which $ = 0, and which may be said to belong to both regions, r 1 + c=%, (2) where C is the constant excess of potential, if any, in the substance on the positive side of the surface. Let I, m, n be the direction-cosines of the normal v 2 drawn from a given point of the surface into the positive region. Those of the normal v l drawn from the same point into the negative region will be I, m, and n. The rates of variation of V along the normals are (3) (4) dv ax ay dz dK 7 dK d\ dK T*-= l-r^ + m-^- +n-jl-. dv 2 dx dy dz Let any line be drawn on the surface, and let its length, measured from a fixed point in it, be s, then at every point of the surface, and therefore at every point of this line, V^ T[ = C. Differentiating this equation with respect to s, we get _ dx " dx dsdy " dy ds \4& " dz > ds and since the normal is perpendicular to this line , dx dy dz 786.] POTENTIAL NEAR A CHARGED SURFACE. 87 From (3), (4), (5), (6) we find AV t AT, djr dr t . _ dy dy dV, ,dV, dV^ -- r 1 = n 13-T + -T^} ' ( 9 ) dz dz \dv dv^' If we consider the variation of the electromotive intensity at a point in passing through the surface, that component of the in- tensity which is normal to the surface may change abruptly at the surface, but the other two components parallel to the tangent plane remain continuous in passing through the surface. 783.] To determine the charge of the surface, let us consider a closed surface which is partly in the positive region and partly in the negative region, and which therefore encloses a portion of the surface of discontinuity. The surface integral, r r II R cos c ffS t extended over this surface, is equal to 4 ire, where e is the quantity of electricity within the closed surface. Proceeding as in Art. 2 1 , we find S, (2) where the triple integral is extended throughout the closed surface, and the double integral over the surface of discontinuity. Substituting for the terms of this equation their values from But by the definition of the volume- density, p, and the surface- density, , + ^ ^ (J g) Hence, comparing the last terms of these two equations, . (13) This equation is called the characteristic equation of V at an elec- trified surface of which the surface-density is o-. 88 ELECTKOSTATICS. [78 C. 78 conductor. This relation between the surface-density and the resultant in- tensity close to the surface of a conductor is known as Coulomb's Law, Coulomb having ascertained by experiment that the intensity of the electric force near a given point of the surface of a conductor is normal to the surface and proportional to the surface-density at the given point. The numerical relation R = 477(7 was established by Poisson. 8 1.] CHARGED WIRE. 91 The force acting on an element, dS, of the charged surface of a conductor is, by Art. 79, (since the intensity is zero on the inner side of the surface,) 8 77 This force acts outwards from the conductor, whether the charge of the surface is positive or negative. Its value in dynes per square centimetre is \E n W'-W=^(e'-e)(Y'+r). (4) VOL. I. H 98 SYSTEM OF CONDUCTOKS. But r=*S(*F) and r'=|S(/r). Substituting these values in equation (4) we find S(F) = S(/F). (5) Hence if, in the same fixed system of electrified conductors, we consider two different states of electrification, the sum of the products of the charges in the first state into the potentials of the corresponding portions of the conductors in the second state, is equal to the sum of the products of the charges in the second state into the potentials of the corresponding conductors in the first state. This result corresponds, in the elementary theory of electricity, to Green's Theorem in the analytical theory. By properly choosing the initial and final state of the system, we may deduce a number of useful results. 85 $.] From (4) and (5) we find another expression for the in- crement of the energy, in which it is expressed in terms of the increments of potential, w-w-=\^(e'+e)(V'-v). (6) If the increments are infinitesimal, we may write (4) and (6) dr=2(F6tf) = S( " The number of different coefficients with double suffix is there- fore J n (n 1), being one for each pair of conductors. By solving the equations (16) for e lt e 2 &c., we obtain n equations giving the charges in terms of the potentials (18) 102 SYSTEM OF CONDUCTORS. [87. We have in this case also q r8 = q sr , for de r d dWy d dW v de s , . By substituting the values of the charges in the equation for the electric energy r=iOi*i+-+r5. ..+.], (20) we obtain an expression for the energy in terms of the potentials (21) A coefficient in which the two suffixes are the same is called the Electric Capacity of the conductor to which it belongs. Definition. The Capacity of a conductor is its charge when its own potential is unity, and that of all the other conductors is zero. This is the proper definition of the capacity of a conductor when no further specification is made. But it is sometimes convenient to specify the condition of some or all of the other conductors in a different manner, as for instance to suppose that the charge of certain of them is zero, and we may then define the capacity of the conductor under these conditions as its charge when its potential is unity. The other coefficients are called coefficients of induction. Any one of them, as q ra denotes the charge of A r when A g is raised to potential unity, the potential of all the conductors except A 8 being zero. The mathematical calculation of the coefficients of potential and of capacity is in general difficult. We shall afterwards prove that they have always determinate values, and in certain special cases we shall calculate these values. We shall also shew how they may be determined by experiment. When the capacity of a conductor is spoken of without specifying the form and position of any other conductor in the same system, it is to be interpreted as the capacity of the conductor when no other conductor or electrified body is within a finite distance of the conductor referred to. It is sometimes convenient, when we are dealing with capacities and coefficients of induction only, to write them in the form [ A . P], this symbol being understood to denote the charge on A when P is raised to unit potential. In like manner [(A + H) . (P + Q)] would denote the charge on 89 #] PROPERTIES OF THE COEFFICIENTS. 103 A + B when P and Q are both raised to potential 1, and it is manifest that since [(A+S)(P+Q)-\ = the compound symbols may be combined by addition and multipli- cation as if they were symbols of quantity. The symbol [A . A] denotes the charge on A when the potential of A is 1, that is to say, the capacity of A. In like manner [( A + B)(A + Q)] denotes the sum of the charges on A and B when A and Q are raised to potential 1, the potential of all the conductors except A and Q being zero. It may be decomposed into [A.A-] + [A.] + [A.Q] + [B.Q]. The coefficients of potential cannot be dealt with in this way. The coefficients of induction represent charges, and these charges can be combined by addition, but the coefficients of potential represent potentials, and if the potential of A is \ and that of B is T'g, the sum ^-t-7J has no physical meaning bearing on the phenomena, though V^ P 2 represents the electromotive force from A to B. The coefficients of induction between two conductors may be expressed in terms of the capacities of the conductors and that of the two conductors together, thus : [A.S] = Dimensions of the coefficients. & 88.] Since the potential of a charge e at a distance r is -, the dimensions of a charge of electricity are equal to those of the product of a potential into a line. The coefficients of capacity and induction have therefore the same dimensions as a line, and each of them may be represented by a straight line, the length of which is independent of the system of units which we employ. For the same reason, any coefficient of potential may be repre- sented as the reciprocal of a line. On certain conditions which the coefficients must satisfy. 890.] In the first place, since the electric energy of a system is an essentially positive quantity, its expression as a quadratic 104 SYSTEM OF CONDUCTORS. function of the charges or of the potentials must be positive, whatever values, positive or negative, are given to the charges or the potentials. Now the conditions that a homogeneous quadratic function of n variables shall be always positive are n in number, and may be written > 0, Pll '"Pin , - - - - > 0. (22) Pnl ' ' ' Pnn , These n conditions are necessary and sufficient to ensure that W shall be essentially positive *. But since in equation (16) we may arrange the conductors in any order, every determinant must be positive which is formed sym- metrically from the coefficients belonging to any combination of the n conductors, and the number of these combinations is 2 M 1. Only n, however, of the conditions so found can be independent. The coefficients of capacity and induction are subject to con- ditions of the same form. 89 .] The coefficients of potential are all positive , lut none of the coefficients p rs is greater than p rr or p ss . JPor let a charge unity be communicated to A r , the other con- ductors being uncharged. A system of equipotential surfaces will be formed. Of these one will be the surface of A r , and its potential will be p rr . If A s is placed in a hollow excavated in A r so as to be completely enclosed by it, then the potential of A s will also be p rr . If, however, A 8 is outside of A r its potential p rs will lie between p rr and zero. For consider the lines of force issuing from the charged con- ductor A r . The charge is measured by the excess of the number of lines which issue from it over those which terminate in it. i Hence, if the conductor has no charge, the number of lines which \ enter the conductor must be equal to the number which issue from it. The lines which enter the conductor come from places of greater potential, and those which issue from it go to places of less poten- * See Williamson's Differential Calculus, 3rd edition, p. 407. 89 dJ] PROPERTIES OF THE COEFFICIENTS. 105 tial. Hence the potential of an uncharged conductor must be intermediate between the highest and lowest potentials in the field, and therefore the highest and lowest potentials cannot belong to any of the uncharged bodies. The highest potential must therefore be p rr , that of the charged body A r , the lowest must be that of space at an infinite distance, which is zero, and all the other potentials such as p r8 must lie between p rr and zero. If A s completely surrounds A t , thenj9 r8 =p rt . 89 receives a charge e l9 the distribution of the electricity on the con- ductors of the system will not be disturbed by JB, as B is still without charge in any part, and the electric energy of the system will be simply K^i = W^n- Now let B become a conductor. Electricity will flow from places of higher to places of lower potential, and in so doing will diminish the electric energy of the system, so that the quantity 4 e i 2 Pn must diminish. But e l remains constant, therefore p li must diminish. Also if B increases by another body b being placed in contact with it, /? n will be further diminished. For let us first suppose that there is no electric communication between B and 6 ; the introduction of the new body b will ,' diminish /> n . Now let a communication be opened between B 906.] APPEOXIMATE VALUES OF THE COEFFICIENTS. 107 and b. If any electricity flows through it, it flows from a place of higher to a place of lower potential, and therefore, as we have shown, still further diminishes jO n . Hence the diminution of jt? n by the body B is greater than that which would be produced by any body the surface of which can be inscribed in B } and less than that produced by any body the surface of which can be described about B. We shall shew in Chapter XI, that a sphere of diameter 6 at a distance r diminishes the value of p tl by a quantity which is approximately J -^ Hence if the body B is of any other figure, and if b is its greatest diameter, the diminution of the value of jt? 11 must be less than . Hence if the greatest diameter of B is so small compared with its distance from A l that we may neglect quantities of the order b 5 -4- 5 we may consider the reciprocal of the capacity of A when alone in the field as a sufficient approximation to p l t . 90 a.~\ Let us therefore suppose that the capacity of A 1 when alone in the field is K 19 and that of A^ K 2 , and let the mean distance between A l and A 2 be r, where r is very great compared with the dimensions of A l and A 2) then we may write JL - i _! F^e.K^ + e.r-i, ? t,* firf + fa ^ F 2 = e^ + e 2 K 2 -\ Hence 9n = K^ (l _J ^^^ ^ i s . * 2 22 = K 2 ( 1 - JTj KI r- 2 )- 1 . Of these coefficients q n and q 22 are the capacities of A and A 2 when, instead of being each alone at an infinite distance from any other body, they are brought so as to be at a distance r from each other. 90 b.~\ When two conductors are placed so near together that their coefficient of mutual induction is large, the combination is called a Condenser. Let A and B be the two conductors or electrodes of a con- denser. & f z - A* r 1 108 SYSTEM OF CONDUCTORS. [906. Let L be the capacity of A, N that of j5, and M the coefficient of mutual induction. (We must remember that M is essentially negative, so that the numerical value of L + M and M+ N is less than L or N.) Let us suppose that a and I are the electrodes of another con- denser at a distance R from the first, R being very great com- pared with the dimensions of either condenser, and let the coefficients of capacity and induction of the condenser al when alone be I, m, n. Let us calculate the effect of one of the condensers on the coefficients of the other. Let D = LN-M 2 and d=ln-m 2 -, then the coefficients of potential for each condenser by itself are PAB = PBB = The values of these coefficients will not be sensibly altered when the two condensers are at a distance R. The coefficient of potential of any two conductors at distance R is R~ l , so that The equations of potential are therefore V A = V = Solving these equations for the charges, we find 1 R*-(L+2M+N)(l+2m + (L q AB =M'=M+ ^- R(L+M)(l+m) * where If, M\ N' are what L, M, N become when the second con- denser is brought into the field. 91.] APPROXIMATE VALUES OF THE COEFFICIENTS. 109 If only one conductor, a, is brought into the field, m=n=0 ) and If T i " - h Rl(L+M) If there are only the two simple conductors, A and a, M N=m = n = 0, L 2 l ELI and q AA = L R*-Ll' expressions which are the same as those found in Art. 90 a. The quantity L + 2 H + N is the total charge of the condenser when its electrodes are at potential 1. It cannot exceed half the greatest diameter of the condenser. L-t-Mis the charge of the first electrode, and M + N that of the second when both are at potential 1. These quantities must be each of them positive and less than the capacity of the electrode by itself. Hence the corrections to be applied to the coefficients of capacity of a condenser are much smaller than those for a simple conductor of equal capacity. Approximations of this kind are often useful in estimating the capacities of conductors of irregular form placed at a finite distance from other conductors. 91.] When a round conductor, J.%, of small size compared with the distances between the conductors, is brought into the field, the coefficient of potential of A l on A 2 will be increased when A 5 is inside and diminished when A 3 is outside of a sphere whose diameter is the straight line A L A 2 . For if A L receives a unit charge there will be a distribution of electricity on A 3 , -\-e being on the side furthest from A lt and e on the side nearest A . The potential at A 2 due to this distribution A/ on A 3 will be positive or negative as +e or e is nearest to A 2 , and if the form of A 3 is not very elongated this will depend on whether the angle A l A 3 A 2 is obtuse or acute, and therefore on whether A z is inside or outside the sphere described on A l A 2 as ., ^ ^ diameter. If A 3 is of an elongated form it is easy to see that if it is placed with its longest axis in the direction of the tangent to the circle 110 SYSTEM OF CONDUCTORS. [92. ^ drawn through the points A lf A z , A 2 it may increase the potential of A 2 , even when it is entirely outside the sphere, and how by placing it with its longest axis in the direction of the radius of ; .the sphere, it may diminish the potential of J 2 , even when entirely within the sphere. But this proposition is only intended for forming a rough estimate of the phenomena to be expected in a given arrangement of apparatus. 92.] If a new conductor, A 3 , is introduced into the field, the capacities of all the conductors already there are increased, and the numerical values of the coefficients of induction between every pair of them are diminished. Let us suppose that A 1 is at potential unity and all the rest at potential zero. Since the charge of the new conductor is negative it will induce a positive charge on every other conductor, and will therefore increase the positive charge of A l and diminish the negative charge of each of the other conductors. 93 a.~\ Work done ly the electric forces during the displacement of a system of insulated charged conductors. Since the conductors are insulated, their charges remain constant during the displacement. Let their potentials be 7^, 7, . . . ^ n before and Tj', 7^', ... V after the displacement. The electrical energy is before the displacement, and r'-i after the displacement. The work done by the electric forces during the displacement is the excess of the initial energy W over the final energy W 9 or This expression gives the work done during any displacement, small or large, of an insulated system. To find the force tending to produce a particular kind of dis- placement, let $ be the variable whose variation corresponds to the kind of displacement, and let 3> be the corresponding force, reckoned positive when the electric force tends to increase <, then where W t denotes the expression for the electric energy as a quadratic function of the charges. 93 C -] MECHANICAL FORCES. Ill 93 a.] To prove that - + = 0. d(f> d$ We have three different expressions for the energy of the system, (i) r=i2(n, a definite function of the n charges and n potentials (2) ^=J2S (,./), where r and s may be the same or different, and both rs and sr are to be included in the summation. This is a function of the n charges and of the variables which define the configuration. Let (/> be one of these. (3) r r =iss(^.?r.), where the summation is to be taken as before. This is a function of the n potentials and of the variables which define the configura- tion of which = 112 SYSTEM Or CONDUCTORS. [94. and if the system is displaced under the condition that all the potentials remain constant, the work done by the electric forces is or the work done by the electric forces in this case is equal to the increment of the electric energy. Here,, then, we have an increase of energy together with a quan- tity of work done by the system. The system must therefore be supplied with energy from some external source, such as a voltaic battery, in order to maintain the potentials constant during the displacement. The work done by the battery is therefore equal to the sum of the work done by the system and the increment of energy, or, since these are equal, the work done by the battery is twice the work done by the system of conductors during the displacement. On the comparison of similar electrified systems. 94.] If two electrified systems are similar in a geometrical sense, so that the lengths of corresponding lines in the two systems are as L to If, then if the dielectric which separates the conducting bodies is the same in both systems, the coefficients of induction and of capacity will be in the proportion of L to L '. For if we consider corresponding portions, A and A ', of the two systems, and suppose the quantity of electricity on A to be , and that on A' to be /, then the potentials 7 and 7' at corresponding points B and _Z?', due to this electrification, will be * / Y - and V - ~ AB ' " A'ff But AB is to A'B' as L to L' ', so that we must have e:e'::L7'. L'7'. But if the inductive capacity of the dielectric is different in the two systems, being K in the first and K' in the second, then if the potential at any point of the first system is to that at the cor- responding point of the second as 7 to 7', and if the quantities of electricity on corresponding parts are as E to E', we shall have ei, let us write ***!;* dx dx dy dy dz dz The reader who is not acquainted with the method of Quater- nions may, if it pleases him, regard the expressions V 2 ^ and tf.V^V^ as mere conventional abbreviations for the quantities to which they are equated above, and as in what follows we shall employ ordinary Cartesian methods, it will not be necessary to remember the Quaternion interpretation of these expressions. The 96 a.] GREEN'S THEOREM. 119 reason, however, why we use as our abbreviations these expressions and not single letters arbitrarily chosen, is, that in the language of Quaternions they represent fully the quantities to which they are equated. The operator V applied to the scalar function ^ gives the space-variation of that function, and the expression xS.V^V^ is the scalar part of the product of two space- variations, or the product of either space-variation into the resolved part of the dty other in its own direction. The expression -j- is usually written in Quaternions S. UvW, Uv being a unit- vector in the direction of the normal. There does not seem much advantage in using this notation here, but we shall find the advantage of doing so when we come to deal with anisotropic media. Statement of Green's Theorem. Let # and be two functions of x, y, z, which, with their first derivatives, are finite and continuous within the acyclic region s, bounded by the closed surface s, then = ff *ll;*-fff where the double integrals are to be extended over the whole closed surface *, and the triple integrals throughout the field, s, enclosed by that surface. To prove this, let us write, in Art. 21, Theorem III, x=, r=*, z=* d , (5) then (l); (6) dX dY dZ ^ ~dx llx dy dy dz dz = _^v 2 4> /S'.V^V*, by (2) and (3). (7) But by Theorem III dY dZ 120 GENERAL THEOREMS. [966. or by (6) and (7) (8) Since in the second member of this equation ^ and 4> may be interchanged,, we may do so in the first, and we thus obtain the complete statement of Green's Theorem, as given in equation (4). 96.] We have next to shew that Green's Theorem is true when one of the functions, say ^, is a many-valued one, provided that its first derivatives are single-valued, and do not become infinite within the acyclic region s. Since V# and V4> are single- valued, the second member of equa- tion (4) is single-valued ; but since ^ is many-valued^ any one element of the first member, as ^ V 2 4>, is many-valued. If, however, we select one of the many values of #, as ^ , at the point A within the region ?, then the value of # at any other point, P, will be definite. For, since the selected value of ^ is continuous within the region, the value of ^ at P must be that which is arrived at by continuous variation along any path from A to P, beginning with the value % at A. If the value at P were different for two paths between A and P, then these two paths must embrace between them a closed curve at which the first derivatives of ^ become infinite. Now this is contrary to the specification, for since the first derivatives do not become infinite within the region s, the closed curve must be entirely without the region ; and since the region is acyclic, two paths within the region cannot embrace anything outside the region. Hence, if # is given as the value of ^ at the point A, the value at P is definite. If any other value of 3*, say 4^4- K, had been chosen as the value at A, then the value at P would have been ^ + ^/c. But the value of the first member of equation (4) would be the same as before, for the change amounts to increasing the first member by n*. and this, by Theorem III, is zero. 96 c.~\ If the region s is doubly or multiply connected, we may reduce it to an acyclic region by closing each of its circuits with a diaphragm. Let S-L be one of these diaphragms, and ^ the corresponding cyclic constant, that is to say, the increment of ^ in going once 96 d.] GREEN'S THEOREM. 121 round the circuit in the positive direction. Since the region s lies on both sides of the diaphragm s 1} every element of ^ will occur twice in the surface integral. If we suppose the normal z^ drawn towards the positive side of ds lt and r/ drawn towards the negative side, and ^ = ^ + K , so that the element of the surface-integral arising from ds l will be Hence if the region s is multiply connected, the first term of equa- tion (4) must be written //* *-*// *-*--// *-///**' w where the first surface-integral is to be taken over the bounding surface, and the others over the different diaphragms, each element of surface of a diaphragm being taken once only, and the normal being drawn in the positive direction of the circuit. This modification of the theorem in the case of multiply- connected regions was first shewn to be necessary by Helmholtz *, and was first applied to the theorem by Thomson f. 96 dj\ Let us now suppose, with Green, that one of the functions, say , does not satisfy the condition that it and its first derivatives do not become infinite within the given, region, but that it becomes infinite at the point P, and at that point only, in that region, and that very near to P the value of is 4> + e/r%, where 4> is a finite and continuous quantity, and r is the distance from P. This will be the case if 4> is the potential of a quantity of electricity e concen- trated at the point P, together with any distribution of electricity the volume density of which is nowhere infinite within the region considered. Let us now suppose a very small sphere whose radius is a to be described about P as centre ; then since in the region outside this sphere, but within the surface s, 4> presents no singularity, we * ' Ueber Integrals der hydrodynamischen Gleichungen welche den Wirbelbewe- gungen entsprechen,' Crelle, 1858. Translated by Prof. Tait, Phil. Mag., 1867 (I). t ' On Vortex Motion,' Trans. E. S. Edin. xxv part i. p. 241 (1867). J The mark / separates the numerator from the denominator of a fraction. 122 GENERAL THEOREMS. [96 ^ may apply Green's Theorem to this region, remembering that the surface of the small sphere is to be taken account of in forming the surface-integral. In forming the volume-integrals we have to subtract from the volume-integral arising from the whole region that arising from the small sphere. Now / / / -=- ds cannot be numerically greater than $> II ds. y J J dv Now by Theorem III and this cannot be numerically greater than (V 2 ^-Jtffl 8 , and $ g 6 C C dty at the surface is approximately -, so that / / 4> j- ds cannot be nu- merically greater than and is therefore of the order a 2 , and may be neglected when a vanishes. But the surface-integral on the other side of the equation, namely -j- ds > GREEN'S THEOREM. 123 does not vanish, for / / -- ds = 4^; J J dv and if ^ be the value of # at the point P, dv Equation (4) therefore becomes in this case 97 .] We may illustrate this case of Green's Theorem by em- ploying it as Green does to determine the surface-density of a distribution which will produce a potential whose values inside and outside a given closed surface are given. These values must coincide at the surface, also within the surface V 2 = 0, and outside V 2 *'= 0. Green begins with the direct process, that is to say, the distribu- tion of the surface density, o-, being given, the potentials at an internal point P and an external point P' are found by integrating the expressions where r and / are measured from the points P and P' respectively. Now let 4> = 1/r, then applying Green's Theorem to the space within the surface, and remembering that V 2 3> = and V 2 * = 0, we find 1 where V P is the value of ty at P. Again, if we apply the theorem to the space between the surface s and a surface surrounding it at an infinite distance a, the part of the surface-integral belonging to the latter surface will be of the order I/a and may be neglected, and we have Now at the surface, * = * / , and since the normals v and v are drawn in opposite directions, fy 124 GENERAL THEOREMS. [97 & Hence on adding equations (10) and (11), the left-hand members destroy each other, and we have 97 #.] Green also proves that if the value of the potential at every point of a closed surface s be given arbitrarily, the potential at any point inside or outside the surface may be determined. For this purpose he supposes the function 4> to be such that near the point P its value is sensibly 1/r, while at the surface s its value is zero, and at every point within the surface V 2 = 0. That such a function must exist. Green proves from the physical consideration that if s is a conducting surface connected to the earth, and if a unit of electricity is placed at the point P, the potential within s must satisfy the above conditions. For since s is connected to the earth the potential must be zero at every point of s, and since the potential arises from the electricity at P and the electricity induced on s, V 2< J> = at every point within the surface. Applying Green's Theorem to this case, we find P = where, in the surface-integral, # is the given value of the potential at the element of surface ds ; and since, if o> is the density of the electricity induced on s by unit of electricity at P, 47r where the suffix a indicates that a point A on the surface s is taken instead of Q. Let , we find If *P is the potential of a distribution of electricity in space with a volume-density p and on conductors whose surfaces are s v * 2 , &c., and whose potentials are ^u^ & c '> w ^ n surface-densities cr 1} (7 2 , &c., then V 2 ^P =477/3, (17) where , is the charge of the surface * r * Thomson and Tait's Natural Philosophy, 526. UNIQUE MINIMUM OF W+. 127 Dividing (16) by 877, we find - (* x e, + V 2 e 2 + &e.) -f 1 *p dx dydz The first term is the electric energy of the system arising from the surface-distributions, and the second is that arising from the distri- bution of electricity through the field, if such a distribution exists. Hence the second member of the equation expresses the whole electric energy of the system, the potential # being a given function of a?, y, z. As we shall often have occasion to employ this volume-integral, we shall denote it by the abbreviation W^ so that >+( >'*()>** () If the only charges are those on the surfaces of the conductors, p=0, and the second term of the first member of equation (20) disappears. The first term is the expression for the energy of the charged system expressed, as in Art. 84, in terms of the charges and the potentials of the conductors, and this expression for the energy we denote by W. 99 d.~\ Let ^ be a function of a?, y^ z, subject to the condition that its value at the closed surface s is ^f, a known quantity for every point of the surface. The value of * at points not on the surface s is perfectly arbitrary. Let us also write the integration being extended throughout the space within the surface ; then we shall prove that if ^ is a particular form of V which satisfies the surface condition and also satisfies Laplace's Equation V 2 ^ _ (23) at every point within the surface, then W[, the value of W corre- sponding to * ls is less than that corresponding to any function which differs from S^ at any point within the surface. For let ^ be any function coinciding with ^ at the surface but not at every point within it, and let us write ^ = ^ + ^2; (24) then ^ 2 is a function which is zero at every point of the surface. 128 GENERAL THEOREMS. [99 b. The value of W for ^ will be evidently i By Green's Theorem the last term may be written The volume-integral vanishes because V 2 ^ = within the surface, and the surface-integral vanishes because at the surface ^ 2 =0. Hence equation (25) is reduced to the form r=r 1+ r 2 . (2?) Now the elements of the integral W^ being sums of three squares, are incapable of negative values, so that the integral itself can only be positive or zero. Hence if W^ is not zero it must be positive, and therefore W greater than W r But if W z is zero, every one of its elements must be zero, and therefore dx dy dz i7 at every point within the surface, and V 2 must be a constant within the surface. But at the surface ^ = 0, therefore 4^ at every point within the surface, and ^ = V lt so that if W is not greater than W v ty must be identical with ^ at every point within the surface. It follows from this that ^ is the only function of #, y, z which becomes equal to ^ at the surface, and which satisfies Laplace's Equation at every point within the surface. For if these conditions are satisfied by any other function ^ 3 , ; 4hen W^ must be less than any other value of W. But we have already proved that W^ is less than any other value, and therefore V/ t ^an W y Hence no function different from ^ can satisfy the conditions. The case which we shall find most useful is that in which the k field is bounded by one exterior surface, s, and any number of /interior surfaces, S L , s 2) &c., and when the conditions are that the * value of ^ shall be zero at s } 4^ at * 1; ^ 2 at * 2 , and so on, where 4 r 1 , ^ 2 , &c. are constant for each surface, as in a system of conductors, the potentials of which are given. Of all values of ^ satisfying these conditions, that gives the minimum value of W^ for which V 2v f = at every point in the field. 1 00 &.] LEMMA. 129 Thomson's Theorem. Lemma. 100 #.] Let ^ be any function of #, y, z which is finite and continuous within the closed surface s, and which at certain closed surfaces, * lf s z , s p , &c., has the values V 19 ^ 2) ty p , &c. constant for each surface. Let Uj v, w be functions of #, y, z, which we may consider as the components of a vector ( subject to the solenoidal condition r, _n- du dv dw 8. V( = -j- + -j- + ~r = 0, (28) dx dy dz and let us put in Theorem III X=*, Y=Vv, Z=3>w, (29) we find as the result of these substitutions /du dv dw\ . * (+ + ) todyd* the surface-integrals being extended over the different surfaces and the volume-integrals being taken throughout the whole field. Now the first volume-integral vanishes in virtue of the solenoidal condition for u^ v, w, and the surface-integrals vanish in the follow- ing cases : (1) When at every point of the surface ^ =.0. (2) When at every point of the surface lu + mv + nw = 0. (3) When the surface is entirely made up of parts which satisfy either (l) or (2). (4) When ^ is constant over the whole closed surface, and nw)ds = 0. Hence in these four cases the volume-integral / / 100 .] Now consider a field bounded by the external closed surface s, and the internal closed surfaces * 1} s 2 , &c. Let ^ be a function of #, y, z, which within the field is finite and continuous and satisfies Laplace's Equation V 2 *=0, (32) and has the constant, but not given, values * 15 # 25 &c. at the surfaces s l3 s 2 , &c. respectively, and is zero at the external surface s. VOL. i. K 130 GENERAL THEOREMS. [lOOC. The charge of any of the conducting surfaces, as s lf is given by the surface-integral the normal v being drawn from the surface s 1 into the electric field. 100 c.] Now let y ff, h be functions of #, y> z, which we may consider as the components of a vector 2), subject only to the conditions that at every point of the field they must satisfy the solenoidal equation df da dk . x + + = (34) dx dy dz and that at any one of the internal closed surfaces, as s lt the surface- integral (35) where I, m, n are the direction cosines of the normal v drawn outwards from the surface s l into the electric field, and e is the same quantity as in equation (33), being, in fact, the electric charge of the conductor whose surface is s 1 . We have to consider the value of the volume-integral (36) extended throughout the whole of the field within s and without 1 dy u=f+ -- =- > v = g-\ --- , w k-\ ---- =-> (38) ITT dx 4ir dy ITT dz and W% = 2 *(u 2 + & + w*)dxdydz-, (39) then since dy TOO c.] THOMSON'S THEOKEM. 131 Now in the first place, n, v, w satisfy the solenoidal condition at every point of the field, for by equations (38) _. da dy dz ~ dx dy dz 4 and by the conditions expressed in equations (34) and (32), both parts of the second member of (41) are zero. In the second place, the surface-integral ^ w) ds l / / but by (35) the first term of the second member is e, and by (33) the second term is e, so that / / = 0. (43) Hence, since * t is constant, the fourth condition of Art. 1 00 a is satisfied, and the last term of equation (40) is zero, so that the equation is reduced to the form #5,= ^+^. (44) Now since the element of the integral W is the sum of three squares, & 2 +# 2 +^ 2 , it must be either positive or zero. If at any point within the field %, v, and w are not each of them equal to zero, the integral W must have a positive value, and W^ must therefore be greater than W*. But the values u = v = w = at every point satisfy the conditions. Hence, if at every point Id* Id* Id* f . -' * =-' ^- then Wv=.W^ (46) and the value of W corresponding to these values of f, g, ft, is less than the value corresponding to any values of f t g> h, differing from these. Hence the problem of determining the displacement and po- tential, at every point of the field, when the charge on each conductor is given, has one and only one solution. This theorem in one of its more general forms was first stated by Sir W. Thomson*. We shall afterwards show of what gene- ralization it is capable. * Cumbridye and Dublin Mathematical Journal, February, 1848. 132 GENERAL THEOREMS. [lOO d. 100^.] This theorem may be modified by supposing that the vector 5), instead of satisfying the solenoidal condition at every point of the field, satisfies the condition . . where p is a finite quantity, whose value is given at every point in the field, and may be positive or negative, continuous or discon- tinuous, its volume-integral within a finite region being, however, finite. We may also suppose that at certain surfaces in the field lf+ mg + nh + I'f + my + n' h' = ( 50 ) da;* dy* dsP and the surface condition + 77 For if, as before, 1 d* I then ^, v t w will satisfy the general solenoidal condition du dv dw _ fa + dj + fa = } and the surface condition lu+mv+nw+l'u'+m'v'+n'w' = 0, IOI &.] INTENSITY AND DISPLACEMENT. 133 and at the bounding surface lu -f mv + nw = 0, whence we find as before that and that #5> Hence as before it is shewn that W^ is a unique minimum when 7/5 = 0, which implies that ( is everywhere zero, and therefore 1 d* J_^ z_ _!_<** ~' " ~" 101 a.~\ In our statement of these theorems we have hitherto confined ourselves to that theory of electricity which assumes that the properties of an electric system depend on the form and relative position of the conductors, and on their charges, but takes no account of the nature of the dielectric medium between the conductors. According to that theory, for example, there is an invariable relation between the surface density of a conductor and the electro- motive intensity just outside it, as expressed in the law of Coulomb E = 4. But this is true only in the standard medium, which we may take to be air. In other media the relation is different, as was proved experimentally, though not published, by Cavendish, and afterwards rediscovered independently by Faraday. In order to express the phenomenon completely, we find it necessary to consider two vector quantities, the relation between which is different in different media. One of these is the electro- motive intensity, the other is the electric displacement. The electromotive intensity is connected by equations of invariable form with the potential, and the electric displacement is connected by equations of invariable form with the distribution of electricity, but the relation between the electromotive intensity and the electric displacement depends on the nature of the dielectric medium, and must be expressed by equations, the most general form of which is as yet not fully determined, and can be determined only by ex- periments on dielectrics. 101 b.] The electromotive intensity is a vector defined in Art. 68, as the mechanical force on a small quantity e of electricity divided by e. We shall denote its components by the letters P, Q, E, and the vector itself by ($. In electrostatics, the line integral of @ is always independent 134 GENERAL THEOREMS. [lOI C. of the path of integration, or in other words ( is the space- variation of a potential. Hence p d* d* d* r = -- 7- > U = -- 7 A/ = das dy or more briefly, in the language of Quaternions 101 ^ = -5. In an isotropic medium whose dielectric constant is K There are some media, however, of which glass has been the most carefully investigated, in which the relation between 2) and ( 136 GENERAL THEOREMS. is more complicated, and involves the time variation of one or both of these quantities, so that the relation must be of the form We shall not attempt to discuss relations of this more general kind at present, but shall confine ourselves to the case in which 2) is a linear and vector function of (. The most general form of such a relation may be written 477$) =4> ((), where $ during the present investigation always denotes a linear and vector function. The components of 2) are therefore homo- geneous linear functions of those of (, and may be written in the form 4 TT/= K xx P + K xy Q + K XX E ; where the first suffix of each coefficient K indicates the direction of the displacement, and the second, that of the electromotive intensity. The most general form of a linear and vector function involves nine independent coefficients. When the coefficients which have the same pair, of suffixes are equal, the function is said to be self-conjugate. If we express ( in terms of 3) we shall have or P = 4 TT (k iex f+ 7c yx g + Jc K 101 y.] The work done by the electromotive intensity whose components are P f Q, R, in producing a displacement whose com- ponents are df, dg> and dk, in unit of volume of the medium, is Since a dielectric under electric displacement is a conservative system, W must be a function of f t g> h, and since f, g, 7i may vary independently, we have aw AW aw = J/' Q= '~W' R = ~dh' Hence dP = W = ffiW =^ Ag AfAg AgAf Af fl ~P But = lirfcyx, the coefficient of g in the expression for P, and ~Y ^v7c xy > the coefficient of f'm the expression for Q, 101 L] EXTENSION OF GREEN'S THEOREM. 137 Hence if a dielectric is a conservative system, (and we know that it is so, because it can retain its energy for an indefinite time), &xv & V x, and M 2 + 2 krfk + 2 M A/+ ^Jc xy fg\ dxdydz, where the suffix denotes the vector in terms of which TFis to be expressed. When there is no suffix, the energy is understood to be expressed in terms of both vectors. We have thus, in all, six different expressions for the energy of the electric field. Three of these involve the charges and poten- tials of the surfaces of conductors, and are given in Art. 87. The other three are volume-integrals taken throughout the electric field, and involve the components of electromotive intensity or of electric displacement, or of both. The first three therefore belong to the theory of action at a distance, and the last three to the theory of action by means of the intervening medium. These three expressions for W may be written, 101 hJ\ To extend Green's Theorem to the case of a hetero- geneous anisotropic medium, we have only to write in Theorem III, *-.**_+*,. - + *". 138 GENERAL THEOREMS. [lO2 a. and we obtain (remembering that the order of the suffixes of the coefficients is indifferent), d ( K d ** dx d%~ KVV dy dy ' M ~fa~d^ (d^d$_ ^^\ T7 /_^*^* -+ + A| "" rr r = J J * ^ ^-: -- = 7 v \ doc dy dy -dx v r-j \dxdydz KVX m Using quaternion notation the result may be written more briefly, jfy s. Uv ^ (v) fa- Limits between which the electric capacity of a conductor must lie. 102 #.] The capacity of a conductor or system of conductors has been already defined as the charge of that conductor or system 102 a.] LIMITING VALUES OF CAPACITY. 139 of conductors when raised to potential unity, all the other con- ductors in the field being at potential zero. The following method of determining limiting values between which the capacity must lie, was suggested by a paper ' On the Theory of Resonance/ by the Hon. J. W. Strutt, Phil. Trans. 1871. See Art. 308. Let s l denote the surface of the conductor, or system of con- ductors, whose capacity is to be determined, and s the surface of all other conductors. Let the potential of ^ be 1} and that of * , ^ . Let the charge of s 1 be ^. That of S Q will be e lt Then if q is the capacity of s lt . and if W is the energy of the system with its actual distribution of electricity W=\e^ (^-^ ), (2) 2W e^ , . * = (4v=*rp = 2F' To find an upper limit of the value of the capacity. Assume any value of # which is equal to 1 at s 1 and equal to zero at s , and calculate the value of the volume-integral extended over the whole field. Then as we have proved (Art. 99 6) that W cannot be greater than W*, the capacity, q, cannot be greater than 2%. To find a lower limit of the value of the capacity. Assume any system of values of f> g, h, which satisfies the equation ^+^+^-0 (5) dx + dy + dz " ( } and let it make / / (l^f-\- m^g + n-Ji) ds = e lt (6) Calculate the value of the volume-integral = tiff / extended over the whole field ; then as we have proved (Art. 100 c) that W cannot be greater than #J, the capacity, , cannot be less than e-f , . 2%' V ; The simplest method of obtaining a system of values of/, g, h, which will satisfy the solenoidal condition, is to assume a distribu- tion of electricity on the surface of s lt and another on * , the sum 140 GENERAL THEOREMS. of the charges being zero, then to calculate the potential, #, due to this distribution, and the electric energy of the system thus arranged, which we may call W^. If we then make 1 d* 1 d* 1 d 4 TT h A -7 > dx dy dz i/ where A is to be determined so that at every point of the field df dg dli , } + + = 3 102 C.] POTENTIAL BETWEEN TWO NEARLY PLAT SURFACES. 143 and also so that the line-integral dy , dz \ , - + ' + ** (27) taken along any line of induction from the surface a to the surface #, shall be equal to 1 . Let us assume A= l+A + (z-a) + C(z-ay, (28) and let us neglect powers and products of A> B, C, and at this stage of our work powers and products of the first derivatives of a and b. ^ ^ The solenoidal condition then gives/"" "~*x* **/*/. /t 1 = _V 2 a, (*) where the integration with respect to a l9 y l} g t is extended throughout the region occupied by E 19 and the integration with respect to x^y^z^ is extended throughout the region occupied by -E 2 . Since, however, p 1 is zero except in the system E lt and p 2 is zero except in the system E 29 the value of the integral will not be altered by extending the limits of the integrations, so that we may suppose the limits of every integration to be + - This expression for the force is a literal translation into mathe- matical symbols of the theory which supposes the electric force to act directly between bodies at a distance, no attention being bestowed on the intervening medium. If we now define V 2 , the potential at the point a lt y^ z-^ arising from the presence of the system E 2) by the equation ^ 2 will vanish at an infinite distance, and will everywhere satisfy the equation V 2 * 2 =47rp 2 . (3) 1 04.] MECHANICAL ACTIO^. 145 We may now express A in the form of a triple integral Here the potential ^ 2 is supposed to have a definite value at every point of the field, and in terms of this, together with the distribution, p 15 of electricity in the first system E lt the force A is expressed, no explicit mention being made of the distribution of electricity in the second system E 2 . Now let *j be the potential arising from the first system, expressed as a function of #,y, z t and defined by the equation *l*kAk, (5) *! will vanish at an infinite distance, and will everywhere satisfy the equation V 2 # 1 = 47rp 1 . (6) We may now eliminate ^ from A and obtain in which the force is expressed in terms of the two potentials only. 104.] In all the integrations hitherto considered, it is indifferent what limits are prescribed, provided they include the whole of the system E lf In what follows we shall suppose the systems E and E 2 to be such that a certain closed surface a contains within it the whole of E l but no part of E 2 . Let us also write P = /VH> 2 , * = *!+* (8) then within s, p 2 = 0, p =p 1 , and without s p t = 0, p = /o a . (9) Now A 11 = - Pl d a;i ^ 1 ^ 1 (10) represents the resultant force, in the direction #, on the system E^ arising from the electricity in the system itself. But on the theory of direct action this must be zero, for the action of any particle P on another Q is equal and opposite to that of Q on P, and since the components of both actions enter into the integral, they will destroy each other. We may therefore write VOL. I. 146 MECHANICAL ACTION. [105. where ^ is the potential arising from both systems, the integration being now limited to the space within the closed surface s, which includes the whole of the system E l but none of E 2 . 105.] If the action of E 2 on E l is effected, not by direct action at a distance, but by means of a distribution of stress in a medium extending continuously from E 2 to E 1 , it is manifest that if we know the stress at every point of any closed surface * which completely separates E l from E 2 , we shall be able to determine completely the mechanical action of E 2 on E. For if the force on E l is not completely accounted for by the stress through $, there must be direct action between something outside of s and some- thing inside of s. Hence if it is possible to account for the action of E 2 on E 1 by means of a distribution of stress in the intervening medium, it must be possible to express this action in the form of a surface- integral extended over any surface s which completely separates E 2 from E lu Let us therefore endeavour to express d 2 * A = in the form of a surface integral. By Theorem III we may do so if we can determine X, Y and Z, so that _dX dY dZ " " ~~~~~~~ ~*~ ~ Taking the terms separately, 2 d dx dy 2 dy dx dy' dy dxdy d ,d3> d*^ 1 d /^ _. dy ^dx dy ' 2 dx ^dy i/ \ 2 dz* ^dx' Wy' __ IV 1 = ^Pi (14) r r f /dp~~ dp,. v dp.- ttenA=JJJ(-Lj? + J^ + .J ! - (15) v ' the integration being extended throughout the space within Transforming the volume-integral by Theorem III, Art. 2.1, A = (16) where ds is an element of any closed surface including the whole of E l but none of S 2 , and Imn are the direction cosines of the normal drawn from ds outwards. For the components of the force on E^ in the directions of y and 2, we obtain in the same way = JJ C = (lp (17) (18) If the action of the system E 2 on E l does in reality take place by direct action at a distance, without the intervention of any medium, we must consider the quantities p xx &c. as mere abbreviated forms for certain symbolical expressions, and as having no physical significance. But if we suppose that the mutual action between E 2 and E is kept up by means of stress in the medium between them, then since the equations (16), (17), (18) give the components of the resultant force arising from the action, on the outside of the surface s, of the stress whose six components are p xx &c., we must consider p xx &c. as the components of a stress actually existing in the medium. L 2 148 MECHANICAL ACTION. [106. 106.] To obtain a clearer view of the nature of this stress let us alter the form of part of the surface s so that the element ds may become part of an equipotential surface. (This alteration of the surface is legitimate provided we do not thereby exclude any part of E-L or include any part of E 2 ). Let v be a normal to ds drawn outwards. city Let E = j be the intensity of the electromotive force in the direction of v, then dty dty dty -j = El, r- = Em, -= = En. dx dy dz Hence the six components of stress are A. = ff (I 2 --), Pt . = J- R*mn, t n = -^ K-"-P), p, x = _L JBnl, If a, I, c are the components of the force on ds per unit of area 87T c = E*n. Sir Hence the force exerted by the part of the medium outside ds on the part of the medium inside ds is normal to the element and directed outwards, that is to say, it is a tension like that of a rope, and its value per unit of area is E 2 . O7T Let us next suppose that the element ds is at right angles to the equipotential surfaces which cut it, in which case 7 dty dty dty l-j-^.m~ T - + n- 7 - = 0. (19) dx dy dz v ' ,r/d*\ 2 /d*\ 2 /^\ 2 1 Now 8,- = I - - -) - (-g.) J (20) x dz Multiplying (19) by 2 -- and subtracting from (20), we. find I07-] COMPONENTS OF STRESS. 149 *\ 2 /^fx 2 Hence the components of the tension per unit of area of ds are * = -m, Hence if the element ^ is at right angles to an equipotential surface, the force which acts on it is normal to the surface, and its numerical value per unit of area is the same as in the former case, but the direction of the force is different, for it is a pressure instead of a tension. We have thus completely determined the type of the stress at any given point of the medium. The direction of the electromotive intensity at the point is a principal axis of stress, and the stress in this direction is a tension whose numerical value is t =^ ^ (22) where E is the electromotive intensity. Any direction at right angles to this is also a principal axis of stress, and the stress along this axis is a pressure whose numerical magnitude is also p. The stress as thus defined is not of the most general type, for it has two of its principal stresses equal to each other, and the third has the same value with the sign reversed. These conditions reduce the number of independent variables which determine the stress from six to three, and accordingly it is completely determined by the three components of the electro- motive intensity ~ dx dy dz The three relations between the six components of stress are ).) l\ ( 23 ) }' ) 107.] Let us now examine whether the results we have obtained 150 MECHANICAL ACTION. [107. will require modification when a finite quantity of electricity is collected on a finite surface so that the volume-density becomes infinite at the surface. In this case, as we have shown in Art. 78, the components of the electromotive intensity are discontinuous at the surface. Hence the components of stress will also be discontinuous at the surface. Let I m n be the direction cosines of the normal to ds. Let P, Q, R be the components of the electromotive intensity on the side on which the normal is drawn, and P' ; Q\R' their values on the other side. Then by Art. (78 a) if z which satisfies the equation ^^ o at every point outside the closed surface where C is always positive. Now we have shewn that dM/dr is negative for certain directions of r, hence when the electricity is free to move the instability in these directions will be increased. M ^ CHAPTER VII. FORMS OF THE EQUIPOTENTIAL SURFACES AND LINES OF INDUCTION IN SIMPLE CASES. 117.] WE have seen that the determination of the distribution of electricity on the surface of conductors may be made to depend on the solution of Laplace's equation V being a function of x 9 y, and #, which is always finite and con- tinuous, which vanishes at an infinite distance, and which has a given constant value at the surface of each conductor. It is not in general possible by known mathematical methods to solve this equation so as to fulfil arbitrarily given conditions, but it is easy to write down any number of expressions for the function V which shall satisfy the equation, and to determine in each case the forms of the conducting surfaces, so that the function V shall be the true solution. It appears, therefore, that what we should naturally call the inverse problem of determining the forms of the conductors when the expression for the potential is given is more manageable than the direct problem of determining the potential when the form of the conductors is given. In fact, every electrical problem of which we know the solution has been constructed by this inverse process. It is therefore of great importance to the electrician that he should know what results have been obtained in this way, since the only method by which he can expect to solve a new problem is by reducing it to one of the cases in which a similar problem has been con- structed by the inverse process. This historical knowledge of results can be turned to account in two ways. If we are required to devise an instrument for making electrical measurements with the greatest accuracy, we may select those forms for the electrified surfaces which correspond to cases of which we know the accurate solution. If, on the other hand, we are required to estimate what will be the electrification of bodies 1 1 8.] USE OF DIAGRAMS. 165 whose forms are given, we may begin with some case in which one of the equipotential surfaces takes a form somewhat resembling the given form, and then by a tentative method we may modify the pro- blem till it more nearly corresponds to the given case. This method is evidently very imperfect considered from a mathematical point of view, but it is the only one we have, and if we are not allowed to choose our conditions, we can make only an approximate cal- culation of the electrification. It appears, therefore, that what we want is a knowledge of the forms of equipotential surfaces and lines of induction in as many different cases as we can collect together and remember. In certain classes of cases, such as those relating to spheres, there are known mathematical methods by which we may proceed. In other cases we cannot afford to despise the humbler method of actually drawing tentative figures on paper, and selecting that which appears least unlike the figure we require. This latter method I think may be of some use, even in cases in which the exact solution has been obtained, for I find that an eye- knowledge of the forms of the equipotential surfaces often leads to a right selection of a mathematical method of solution. I have therefore drawn several diagrams of systems of equi- potential surfaces and lines of induction, so that the student may make himself familiar with the forms of the lines. The methods by which such diagrams may be drawn will be explained in Art. 123. 118.] In the first figure at the end of this volume we have the sections of the equipotential surfaces surrounding two points charged with quantities of electricity of the same kind and in the ratio of 20 to 5. Here each point is surrounded by a system of equipotential surfaces which become more nearly spheres as they become smaller, though none of them are accurately spheres. If two of these sur- faces, one surrounding each point, be taken to represent the surfaces of two conducting bodies, nearly but not quite spherical, and if these bodies be charged with the same kind of electricity, the charges being as 4 to 1, then the diagram will represent the equipotential surfaces, provided we expunge all those which are drawn inside the two bodies. It appears from the diagram that the action between the bodies will be the same as that between two points having the same charges, these points being not exactly in the middle of the axis of each body, but each somewhat more remote than the middle point from the other body. The same diagram enables us to see what will be the distribution 166 EQUIPOTENTIAL SURFACES [119. of electricity on one of the oval figures, larger at one end than ^ De ther, which surround both centres. Such a body, if charged with 25 units of electricity and free from external influence, will have the surface-density greatest at the small end, less at the large end, and least in a circle somewhat nearer the smaller than the ^ t larger end. There is one equipotential surface, indicated by a dotted line, which consists of two lobes meeting at the conical point P. That point is a point of equilibrium, and the surface-density on a body of the form of this surface would be zero at this point. The lines of force in this case form two distinct systems, divided from one another by a surface of the sixth degree, indicated by a dotted line, passing through the point of equilibrium, and some- what resembling one sheet of the hyperboloid of two sheets. This diagram may also be taken to represent the lines of force and equipotential surfaces belonging to two spheres of gravitating matter whose masses are as 4 to 1. 119.] In the second figure we have again two points whose charges are as 20 to 5, but the one positive and the other negative. In this case one of the equipotential surfaces, that, namely, corre- sponding to potential zero, is a sphere. It is marked in the diagram by the dotted circle Q. The importance of this spherical surface will be seen when we come to the theory of Electrical Images. We may see from this diagram that if two round bodies are charged with opposite kinds of electricity they will attract each other as much as two points having the same charges but placed somewhat nearer together than the middle points of the round bodies. Here, again, one of the equipotential surfaces, indicated by a dotted line, has two lobes, an inner one surrounding the point whose charge is 5 and an outer one surrounding both bodies, the two lobes meeting in a conical point P which is a point of equilibrium. If the surface of a conductor is of the form of the outer lobe, a roundish body having, like an apple, a conical dimple at one end of its axis, then, if this conductor be electrified, we shall be able to determine the surface-density at any point. That at the bottom of the dimple will be zero. Surrounding this surface we have others having a rounded' dimple which flattens and finally disappears in the equipotential surface passing through the point marked M. The lines of force in this diagram form two systems divided by a surface which passes through the point of equilibrium. 121.] AND LINES OF INDUCTION. 167 If we consider points on the axis on the further side of the point J2, we find that the resultant force diminishes to the double point P, where it vanishes. It then changes sign, and reaches a maximum at Jf, after which it continually diminishes. This maximum, however, is only a maximum relatively to other points on the axis, for if we consider a surface through H per- pendicular to the axis, M is a point of minimum force relatively to neighbouring points on that surface. 120.] Figure III represents the equipotential surfaces and lines of induction due to a point whose charge is 10 placed at A, and surrounded by a field of force, which, before the introduction of the charged point, was uniform in direction and magnitude at every part. The equipotential surfaces have each of them an asymptotic plane. One of them, indicated by a dotted line, has a conical point and a lobe surrounding the point A. Those below this surface have one sheet with a depression near the axis. Those above have a closed portion surrounding A and a separate sheet with a slight depression near the axis. If we take one of the surfaces below A as the surface of a con- ductor, and another a long way below A as the surface of another conductor at a different potential, the system of lines and surfaces between the two conductors will indicate the distribution of electric force. If the lower conductor is very far from A its surface will be very nearly plane, so that we have here the solution of the distribution of electricity on two surfaces, both of them nearly plane and parallel to each other, except that the upper one has a protuberance near its middle point, which is more or less prominent according to the particular equipotential surface we choose. 121.] Figure IV represents the equipotential surfaces and lines of induction due to three points A, B and C, the charge of A being 15 units of positive electricity, that of B 12 units of negative electricity, and that of C 20 units of positive electricity. These points are placed in one straight line, so that AB = 9, BC=16, AC =25. In this case, the surface for which the potential is zero consists of two spheres whose centres are A and and their radii 15 and 20. These spheres intersect in the circle which cuts the plane of the paper at right angles in D and J7, so that B is the centre of this circle and its radius is 12. This circle is an example of a line 168 EQUIPOTENTIAL SUEFACES [l22. of equilibrium, for the resultant force vanishes at every point of this line. If we suppose the sphere whose centre is A to be a conductor with a charge of 3 units of positive electricity, and placed under the influence of 20 units of positive electricity at C, the state of the case will be represented by the diagram if we leave out all the lines within the sphere A. The part of this spherical surface within the small circle DIf will be negatively charged by the influence of C. All the rest of the sphere will be positively charged, and the small circle Dl/ itself will be a line of no charge. We may also consider the diagram to represent the sphere whose centre is (?, charged with 8 units of positive electricity, and in- fluenced by 15 units of positive electricity placed at A. The diagram may also be taken to represent a conductor whose surface consists of the larger segments of the two spheres meeting in DD', charged with 23 units of positive electricity. We shall return to the consideration of this diagram as an illustration of Thomson's Theory of Electrical Images. See Art. 168. 122.] These diagrams should be studied as illustrations of the language of Faraday in speaking of ' lines of force,' the ' forces of an electrified body,' &c. The word Force denotes a restricted aspect of that action between two material bodies by which their motions are rendered different from what they would have been in the absence of that action. The whole phenomenon, when both bodies are contemplated at once, is called Stress, and may be described as a transference of momentum from one body to the other. When we restrict our attention to the first of the two bodies, we call the stress acting on it the Moving Force, or simply the Force on that body, and it is measured by the momentum which that body is receiving per unit of time. The mechanical action between two charged bodies is a stress^ and that on one of them is a force. The force on a small charged body is proportional to its own charge, and the force per unit of charge is called the Intensity of the force. The word Induction was employed by Faraday to denote the mode in which the charges of electrified bodies are related to each other, every unit of positive charge being connected with a unit of negative charge by a line, the direction of which, in fluid dielectrics, coincides at every part of its course with that of the electric intensity. Such a line is often called a I23-] AND LINES OF INDUCTION. 169 line of Force, but it is more correct to call it a line of In- duction. Now the quantity of electricity in a body is measured, according to Faraday's ideas, by the number of lines of force, or rather of induction, which proceed from it. These lines of force must all terminate somewhere, either on bodies in the neighbourhood, or on the walls and roof of the room, or on the earth, or on the heavenly bodies, and wherever they terminate there is a quantity of elec- tricity exactly equal and opposite to that on the part of the body from which they proceeded. By examining the diagrams this will be seen to be the case. There is therefore no contradiction between Faraday's views and the mathematical results of the old theory, but, on the contrary, the idea of lines of force throws great light on these results, and seems to afford the means of rising by a con- tinuous process from the somewhat rigid conceptions of the old theory to notions which may be capable of greater expansion, so as to provide room for the increase of our knowledge by further researches. 123.] These diagrams are constructed in the following manner : First, take the case of a single centre of force, a small electrified body with a charge e. The potential at a distance r is Ve/r\ hence, if we make r = e/P~ t we shall find r, the radius of the sphere for which the potential is V. If we now give to V the values 1, 2, 3, &c., and draw the corresponding spheres, we shall obtain a series of equipotential surfaces, the potentials corresponding to which are measured by the natural numbers. The sections of these spheres by a plane passing through their common centre will be circles, each of which we may mark with the number denoting its potential. These are indicated by the dotted semi-circles on the right hand of Fig. 6. If there be another centre of force, we may in the same way draw the equipotential surfaces belonging to it, and if we now wish to find the form of the equipotential surfaces due to both centres together, we must remember that if T[ be the potential due to one centre, and 7 that due to the other, the potential due to both will be 7^-f- J^=F. Hence, since at every intersection of the equipotential surfaces belonging to the two series we know both ^ and 7, we also know the value of V. If therefore we draw a surface which passes through all those intersections for which the value of Pis the same, this surface will coincide with a true equipotential surface at all these intersections, and if the original systems of surfaces 170 EQUIPOTENTIAL SURFACES. [123. are drawn sufficiently close, the new surface may be drawn with any required degree of accuracy. The equipotential surfaces due to two points whose charges are equal and opposite are represented by the continuous lines on the right hand side of Fig. 6. This method may be applied to the drawing of any system of equipotential surfaces when the potential is the sum of two potentials, for which we have already drawn the equipotential surfaces. The lines of force due to a single centre of force are straight lines radiating from that centre. If we wish to indicate by these lines the intensity as well as the direction of the force at any point, we must draw them so that they mark out on the equipotential surfaces portions over which the surface-integral of induction has definite values. The best way of doing this is to suppose our plane figure to be the section of a figure in space formed by the revolution of the plane figure about an axis passing through the centre of force. Any straight line radiating from the centre and making an angle 6 with the axis will then trace out a cone, and the surface-integral of the induction through that part of any surface which is cut off by this cone on the side next the positive direction of the axis is 2ne (l cos0). If we further suppose this surface to be bounded by its inter- section with two planes passing through the axis, and inclined at the angle whose arc is equal to half the radius, then the induction through the surface so bounded is e (l cos0) = 23>, say; and = cos" 1 ( 1 2 V v e ' If we now give to 4> a series of values 1, 2, 3 ... 2 , and then, by drawing lines through the consecutive intersections of these lines for which the value of 4>j -f 4> 2 is the same, we may find the lines of force due to both centres, and in the same way we may combine any two systems of lines of force which are symmetrically situated about the same axis. The con- tinuous curves on the left hand side of Fig. 6 represent the lines of force due to the two charged points acting at once. After the equipotential surfaces and lines of force have been constructed by this method the accuracy of the drawing may be tested by observing whether the two systems of lines are every- where orthogonal, and whether the distance between consecutive equipotential surfaces is to the distance between consecutive lines of force as half the mean distance from the axis is to the assumed unit of length. In the case of any such system of finite dimensions the line of force whose index number is 4> has an asymptote which passes through the electric centre (Art. 89 d) of the system, and is inclined to the axis at an angle whose cosine is 1 2 is less than e. Lines of force whose index is greater than e are finite lines. If e is zero, they are all finite. The lines of force corresponding to a field of uniform force parallel to the axis are lines parallel to the axis, the distances from the axis being the square roots of an arithmetical series. The theory of equipotential surfaces and lines of force in two dimensions will be given- when we come to the theory of conjugate functions*. * See a paper 'On the Flow of Electricity in Conducting Surfaces,' by Prof. W. R. Smith, Proc. M.S. Edin., 1869-70, p. 79. CHAPTER VIII. SIMPLE CASES OF ELECTRIFICATION. Two Parallel Planes. 124.] WE shall consider, in the first place, two parallel plane conducting surfaces of infinite extent, at a distance e from each other, maintained respectively at potentials A and B. It is manifest that in this case the potential V will be a function of the distance z from the plane A, and will be the same for all points of any parallel plane between A and .Z?, except near the boundaries of the electrified surfaces, which by the supposition are at an infinitely great distance from the point considered. Hence, Laplace's equation becomes reduced to the integral of which is 7= Q+<7 2 *; and since when z = 0, 7= A, and when z = c, 7= B, For all points between the planes, the resultant intensity is normal to the planes, and its magnitude is -n A B R = -- j- fi^ **v In the substance of the conductors themselves, R = 0. Hence the distribution of "electricity on the first plane has a surface- density o-, where /y- 477(7=72 = - On the other surface, where the potential is B, the surface- density and if p _ ^ ( 3 ) p is the resolved part of r in the direction of the axis k. Different axes are distinguished by different suffixes, and the cosine of the angle between two axes is denoted by A mn , where m and n are the suffixes specifying the axes. Differentiation with respect to an axis, Ji> whose direction cosines are Z, M, N, is denoted by d d ,_ d ^ T d -- = 1 +M- r +N- r . (4) dh dx dy dz From these definitions it is evident that , fi v dk If we now suppose that the potential at the point (#, y, z) due to a singular point of any order placed at the origin is I29C.] INFINITE POINTS. 181 then if such a point be placed at the extremity of the axis /&, the potential at (#, y, z) will be Af \fr-Lk), (y-Mk), (z-Nh)l and if a point in all respects the same, except that the sign of A is reversed, be placed at the origin, the potential due to the pair of points will be V = Af[(x-Lk), (y-MA), (z-Nh)]-Af(x,y,z), = Ah f(x, y, z) + terms containing k 2 . If we now diminish h and increase A without limit, their pro- duct continuing finite and equal to A', the ultimate value of the potential of the pair of points will be P =-^JS/(*>*>- (") If f (#, y> z) satisfies Laplace's equation, then, since this equation is linear, V , which is the difference of two functions, each of which separately satisfies the equation, must itself satisfy it. 1296*.] Now the potential due to an infinite point of order zero V^A,\, (9) satisfies Laplace's equation, therefore every function formed from this by differentiation with respect to any number of axes in suc- cession must also satisfy that equation. A point of the first order may be formed by taking two points of order zero, having equal and opposite charges A Q and A Q) and placing the first at the origin and the second at the extremity of the axis h^. The value of h^ is then diminished and that of A increased indefinitely, but so that the product A h^ is always equal to A lt The ultimate result of this process, when the two points coincide, is a point of the first order whose moment is A 1 and whose axis is 7^. A point of the first order is therefore a double point. Its potential is By placing a point of the first order at the origin, whose moment is A 19 and another at the extremity of the axis h% whose moment is A lt and then diminishing ^ 2 and increasing A 19 so that 182 SPHERICAL HARMONICS. [129 d. we obtain a point of the second order, whose potential is --**,-- 7 < We may call a point of the second order a quadruple point because it is constructed by making four points of order zero ap- proach each other. It has two axes h- and k% and a moment A 2 . The directions of these axes and the magnitude of the moment completely define the nature of the point. By differentiating with respect to n axes in succession we obtain the potential due to a point of the n ih order. It will be the product of three factors, a constant, a certain combination of cosines, and r ~( n+1 ). It is convenient, for reasons which will appear as we go on, to make the numerical value of the constant such that when all the axes coincide with the vector, the coefficient of the moment is r~( n+1 \ We therefore divide by n when we differ- entiate with respect to h n . In this way we obtain a definite numerical value for a particular potential, to which we restrict the name of The Solid Harmonic of degree (?+ 1), namely 1 d d 1.2.3. ..nd^ dhz dh n r If this quantity is multiplied by a constant it is still the poten- tial due to a certain point of the n ih order. 129 d.] The result of the operation (13) is of the form F= Y n r~( n+l \ ' (14) where T n is a function of the n cosines fa... i* n of the angles between r and the n axes, and of the \ n (nl) cosines A 12 , &c. of the angles between pairs of the axes. If we consider the directions of r and the n axes as determined by points on a spherical surface, we may regard Y n as a quantity varying from point to point on that surface, being a function of the \n(n+l] distances between the n poles of the axes and the pole of the vector. We therefore call Y n the Surface Harmonic of order n. 130#.] We have next to shew that to every surface-harmonic of order n there corresponds not only a solid harmonic of degree (n+ 1) but another of degree n, or that js.= r,,i- = ^ 1 f- (15) satisfies Laplace's equation. 1 30 6.] SOLID HARMONIC OF POSITIVE DEGREE. 183 For Hence Now, since ^ is a homogeneous^ function of. ..0v...#,.jind 2, of negative degree n + 1~ r.. (17) dz J ' The first two terms therefore of the right-hand member of equation (16) destroy each other, and, since V n satisfies Laplace's equation, the third term is zero, so that // also satisfies LaplaceV equation, and is therefore a solid harmonic of degree n. This is a particular case of the more general theorem of electrical inversion, which asserts that if F (%, y> z) is a function of a?, y, and z which satisfies Laplace's equation, then there exists another function, a a * x a z y a 2 z - 4 { -$-> ;-) -~S~j* r \ r 2 r 2 r 2 ' which also satisfies Laplace's equation. See Art. 162. 1303.] The surface harmonic T n contains 2n arbitrary variables, for it is defined by the positions of its n poles on the sphere, and each of these is defined by two coordinates. Hence the solid harmonics V n and H n also contain 2n arbitrary variables. Each of these quantities, however, when multiplied by a constant, will still satisfy Laplace's equation. To prove that AH n is the most general rational homogeneous function of degree n which can satisfy Laplace's equation, we observe that K, the general rational homogeneous function of degree n, contains J (-f l)(4-2) terms. But V 2 K is a homo- geneous function of degree w 2, and therefore contains \n{n -1) terms, and the condition V 2 K= requires that each of these must vanish. There are therefore \n(n 1) equations between the 184 SPHERICAL HARMONICS. [131 a. coefficients of the \ (n+ 1) (n + 2) terms of the function K, leaving 2^ + 1 independent constants in the most general form of the homo- geneous function of degree n which satisfies Laplace's equation. But H n , when multiplied by an arbitrary constant, satisfies the required conditions, and has 2 # + 1 arbitrary constants. It is therefore of the most general form. 131 aJ\ We are now able to form a distribution of potential such that neither the potential itself nor its first derivatives become infinite at any point. The function F n = Y n r~( n+l ) satisfies the condition of vanishing at infinity, but becomes infinite at the origin. The function H n =Y n r n is finite and continuous at finite dis- tances from the origin, but does not vanish at an infinite distance. But if we make a n Y n r~( n+l ) the potential at all points outside a sphere whose centre is the origin, and whose radius is a, and -( ri + 1 ) Y n r n the potential at all points within the sphere, and if on the sphere itself we suppose electricity spread with a surface density cr such that 47T(T 2 = (2n+l)Y nt (18) then all the conditions will be satisfied for the potential due to a shell charged in this manner. For the potential is everywhere finite and continuous, and vanishes at an infinite distance ; its first derivatives are everywhere finite and are continuous except at the charged surface, where they satisfy the equation + f: +4 = <>, (19) and Laplace's equation is satisfied at all points both inside and outside of the sphere. This, therefore, is a distribution of potential which satisfies the conditions, and by Art. 100 a it is the only distribution which can satisfy them. 131 #.] The potential due to a sphere of radius a whose surface density is given by the equation 47Tfl 2 (7=:(2^+l)7 n , (20) is, at all points external to the sphere, identical with that due to the corresponding singular point of order n. Let us now suppose that there is an electrical system which we may call U, external to the sphere, and that ^ is the potential due to this system, and let us find the value of 2 (^tf) for the 131 C.] SINGULAR POINT EQUIVALENT TO CHARGED SHELL. 185 singular point. This is the part of the electric energy depending on the action of the external system on the singular point. If AQ is the charge of a single point of order zero, then the potential energy in question is = 4>*. (21) If there are two such points, a negative one at the origin and a positive one of equal numerical value at the extremity of the axis h l} then the potential energy will be + 4 VJT + *' and when A Q increases and h^ diminishes indefinitely, but so that AQ&! = A 13 the value of the potential energy for a point of the first order will be (22) Similarly for a point of order n the potential energy will be 1 fJn j/ ^ = r 4. 7l * (23) 1.2. ..n n dh^...dh n 131 and the singular point to be made up of parts any one of which is de, then (24) But if V n is the potential due to the singular point, 5 = S(1&), (25) and the potential energy due to the action offione is = 22 (-dEde) = 2r n dJ$, (26) the last expression being the potential energy due to the action of e on E. Similarly, if \ , X/.. / j #*tA J. **"' 190 SPHERICAL HARMONICS. 135 aJ\ The expression (46) for the surface-integral of the product of two surface-harmonics assumes a remarkable form if we suppose all the axes of one of the harmonics, Y m , to coincide with each other, so that Y m becomes what we shall afterwards define as the zonal harmonic of order m, denoted by the symbol P m . In this case all the cosines of the form \ nm may be written JU H , where p n denotes the cosine of the angle between the common axis of P m and one of the axes of T n . The cosines of the form \ mm will all become equal to unity, so that for 2A s mm we must put the number of combinations of s symbols, each of which is distinguished by two suffixes out of n, no suffix being repeated. Hence The number of permutations of the remaining n2s indices of the axes of P m is (n 2 s) ! Hence 2 (^~ 2s ) = (n2s) ! ju w - 28 . (48) Equation (46) therefore becomes, when all the axes of Y m coincide with each other, ; : -..-. A 2 ), ty equation (43), (50) 2n+l where Y n ( m ) denotes the value of Y n at the pole of P m . We may obtain the same result by the following shorter pro- cess : Let a system of rectangular coordinates be taken so that the axis of z coincides with the axis of P m) and let Y n r n be expanded as a homogeneous function of #, y> z of degree n. At the pole of P m , x = y = and z = r, so that if Cz n is the term not involving x or y, C is the value of Y n at the pole of P m . Equation (31) becomes in this case If m is equal to n, the result of differentiating Cz n is n\ C, and is zero for the other terms. Hence C being the value of Y n at the pole of P m . 135 b.] This result is- a very important one in the theory of I35&-] EXPANSION IN SPHERICAL HARMONICS. 191 spherical harmonics, as it shews how to determine a series of spherical harmonics which expresses the value of a quantity having any arbitrarily assigned finite and continuous value at each point of a spherical surface. For let F be the value of the quantity and ds the element of surface at a point Q of the spherical surface, then if we multiply Ids by P n , the zonal harmonic whose pole is the point P of the same surface, and integrate over the surface, the result, since it depends on the position of the point P, may be considered as a function of the position of P. But since the value at P of the zonal harmonic whose pole is Q is equal to the value at Q of the zonal harmonic of the same order whose pole is P, we may suppose that for every element ds of the surface a zonal harmonic is constructed having its pole at Q and having a coefficient Fds. We shall thus have a system of zonal harmonics superposed on each other with their poles at every point of the sphere where F has a value. Since each of these is a multiple of a surface harmonic of order n, their sum is a multiple of a surface harmonic (not necessarily zonal) of order n. The surface integral / / FP n ds considered as a function of the point P is therefore a multiple of a surface harmonic Y n ; so that is also that particular surface harmonic of the n ih order which belongs to the series of harmonics which expresses F, provided F can be so expressed. For if F can be expressed in the form then if we multiply by P n ds and take the surface integral over the whole sphere, all terms involving products of harmonics of different orders will vanish, leaving Hence the only possible expansion of F in spherical harmonics is (51) 192 SPHEEICAL HARMONICS. [137. Conjugate Harmonics. 136.] We have seen that the surface integral of the product of two harmonics of different orders is always zero. But even when the two harmonics are of the same order, the surface integral of their product may be zero. The two harmonics are then said to be conjugate to each other. The condition of two harmonics of the same order being conjugate to each other is expressed in terms of equation (46) by making its members equal to zero. If one of the harmonics is zonal, the condition of conjugacy is that the value of the other harmonic at the pole of the zonal harmonic must be zero. If we begin with a given harmonic of the n ih order, then, in order that a second harmonic may be conjugate to it, its 2n variables must satisfy one condition. If a third harmonic is to be conjugate to both, its 2 n variables must satisfy two conditions. If we go on constructing harmonics, each of which is conjugate to all those before it, the number of conditions for each will be equal to the number of harmonics already in existence, so that the (2^+l) th harmonic will have 2n conditions to satisfy by means of its 2 n variables, and will therefore be completely determined. Any multiple A T n of a surface harmonic of the n ih order can be expressed as the sum of multiples of any set of 2 n + 1 conjugate harmonics of the same order, for the coefficients of the 2^+1 conjugate harmonics are a set of disposable quantities equal in number to the 2 n variables of Y n and the coefficient A. In order to find the coefficient of any one of the conjugate harmonics, say Y n *, suppose that Multiply by Y n *ds and find the surface integral over the sphere. All the terms involving products of harmonics conjugate to each other will vanish, leaving AJJT. r n 'd s = 4,//(r/)" 4, (52) an equation which determines A . Hence if we suppose a set of 2ti+l conjugate harmonics given, any other harmonic of the n ih order can be expressed in terms of them, and this only in one way. Hence no other harmonic can be conjugate to all of them. 137.] We have seen that if a complete system of 2#+l har- 138.] ZONAL HARMONICS. 193 monies of the n ih order, all conjugate to each other, be given, any other harmonic of that order can be expressed in terms of these. In such a system of 2 n -{- 1 harmonics there are 2n(2n+l) variables connected by n(2n+l) equations, n(2n+l) of the variables may therefore be regarded as arbitrary. We might, as Thomson and Tait have suggested, select as a system of conjugate harmonics one in which each harmonic has its n poles distributed so that j of them coincide at the pole of the axis of x, Jc at the pole of ^, and l(= njJc) at the pole of z. The n -f 1 distributions for which I = and the n distributions for which 1=1 being given, all the others may be expressed in terms of these. The system which has been actually adopted by all mathe- maticians (including Thomson and Tait) is that in which n a- of the poles are made to coincide at a point which we may call the Positive Pole of the sphere, and the remaining a poles are placed at equal distances round the equator when their number is odd, or at equal distances round one half of the equator when their number is even. In this case ju 1} /u 2 , . . . /tx n _ a are each of them equal to cos 0, which we shall denote by /u. If we also write v for sin 0, /ot n _ '), then the value of the zonal harmonic at the point (0, $) is a function of the four angles 0', $', 0, $, and because it is a function of jut, the cosine of the arc joining the points (0, ) and (0', <'), ii> will be unchanged in value if and 0', and also < and $', are made to change places. The zonal harmonic so expressed has been called Laplace's Coefficient. Thomson and Tait call it the Biaxal Harmonic. Any homogeneous function of as, y, z- which satisfies Laplace's equa- tion may be called a Solid harmonic, and the value of a solid harmonic at the surface of a sphere whose centre is the origin may be called a Surface harmonic* In this book we have defined a surface harmonic by means of its n poles, so that it has only 2n variables. The more general surface harmonic, which has 2n-\-l variables, is the more restricted surface harmonic multiplied by an arbitrary constant. The more general surface harmonic, when expressed in terms of and $, is called a Laplace's Function. 140 0.] To obtain the other harmonics of the symmetrical system, we have to differentiate with respect to <7 ~ 7^ n -= - Ds = Ds and -j-Dc = Dc' t (62) ^ n -- n ^"-^ ' () W so that i><9 and DC denote the operations of differentiating with n n respect to n axes, n , y = p sin $, Behave I/? I = (-ifM^-^-Ly, (65) ^H-rg^^. <> in which we may write I Or-f*) = P" sin . (67) We have now only to differentiate with respect to z, which we may do either so as to obtain the result in terms of r and z, or as a homogeneous function of z and p divided by a power of r, (2r)l If we write . and (n or, as Todhunter translates it, an ' Associated Function of the First Kind,' is related to ^ by the equation ej?=(-)fpj?. (75) The series of descending powers of /x, beginning with /x"-* 7 , is expressed by Heine by the symbol ^\ and by Todhunter by the symbol ar(o^ n). This series may also be expressed in two other forms, n(r\ d n+). (78) Writing the solid harmonic in the form of a homogeneous func- tion of z and f, 77, viz., r n ls = we find that on performing the differentiations with respect to z, all the terms of the series except the first disappear, and the factor (# z at the origin. 143.] It appears from equation (50) that it is always possible to express a harmonic as the sum of a system of zonal harmonics of the same order, having their poles distributed over the surface of the sphere. The simplification of this system, however, does not appear easy. I have, however, for the sake of exhibiting to the eye some of the features of spherical harmonies, calculated the zonal harmonics of the third and fourth orders, and drawn, by the method already described for the addition of functions, the equi- potential lines on the sphere for harmonics which are the sums of two zonal harmonics. See Figures VI to IX at the end of this volume. Fig. VI represents the difference of two zonal harmonics of the third order whose axes are inclined 120 in the plane of the paper, and this difference is the harmonic of the second type in which o- = 1 , the axis being perpendicular to the paper. In Fig. VII the harmonic is also of the third order, but the axes of the zonal harmonics of which it is the sum are inclined 90, and the result is not of any type of the symmetrical system. One of the nodal lines is a great circle^ but the other two which are intersected by it are not circles. Fig. VIII represents the difference of two zonal harmonics of 144 &] DIAGRAMS OF SPHERICAL HARMONICS. 201 the fourth order whose axes are at right angles. The result is a tesseral harmonic for which n == 4, a- 2. Fig. IX represents the sum of the same zonal harmonics. The result gives some notion of one type of the more general har- monic of the fourth order. In this type the nodal line on the sphere consists of six ovals not intersecting each other. Within these ovals the harmonic is positive, and ia the sextuply connected part of the spherical surface which lies outside the ovals, the har- monic is negative. All these figures are orthogonal projections of the spherical surface. I have also drawn in Fig. V a plane section through the axis of a sphere, to shew the equipotential surfaces and lines of force due to a spherical surface electrified according to the values of a spherical harmonic of the first order. Within the sphere the equipotential surfaces are equidistant planes, and the lines of force are straight lines parallel to the axis, their distances from the axis being as the square roots of the natural numbers. The lines outside the sphere may be taken as a representation of those which would be due to the earth's magnetism if it were distributed according to the most simple type. 144 #.] We are now able to determine the distribution of electricity on a spherical conductor under the action of electric forces whose potential is given. By the methods- already given we expand *, the potential due to the given forces, in a series of solid harmonics of positive degree having their origin at the centre of the sphere. Let A n r n Y n be one of these, then since within the conducting sphere the potential is uniform, there must be a term A n r n Y n arising from the distribution of electricity on the surface of the sphere, and therefore in the expansion of 477(7 there must be a term In this way we can determine the coefficients of the harmonics of all orders except zero in the expression for the surface density. The coefficient corresponding to order zero depends on the charge, 0, of the sphere, and is given by 47ro- = a~ 2 e. The potential of the sphere is 144 .] Let us next suppose that the sphere is placed in the neighbourhood of conductors connected with the earth, and that 202 SPHERICAL HARMONICS. Green's Function, G, has been determined in terms of a?, y, z and #', y, /, the coordinates of any two points in the region in which the sphere is placed. If the surface density on the sphere is expressed in a series of spherical harmonics, then the electrical phenomena outside the sphere, arising from this charge on the sphere, are identical with those arising from an imaginary series of singular points all at the centre of the sphere, the first of which is a single point having a charge equal to that of the sphere and the others are multiple points of different orders corresponding to the harmonics which express the surface density. Let Green's function be denoted by G pp >, where p indicates the point whose coordinates are #, y, z> and p' the point whose co- ordinates are #', y', z'. If a charge A is placed at the point p' , then, considering x', y\ z' as constants, G ppf becomes a function of so, y } z and the potential arising from the electricity induced on surrounding bodies by An is ty =. A. G ( 1 ) pp ' \ / If, instead of placing the charge A Q at the point jo 7 , it were distributed uniformly over a sphere of radius a having its centre at p') the value of at points outside the sphere would be the same. If the charge on the sphere is not uniformly distributed, let its surface density be expressed, as it always can, in a series of spherical harmonics, thus 47r<2 2 n will be n+1 A n Y n for points inside the sphere, and tM+1 ^ w ^n f r points outside the sphere. Now the latter expression, by equations (13), (14), Art. 129, is equal to , , n . a n d n I or the potential outside the sphere, due to the charge on the surface of the sphere, is equivalent to that due to a certain multiple point whose axes are h^.li n and whose moment is A n a\ Hence the distribution of electricity on the surrounding con- ductors and the potential due to this distribution is the same as that which would be due to such a multiple point.. 144 &] GREEN'S FUNCTION. 203 The potential, therefore, at the paint /?, or (a, y, 2), due to the induced electrification of surrounding bodies, is where the accent over the d's indicates that the differentiations are to be performed with respect to x', y ', /. These coordinates are afterwards to be made equal to those of the centre of the sphere. It is convenient to suppose T n broken up into its 2n+\ con- stituents of the symmetrical system . Let A Y^ be one of these, then d '" - J*> (s) Th~Y n - v " It is unnecessary here to supply the affix 8 or c, which indicates whether sino-^ or cosa-0 occurs in the harmonic We may now write the complete expression for *, But within the sphere the potential is constant, or * + 1 4, + SS [ JL f* J] = constant. (7) Now perform on this expression the operation Dr**, where the differentiations are to be with respect to x y> z, and the values of % and Oj are independent of those of n and or. All the terms of (7) will disappear except that in 7, and we find _ o We thus obtain a set of equations, the first member of each of which contains one of the coefficients which we wish to determine. The first term of the second member contains A , the charge of the sphere, and we may regard this as the principal term. Neglecting, for the present, the other terms, we obtain as a first approximation W _ 1 2 2 *V W g () A " - 2 " If the shortest distance from the centre of the sphere to the nearest of the surrounding conductors is denoted by #, 204 SPHERICAL HARMONICS. [145 a. If, therefore, b is large compared with #, the radius of the sphere, the coefficients of the other spherical harmonics are very small compared with A . The terms after the first on the right-hand side of equation (8) will therefore be of an order of magnitude . / similar to -r We may therefore neglect them in a first approximation, and in a second approximation we may insert in these terms the values of the coefficients obtained by the first approximation, and so on till we arrive at the degree of approximation required. Distribution of electricity on a nearly spherical conductor. 145 a.~\ Let the equation of the surface of the conductor be r. = a(l+F), (1) where F is a function of the direction of r, that is to say of 6 and $, and is a quantity the square of which may be neglected in this investigation. Let F be expanded in the form of a series of surface harmonics y=/o+/ 1 r 1 +/ 2 r 2 +&e. +/!. (2) Of these terms, the first depends on the excess of the mean radius above a. If therefore we assume that a is the mean radius, that is to say, approximately the radius of a sphere whose volume is equal to that of the given conductor, the coefficient f Q will disappear. The second term, that in /i , depends on the distance of the centre of mass of the conductor, supposed of uniform density, from the origin. If therefore we take that centre for origin, the coefficient f^ will also disappear. We shall begin by supposing that the conductor has a charge J , and that no external electrical force acts on it. The potential outside the conductor must therefore be of the form where the surface harmonics are not assumed to be of the same types as in the expansion of F. At the surface of the conductor the potential is that of the conductor, namely, the constant quantity a. Hence, expanding the powers of r in terms of a and F, and neglecting the square and higher powers of F, we have 145 a -] NEARLY SPHERICAL CONDUCTORS. 205 (4) Since the coefficients A lf &c. are evidently small compared with A , we may begin by neglecting products of these coefficients into F. If we then write for F in its first term its expansion in spherical harmonics, and 'equate to zero the terms involving harmonics of the same order, we find a =4,1, (5) A.Y,' = A^f.Y, = 0, (6) It follows from these equations that the Y"s must be of the same type as the Y's, and therefore identical with them, and that A l = and A n = A a n f n . To determine the density at any point of the surface, we have the equation &Y dV 477(7= -- Y- = -j- COS j (8) dv dr where v is the normal and e is the angle which the normal makes with the radius. Since in this investigation we suppose F and its first differential coefficients with respect to 6 and to be small, we may put cos e = 1 , so that (9) Expanding the powers of r in terms of a and F, and neglecting products of F into A n , we find A ~(l-2F) + &c. + (n + l)A n ^Y n . (10) Expanding F in spherical harmonics and giving A n its value as already found, we obtain 477(7 = ^ ^[i+/ a r a +2/ 8 r 8 +&c.+(-i)/ ll rj. (n) Hence, if the surface differs from that of a sphere by a thin stratum whose depth varies according to the values of a spherical harmonic of order n, the ratio of the difference of the surface densities at any two points to their sum will be nl times, 206 SPHERICAL HARMONICS. [145 "b. the ratio of the difference of the radii at the same two points to their sum. 1453.] If a nearly spherical conductor is acted on by external electric forces, let the potential, U, arising from these forces be expanded in a series of spherical harmonics of positive degree, having their origin. at the centre of volume of the conductor U= B +3 ir T^+B^Y t '+& .+B.S>T.', (12) where the accent over T indicates that this harmonic is not necessarily of the same type as the harmonic of the same order in the expansion of F. If the conductor had been accurately spherical, the potential arising from its surface charge at a point outside the conductor would have been Let the actual potential arising from the surface charge be Tf-i- W, where CJ^+.^ (H) the harmonics with a double accent being different from those oecurring either in F or in 77, and the coefficients C being small because F is small. The condition to be fulfilled is that, when r = a (1 + F), +W= constant = A tt the potential of the conductor. Expanding the powers of r in terms of a and F, and retaining the first power of F when it is multiplied by A or B, but neglecting it when it is multiplied by the small quantity C, we find + (^ir+to. + Cj I K'=0. (15) To determine the coefficients C, we must perform the multipli- cation indicated in the first term, and express the result in a series of spherical harmonics. This series, with the signs reversed, will be the series for W at the surface of the conductor. The product of two spherical harmonics of orders n and m, is a rational function of degree n + m in sc/r 9 y/r, and z/r, and can therefore be expanded in a series of spherical harmonics of orders not exceeding m+n. If, therefore, F can be expanded in spherical 145 c -] NEARLY SPHERICAL VESSEL. 207 harmonics of orders not exceeding m, and if the potential due to external forces can be expanded in spherical harmonics of orders not exceeding n t the potential arising from the surface charge will involve spherical harmonies of orders not exceeding m -f n. This surface density can then be found from the potential by the equation , )=0. (16) 145 c.~\ A nearly spherical conductor enclosed in a nearly spherical and nearly concentric vessel. Let the equation of the surface of the conductor be r = a(l+F\ (17) where F =/ i Y 1 + &c. +J Y*\ (is) Let the equation of the inner surface of the vessel y be r = 6(l + G), (49) where G = ffl T l + &c. +^> T&, (26) the f 's and ^'s being small compared with unity, and Y^ being the surface harmonic of order n and type o-. Let the potential of the conductor be a, and that of the vessel /3. Let the potential at any point between the conductor and the vessel be expanded in spherical harmonics, thus l ?T , (21) then we have to determine the constants of the forms h and k so that when r = a (1 + F), ^ = a, and when r = I (l + G), * = /8. It is manifest, from our former investigation, that all the ^'s and /fc's except h$ and k Q will be small quantities, the products of which into F may be neglected. We may, therefore, write a = (1 _J>+ &c. + *" + ) 7 , (22) ft = M- * (1 - G) + &c. + ( b" + jj Y (23) We have therefore a = A<> + ^, (24) 208 SPHERICAL HARMONICS. [146. l> (26) whence we find for the charge of the inner conductor *o = (-/3)^ V (28) and for the coefficients of the harmonics of order n "n /QA\ _ a 2n+l ( 30 ) where we must remember that the coefficients/^, g n , h n , k n are those belonging to the same type as well as order. The surface density on the inner conductor is given by the equation _ n - n ' n ~ 2n + l_2n+l 146.] As an example of the application of zonal harmonics, let us investigate the equilibrium of electricity on two spherical conductors. Let a and b be the radii of the spheres, and c the distance between their centres. We shall also, for the sake of brevity, write a ex, and 6 = cy> so that x and y are numerical quantities less than unity. Let the line joining the centres of the spheres be taken as the axis of the zonal harmonics, and let the pole of the zonal harmonics belonging to either sphere be the point of that sphere nearest to the other. Let r be the distance of any point from the centre of the first sphere, and s the distance of the same point from that of the second sphere. Let the surface density, o- lf of the first sphere be given by the equation 47T(r 1 a 2 = ^-M 1 P 1 -f 34>P 2 + &c. + (2m+l)J m P w , (1) so that A is the total charge of the sphere, and A lt &c. are the coefficients of the zonal harmonics P 1} &c. 146.] TWO SPHERICAL CONDUCTORS. 209 The potential due to this distribution of charge may be repre- sented by for points inside the sphere, and by (3) for points outside. Similarly, if the surface density on the second sphere is given by the equation 47r<7 2 2 = J + - B 1 P J + & c . + (2 + l)J5 ll P w , (4) the potential inside and outside this sphere may be represented by equations of the form (5) (6) where the general harmonics are related to the second sphere. The charges of the sphere are A and B respectively. The potential at every point within the first sphere is constant and equal to a, the potential of that sphere, so that within the first sphere U'+F=a. (?) Similarly, if the potential of the second sphere is /3, for points within that sphere, U+ 7'= ft. (8) For points outside both spheres the potential is V, where Z7+r=*. (9) On the axis, between the centres of the spheres, r+s=c. (10) Hence, differentiating with respect to r, and after differentiation making r = 0, and remembering that at the pole each of the zonal harmonics is unity, we find 2! a '"^ where, after differentiation, s is to be made equal to c. VOL. i. P 210 SPHERICAL HARMONICS. [146. If we perform the differentiations, and write a/c x and 6/c = y, these equations become = = A^ + Bx* + 3^ x*y + 6 -# 2 # 3 / + &c. + J (n + l) (+ 2) B = (12) By the corresponding operations for the second sphere we find, = (13) To determine the potentials, a and /3, of the two spheres we have the equations (7) and (8), which we may now write : m . (15) y If, therefore, we confine our attention to the coefficients A 1 to A m and J5 X to B n , we have ^ + ^ equations from which to determine these quantities in terms of A and JB, the charges of the two spheres, and by inserting the values of these coefficients in (14) and (15) we may express the potentials of the spheres in terms of their charges. These operations may be expressed in the form of determinants, but for purposes of calculation it is more convenient to proceed as follows. Inserting in equations (12) the values of B l ...B n from equa- tions (13), we find [2 . 2 + 3 . 3^ 2 + 4 . 4^ 4 + 5 . TWO SPHERICAL CONDUCTORS. 211 +6.1/4- 10. 1/4- 15. 1/ 4. (17) .& 4 4- A a? 4 / [4 . 1 4- 10 . 1/ + 20 . 1/ . 2 4-10.3/ .3. (18) fl /[5.2. (19) By substituting- in the second members of these equations the approximate values of A l &c., and repeating the process for further approximations, we may cany the approximation to the coefficient to any extent in ascending powers and products of x and y. If we write _ - we find [2 + 3/+ + 30/+ 75/+154/ + 280/ + &C. + 90/ + 288/ + 735 [32 + 200/ + 780/ + &c. [144 + &C. (20) 25/+ 36/-f 49/ + 64y 12 + &c. -f 7 / [6 +18^+ 40/+ 75/4-126/4- 196/ + &c. -f 9 /[8 4-30/+ 80/4-175/ + 336/4-&C. 4- n /[l04-45/4-140/4-350/4-&c. 4-^ 13 /[l24-63/-f 224/4-&C. 4- 15 /[144-84/4-&c. V 2 212 SPHERICAL HARMONICS. [146. + afiy*[ 16 + 72/ + 209/+488/ + &C. + # 10 /[ 60+ 342/+1222/ + &C. + a? 12 / [150+1050/ + &C. 64 + &C. (21) It will be more convenient in subsequent operations to write these coefficients in terms of a, b, and c, and to arrange the terms according to their dimensions in c. This will make it easier to differentiate with respect to c. We thus find 19 . (22) (23) (24) -f 40 6 7 k- 13 (25) 146.] TWO SPHERICAL CONDUCTORS. 213 ( 26 ) -f 5250 9 9 + 336a 7 n )c- 18 . (27) 14 (28) :i7 . (29) (30) (31) (32) 15 . (33) (34) (35) (36) (37) The values of the r's and ^'s may be written down by exchanging 1 a and b in the ^'s and jo's respectively. If we now calculate the potentials of the two spheres in terms of these coefficients in the form (38) (39) then , m, n are the coefficients of potential (Art. 87), and of these m = c~ l +jt?i ac~ z +p 2 a? c~* + &c., (40) n = b-i-frac-t-qzatc-* &c., (41) 214 SPHERICAL HARMONICS. or, expanding in terms of a, d, c, (42) c- 22 . (43) The value of I can be obtained from that of n by exchanging a and b. The potential energy of the system is, by Art. 87, jr=:%lA 2 +mAJ]+%nE 2 , (44) and the repulsion between the two spheres is, by Art. 93 a, dW I0 dl The surface density at any point of either sphere is given by equations (1) and (4) in terms of the coefficients A n and J3 n . CHAPTEK X. CONFOCAL QUADKIC SURFACES*. 147.] LET the general equation of a confocal system be # 2 f z 2 A23^2 + Xtl^ + X*^2 - *i I 1 ) where A is a variable parameter, which we shall distinguish by a suffix for the species of quadric, viz. we shall take A 1 for the hyper- boloids of two sheets, A 2 for the hyperboloids of one sheet, and A 3 for the ellipsoids. The quantities a, A 15 b, A 2 , c, A 3 are in ascending order of magnitude. The quantity a is introduced for the sake of symmetry, but in our results we shall always suppose a = 0. If we consider the three surfaces whose parameters are A 15 A 2 , A 3 , we find, by elimination between their equations, that the value of tf 2 at their point of intersection satisfies the equation The values of y 2 and z 2 may be found by transposing a, b, c symmetrically. Differentiating this equation with respect to A 1? we find dx Aj . . ~d^-l^~^ X ' If ds 1 is the length of the intercept of the curve of intersection of A 2 and A 3 cut off between the surfaces A x and A 1 + ^A 1 , then dx -f dy & Z /v l V"2 ~"1 ) V 2 #j' Since ds l3 ds 2 , and ds z are at right angles to each other, the surface-integral over the element of area ds 2 ds 3 is Now consider the element of volume intercepted between the surfaces a, /3, y, and a + ^a, fi + dft, y + dy. There will be eight such elements, one in each octant of space. We have found the surface-integral of the normal component of the force (measured inwards) for the element of surface intercepted from the surface a by the surfaces j3 and /3 -f d{3, y and I49-] TRANSFORMATION OP POISSON's EQUATION. 217 The surface-integral for the corresponding element of the surface a + da will be since D^ is independent of a. The surface-integral for the two opposite faces of the element of volume will be the sum of these quantities, or Similarly the surface-integrals for the other two pairs of faces will be d 2 7D 2 d 2 FD 2 -T-Z =- da dp dy and -j-^ *- da d(B dy. dp 2 c dy* c These six faces enclose an element whose volume is 7) 2 7) 2 7) 2 , , JL/-1 -LJn -L/n i-it ds t ds 2 ds 3 = 2 | - da dp dy, and if p is the volume-density within that element, we find by Art. 77 that the total surface-integral of the element, together with the quantity of electricity within it, multiplied by 4 TT is zero, or, dividing by da dp dy, which is the form of Poisson's extension of Laplace's equation re- ferred to ellipsoidal coordinates. If p = the fourth term vanishes, and the equation is equivalent to that of Laplace. For the general discussion of this equation the reader is referred to the work of Lame already mentioned. 149.] To determine the quantities a, p, y, we may put them in the form of ordinary elliptic integrals by introducing the auxiliary angles 6, , and \^, where A 1 = ^sin^, (12) A 2 = v/, (13) A 3 = csec\lf, (14) If we put b = kc, and k 2 + V 2 = 1, we may call Jc and k' the two complementary moduli of the confocal system, and we find a = 218 CONFOCAL QUADRIC SURFACES. [l5O. an elliptic integral of the first kind, which we may write according to the usual notation F(k,6}. In the same way we find - (ie) where F(k') is the complete function for modulus k', , ., 1 _ ft* COS 2 \{f Here a is represented as a function of the angle 0, which is ac- cordingly a function of the parameter \ lt ft as a function of < and thence of A 2 , and y as a function of \j/ and thence of A 3 . But these angles and parameters may be considered as functions of a, /3, y. The properties of such inverse functions, and of those connected with them, are explained in the treatise of M. Lame on that subject. It is easy to see that since the parameters are periodic functions of the auxiliary angles, they will be periodic functions of the quantities a, /3, y : the periods of A x and A 3 are F(k), and that of A 2 Particular Solutions. 150.] If Y is a linear function of a, /3, or y, the equation is satisfied. Hence we may deduce from the equation the distribution of electricity on any two confocal surfaces of the same family maintained at given potentials, and the potential at any point between them. The Hyperboloids of Two Sheets. When a is constant the corresponding surface is a hyperboloid of two sheets. Let us make the sign of a the same as that of SB in the sheet under consideration. We shall thus be able to study one of these sheets at a time. Let a l9 a 2 be the values of a corresponding to two single sheets, whether of different hyperboloids or of the same one, and let 7[, 7 be the potentials at which they are maintained. Then, if we make the conditions will be satisfied at the two surfaces and throughout the space between them. If we make V constant and equal to ?i in the space beyond the surface a 1} and constant and equal to T^ 15-] DISTRIBUTION OF ELECTRICITY. 219 in the space beyond the surface a 2 , we shall have obtained the complete solution of this particular case. The resultant force at any point of either sheet is -n cW dV da or -i "? 2 ^A - (20) If PI be the perpendicular from the centre on the tangent plane at any point, and P l the product of the semi-axes of the surface, then Hence we find ^ = ^-^ Wi y (2l) a l~~ a 2 -^1 or the force at any point of the surface is proportional to the per- pendicular from the centre on the tangent plane. The surface-density (27) 4 *yi-y 2 P 3 where p 3 is the perpendicular from the centre on the tangent plane, and P 3 is the product of the semi-axes. The whole charge of electricity on either surface is given by and is finite. When y F(k) the surface of the ellipsoid is at an infinite distance in all directions. If we make P 2 = and y 2 = F(Jc}> we find for the quantity of electricity on an ellipsoid maintained at potential V in an infinitely extended field, y The limiting form of the ellipsoids occurs when y = 0, in which case the surface is the part of the plane of xy within the focal ellipse, whose equation is written above, (25). The surface-density on either side of the elliptic plate whose equation is (25), and whose eccentricity is Jc, is = 47rx/^T 2 /pj" /ri^~? r ~ V ~?""^r^ y and its charge is Q = c TX (31) I 5 !] SURFACES OF REVOLUTION. 221 Particular Cases. 151.] If c remains finite, while 6 and therefore k is diminished till it becomes ultimately zero, the system of surfaces becomes transformed in the following manner : The real axis and one of the imaginary axes of each of the hyperboloids of two sheets are indefinitely diminished, and the surface ultimately coincides with two planes intersecting in the axis of z. The quantity a becomes identical with 0, and the equation of the system of meridional planes to which the first system is reduced is (\9 "" / sin a) 2 (cos a As regards the quantity /3, if we take the definition given in page 216 (7) we shall be led to an infinite value of the integral at the lower limit. In order to avoid this we define /3 in this particular case as the value of the integral If we now put A 2 = c sin <, /3 becomes i.e. log cot |0. e?-e-t Whence cos d> = and therefore sin < = If we call the exponential quantity i (eP -f- e~?) the hyperbolic cosine of /3, or more concisely the hypocosine of fi, or cosh , and if we call \ (e&e~P) the hyposine of /3, or sinh /3, and if in the same way we employ functions of a similar character analogous to the other simple trigonometrical ratios, then A 2 = c sech /3, and the equation of the system of hyperboloids of one sheet is (sech/3) 2 (tanh/3) 2 The quantity y is reduced to \/r, so that A 3 = c cosec y, and the equation of the system of ellipsoids is Ellipsoids of this kind, which are figures of revolution about their conjugate axes, are called planetary ellipsoids. 222 CONFOCAL QTJADRIC SURFACES. [152. The quantity of electricity on a planetary ellipsoid maintained at potential V in an infinite field, is where c sec y is the equatorial radius, and c tan y is the polar radius. If y = 0, the figure is a circular disk of radius c, and 22 2 = 2 (c 3) cosh - cos - sinh - a 2 * u (52) When ^ = c we have the case of paraboloids of revolution about the axis of x y and x = a f e 2a e 2 ^), y 20 d^+vcosft (53) z The surfaces for which ft is constant are planes through the axis, ft being the angle which such a plane makes with a fixed plane through the axis. The surfaces for which a is constant are confocal paraboloids. When a= - oo the paraboloid is reduced to a straight line terminat- ing at the origin. 1 54-] CYLINDERS AND PARABOLOIDS. 225 We may also find the values of a, /3, y in terms of r, 0, and $, the spherical polar coordinates referred to the focus as orgin, and the axis of the parabolas as axis of the sphere, a = log (r* cos J 6), = <*>, (54) y = log (r^ sin J0). We may compare the case in which the potential is equal to a, with the zonal solid harmonic r t Q { . Both satisfy Laplace's equa- tion, and are homogeneous functions of a?, y, z, but in the case derived from the paraboloid there is a discontinuity at the axis, and i has a value not differing by any finite quantity from zero. The surface-density on an electrified paraboloid in an infinite field (including the case of a straight line infinite in one direction) is inversely as the square root of the distance from the focus, or, in the case of the line, from the extremity of the line. VOL. I. CHAPTER XI. THEORY OF ELECTRIC IMAGES AND ELECTRIC INVERSION. 155.] WE have already shewn that when a conducting sphere is under the influence of a known distribution of electricity, the distribution of electricity on the surface of the sphere can be determined by the method of spherical harmonics. For this purpose we require to expand the potential of the in- fluencing system in a series of solid harmonics of positive degree, having the centre of the sphere as origin, and we then find a corresponding series of solid harmonics of negative degree, which express the potential due to the electrification of the sphere. By the use of this very powerful method of analysis, Poisson determined the electrification of a sphere under the influence of a given electrical system, and he also solved the more difficult problem to determine the distribution of electricity on two con- ducting spheres in presence of each other. These investigations have been pursued at great length by Plana and others, who have confirmed the accuracy of Poisson. In applying this method to the most elementary case of a sphere under the influence of a single electrified point, we require to expand the potential due to the electrified point in a series of solid har- monics, and to determine a second series of solid harmonics which express the potential, due to the electrification of the sphere, in the space outside. It does not appear that any of these mathematicians observed that this second series expresses the potential due to an imaginary electrified point, which has no physical existence as an electrified point, but which may be called an electrical image, because the action of the surface on external points is the same as that which would be produced by the imaginary electrified point if the spherical surface were removed. 156.] ELECTRIC IMAGES. 227 This discovery seems to have been reserved for Sir W. Thomson, who has developed it into a method of great power for the solution of electrical problems, and at the same time capable of being pre- sented in an elementary geometrical form. His original investigations, which are contained in the Cambridge and Dublin Mathematical Journal, 1848, are expressed in terms of the ordinary theory of attraction at a distance, and make no use of the method of potentials and of the general theorems of Chapter IV; though they were probably discovered by these methods. Instead, however, of following the method of the author, I shall make free use of the idea of the potential and of equipotential surfaces, when- ever the investigation can be rendered more intelligible by such means. Theory of Electric Images. 156.] Let A and B, Figure 7, represent two points in a uniform dielectric medium of infinite extent. Let the charges of A and B be e l and e% respectively. Let P be any point in space whose distances from A and B are r and r 2 respectively. Then the value of the potential at P will be TT e i e z The equipotential surfaces due to this distribution of electricity are represented in Fig. I (at the end of this volume) when e 1 and e 2 are of the same sign, and in Fig. II when they are of opposite signs. We have now to consider that surface for which V '= 0, which is the only spherical surface in the system. When e I and e 2 are of the same sign, this surface is entirely at an infinite distance, but when they are of opposite signs there is a plane or spherical surface at a finite distance for which the potential is zero. The equation of this surface is Its centre is at a point C in AB produced, such that AC: and the radius of the sphere is AB The two points A and B are inverse points with respect to this 228 ELECTRIC IMAGES. [157'. sphere, that is to say, they lie in the same radius, and the radius is a mean proportional between their distances from the centre. Since this spherical surface is at potential zero, if we suppose it constructed of thin metal and connected with the earth, there will be no alteration of the potential at any point either outside or inside, but the electrical action everywhere will remain that due to the two electrified points A and B. If we now keep the metallic shell in connection with the earth and remove the point B, the ^otentiaLwithin the .sphere .jrilLbecome_ everywhere zero^ but outside it will remain the same as before. For the surface of the sphere still remains at the same potential, and no change has been made in the exterior electrification. Hence, if an electrified point A be placed outside a spherical conductor which is at potential zero, the electrical action at all points outside the sphere will be that due to the point A together with another point within the sphere, which we may call the . electrical image of A. In the same way we may shew that if IB is a point placed inside the spherical shell, the electrical action within the sphere is that due to B, together with its image A. 157.] Definition of an Electrical Image. An electrical image is an electrified point or system of points on one side of a surface which would produce on the other side of that surface the same electrical action which the actual electrification of that surface really does produce. In Optics a point or system of points on one side of a mirror or lens which if it existed would emit the system of rays which actually exists on the other side of the mirror or lens, is called a virtual image. Electrical images correspond to virtual images in Optics in being related to the space on the other side of the surface. They do not correspond to them in actual position, or in the merely approximate character of optical foci. There are no real electrical images, that is, imaginary electrified points which would produce, in the region on the same side of the electrified surface, an effect equivalent to that of the electrified surface. For if the potential in any region of space is equal to that due to a certain electrification in the same region it must be actually produced by that electrification. In fact, the electrification at any point may be found from the potential near that point by the application of Poisson's equation. 1 5 7.] INVERSE POINTS. 229 Let a be the radius of the sphere. Lety be the distance of the electrified point A from the centre C. Let e be the charge of this point. Then the image of the point is at B, on the same radius of the sphere at a distance - , and the charge of the image is e '. ' We have shewn that this image will produce the same effect on the opposite side of the surface as the actual electrification^ of the surface does. We shall next determine the surface-density of this electrification at any point P of the spherical sur- face, and for this purpose we shall make use of the theorem of Coulomb, Art. 80, that if R is the resultant force at the surface of a con- ductor, and o- the superficial density, R = 47TO-, R being measured away from the surface. . We may consider R as the resultant of two forces, a repulsion s> (1. \ ^ acting along AP, and an attraction e -^ -^ acting along PB. Ar j -L.D Resolving these forces in the directions of AC and CP, we find that the components of the repulsion are along AC, and - along CP. Those of the attraction are BC along AG ' and ~~ e * along CP ' Now BP = j AP, and BC = y , so that the components of the attraction may be written along AC, and ~* along CP. The components of the attraction and the repulsion in the direction of AC are equal and opposite, and therefore the resultant force is entirely in the direction of the radius CP. This only confirms what we have already proved, that the sphere is an equi- potential surface, and therefore a surface to which the resultant force is everywhere perpendicular. 230 ELECTRIC IMAGES. [158. The resultant force measured along CP, the normal to the surface in the direction towards the side on which A is placed, is E- c f 2 -"* * / 3) a AP* If A is taken inside the sphere f is less than a t and we must measure R inwards. For this case therefore ^ = _ e ^-/^_l a AP Z In all cases we may write AD. Ad 1 where AD, Ad are the segments of any line through A cutting the sphere, and their product is to be taken positive in all cases. 158.] From this it follows, by Coulomb's theorem, Art. 80, that the surface-density at P is AD. Ad 1 The density of the electricity at any point of the sphere varies inversely as the cube of its distance from the point A. The effect of this superficial distribution, together with that of the point A, is to produce on the same side of the surface as the point A a potential equivalent to that due to e at A, and its image e -j at B, and on the other side of the surface the potential is tj everywhere zero. Hence the effect of the superficial distribution by itself is to produce a potential on the side of A equivalent to that due to the image e -^ at B, and on the opposite side a potential equal and opposite to that of e at A. The whole charge on the surface of the sphere is evidently e since it is equivalent to the image at B. We have therefore arrived at the following theorems on the action of a distribution of electricity on a spherical surface, the surface-density being inversely as the cube of the distance from a point A either without or within the sphere. Let the density be given by the equation where C is some constant quantity, then by equation (6) 1 59.] DISTRIBUTION OF ELECTRICITY. 231 The action of this superficial distribution on any point separated from A by the surface is equal to that of a quantity of electricity ' or 1-naC AD. Ad concentrated at A. Its action on any point on the same side of the surface with A is equal to that of a quantity of electricity fAD.Ad concentrated at B the image of A. The whole quantity of electricity on the sphere is equal to the first of these quantities if A is within the sphere, and to the second if A is without the sphere. These propositions were established by Sir W. Thomson in his original geometrical investigations with reference to the distribution of electricity on spherical conductors, to which the student ought to refer. 159.] If a system in which the distribution of electricity is known is placed in the neighbourhood of a conducting sphere of radius a, which is maintained at potential zero Jby connection with^ thejearth^then the electrifications due to the several parts of the system will be superposed. Let A 13 AI, &c. be the electrified points of the system, f-^f^ &c. their distances from the centre of the sphere, e lt e 2 , &c. their charges, then the images J3 13 B^ &c. of these points will be in the O Q same radii as the points themselves, and at distances > -^ , &c. /I /2 from the centre of the sphere, and their charges will be /I /2 The potential on the outside of the sphere due to the superficial electrification will be the same as that which would be produced by the system of images J? 15 .Z? 2 , &c. This system is therefore called the electrical image of the system A , A z , &c. If the sphere instead of being at potential zero is at potential F, we must superpose a distribution of electricity on its outer surface having the uniform surface-density 7 4-Tra The effect of this at all points outside the sphere will be equal to 232 ELECTRIC IMAGES. [l6o. that of a quantity 7a of electricity placed at its centre, and at all points inside the sphere the potential will be simply increased by 7. The whole charge on the sphere due to an external system of influencing points, A 19 A. 2 , &c. is d a E= ra-e lT -e, T -&*., (9) Jl J-2 from which either the charge E or the potential V may be cal- culated when the other is given. When the electrified system is within the spherical surface the induced charge on the surface is equal and of opposite sign to the inducing charge, as we have before proved it to be for every closed surface, with respect to points within it. *160.] The energy due to the mutual action between an elec- trified point e, at a distance /from the centre of the sphere greater than a the radius, and the electrification of the spherical surface due to the influence of the electrified point and the charge of the sphere, is 1/= f^ '" 5 / (/*-*')' (10) where 7 is the potential, and E the charge of the sphere. The repulsion between the electrified point and the sphere is therefore, by Art. 92^ * ["The discussion in the text will perhaps be more easily understood if the problem be regarded as an example of Art.^t Let us then suppose that what is described as an electrified point is really a small spherical conductor, the radius of which is & and the potential v. We have thus a particular case of the problem of two spheres of whiclroiie solution has already been given in Art. 146, and another will be given in Art. 173. In the case before us however the radius 6 is so small that we may consider the electricity of the small conductor to be uniformly distributed over its surface and all the electric images except the first image of the small conductor to be disregarded. We thus have F = - + , // f The energy of the system is therefore, Art. 85, 2a By means of the above equations we may also express the energy in terms of the potentials : to the same order of approximation it is a7 2 a& T _ 1 ,, a& 2 \ o-i ___F, + _(+__).] l6o.] IMAGE OF AN ELECTRIFIED SYSTEM. 233 - e (E ^-' , } ~* ( 2 - 22 > Hence the force between the point and the sphere is always an attraction in the following cases (1) -When the sphere is uninsulated. (2) When the sphere has no charge. (3) When the electrified point is very near the surface. In order that the force may be repulsive, the potential of the / 3 sphere must be positive and greater than e -r^ 2x2 > an< ^ ^e charge of the sphere must be of the same sign as e and greater . At the point of equilibrium the equilibrium is unstable, the force being an attraction when the bodies are nearer and a repulsion when they are farther off. When the electrified point is within the spherical surface the force on the electrified point is always away from the centre of the sphere, and is equal to The surface-density at the point of the sphere nearest to the electrified point where it lies outside the sphere is The surface-density at the point of the sphere farthest from the electrified point is a(fa}} - r ) When E, the charge of the sphere, lies between a*(3f-a) a* and - e the electrification will be negative next the electrified point and 234 ELECTRIC IMAGES. [161. positive on the opposite side. There will be a circular line of division between the positively and the negatively electrified parts of the surface, and this line will be a line of equilibrium. the equipotential surface which cuts the sphere in the line of equi- librium is a sphere whose centre is the electrified point and whose radius is A// 2 # 2 . The lines of force and equipotential surfaces belonging to a case of this kind are given in Figure IV at the end of this volume. Images in an Infinite Plane Conducting Surface. 161.] If the two electrified points A and B in Art. 156 are electrified with equal charges of electricity of opposite signs, the surfaces of zero potential will be the plane, every point of which is equidistant from A and B. Hence, if A be an electrified point whose charge is , A', o-', p the corresponding line surface and volume densities of electricity at the two points, 236 ELECTRIC IMAGES. [163. V the potential at A due to the original system, and V the potential at A' due to the inverse system, then v / If R 2 r' 2 S' R* / 4 K' R & r' 6 -\ ~r = ' L ' '~~ ' ~^~~E^ ~S~~^~~W ~K"~r*~~R^ e - = ?L- !L - = -L ?L */ l* = & *~*~~ & ^*( 18 ) 7' r R 7~!$"V If in the original system a certain surface is that of a conductor, ' and has therefore a constant potential P, then in the transformed system the image of the surface will have , a potential P . But by placing at 0, the centre of inversion, a quantity of electricity equal to PR, the potential of the transformed surface is reduced to zero. Hence, if we know the distribution of electricity on a conductor when insulated in open space and charged to the potential P, we can find by inversion the distribution on a conductor whose form is the image of the first under the influence of an electrified point with a charge PR placed at the centre of inversion, the conductor being in connexion with the earth. 163.] The following geometrical theorems are useful in studying cases of inversion. Every sphere becomes, when inverted, another sphere, unless it passes through the centre of inversion, in which case it becomes a plane. If the distances of the centres of the spheres from the centre of inversion are a and of, and if their radii are a and of, and if we define the power of a sphere with respect to the centre of in- version to be the product of the segments cut off by the sphere from a line through the centre of inversion, then the power of the first sphere is a z a 2 , and that of the second is a 2 of 2 . We have in this case / / "7~O /> /*> - = - = ^ = T> ( 19 ) a a a 2 a 2 R 2 or the ratio of the distances of the centres of the first and second spheres is equal to the ratio of their radii, and to the ratio of the * See Thomson and Tait's Natural Philosophy, 515,. GEOMETRICAL THEOREMS. 237 power of the sphere of inversion to the power of the first sphere, or of the power of the second sphere to the power of the sphere of inversion. The image of the centre of inversion with regard to one sphere is the inverse point of the centre of the other sphere. In the case in which the inverse surfaces are a plane and a sphere, the perpendicular from the centre of inversion on the plane is to the radius of inversion as this radius is to the diameter of the sphere, and the sphere has its centre on this perpendicular and passes through the centre of inversion. Every circle is inverted into another circle unless it passes 7 A through the centre of inversion, in which case it becomes a straight line. The angle between two surfaces, or two lines at their intersec- tion, is not changed by inversion. Every circle which passes through a point, and the image of that point with respect to a sphere, cuts the sphere at right angles. Hence, any circle which passes through a point and cuts the sphere at right angles passes through the image of the point. 164.] We may apply the method of inversion to deduce the distribution of electricity on an uninsulated sphere under the in- fluence of an electrified point from the uniform distribution on an insulated sphere not influenced by any other body. If the electrified point be at A, take it for the centre of inversion, and if A is at a distance f from the centre of the sphere whose radius is a, the inverted figure will be a sphere whose radius is a and whose centre is distant/', where ^_/'_^t_ (20) " ~" The centre of either of these spheres corresponds to the inverse point of the other with respect to A, or if C is the centre and B the inverse point of the first sphere, C' will be the inverse point, and B' the centre of the second. Now let a quantity / of electricity be communicated to the second sphere, and let it be uninfluenced by external forces. It will become uniformly distributed over the sphere with a surface- Its action at any point outside the sphere will be the same as that of a charge / placed at I? the centre of the sphere. 238 ELECTRIC IMAGES. [165. At the spherical surface and within it the potential is r=?> - (22) a constant quantity. Now let us invert this system. The centre Bf becomes in the inverted system the inverse point B, and the charge / at If -n becomes e' -^ at B, and at any point separated from B by the J surface the potential is that due to this charge at B. The potential at any point P on the spherical surface, or on the same side as B, is in the inverted system e' R a' AP' If we now superpose on this system a charge e at A, where e = - g 7 R, (23) the potential on the spherical surface, and at all points on the same side as B, will be reduced to zero. At all points on the same side as A the potential will be that due to a charge e at A, and a charge (24) as we found before for the charge of the image at B. To find the density at any point of the first sphere we have 7?3 Substituting for the value of or' in terms of the quantities be- longing to the first sphere, we find the same value as in Art. 158, <7 = (26) On Finite Systems of Successive Images. 165.] If two conducting planes intersect at an angle which is a submultiple of two right angles, there will be a finite system of images which will completely determine the electrification. For let AOB be a section of the two conducting planes per- pendicular to their line of intersection, and let the angle of inter- section AOB = -, let P be an electrified point, and let PO = r, ti and POB = 6. Then, if we draw a circle with centre and radius 165.] SYSTEMS OF IMAGES. 239 OP, and find points which are the successive images of P in the two planes beginning 1 with OS, we shall find Q 1 for the image of P in OB, P 2 for the image of Q l in OA, Q 3 for that of P 2 in OB, P 3 for that of Q 3 in OA, and Q 2 for that of P 3 in OB. If we had begun with the image of P in AO we should have found the same points in the reverse order Q 2 , P 3 , Q 3 , P 2 , Q lt provided AOB is a submultiple of two right angles. For the alternate images P 1 ,P 2 > ^s are ran g e ^ round the circle at angular intervals equal to 2 AOB, and the intermediate images Q lt Q 2 , Q 3 are at inter- vals of the same magnitude. Hence, if 2 AOB is a submultiple of 2 IT, there will be a finite number of images, and none of these will fall within the angle AOB. If, however, AOB is not a submultiple of TT, it will be impossible to represent the actual electrification as the re- sult of a finite series of electrified points. If AOB = -, there will be n negative images Q lt (J) 2 , &c., each n equal and of opposite sign to P, and n 1 positive images P 2 , P 35 &c., each equal to P, and of the same sign. The angle between successive images of the same sign is If we consider either of the conducting planes as a plane of sym- metry, we shall find the positive and negative images placed symmetrically with regard to that plane, so that for every positive image there is a negative image in the same normal, and at an equal distance on the opposite side of the plane. If we now invert this system with respect to any point, the two planes become two spheres, or a sphere and a plane intersecting at an angle - , the influencing point P being within this angle. The successive images lie on the circle which passes through P and intersects both spheres at right angles. To find the position of the images we may make use of the principle that a point and its image are in the same radius of the sphere, and draw successive chords of the circle beginning at P and passing through the centres of the two spheres alternately. 240 ELECTRIC IMAGES. [166. To find the charge which must be attributed to each image, take any point in the circle of intersection, then the charge of each image is proportional to its distance from this point, and its sign is positive or negative according as it belongs to the first or the second system. 166.] We have thus found the distribution of the images when any space bounded by a conductor consisting of two spherical surfaces meeting at an angle - , and kept at potential zero, is influenced by VL an electrified point. We may by inversion deduce the case of a conductor consisting of two spherical segments meeting at a re-entering angle - , charged to potential unity and placed in free space. For this purpose we invert the system with respect to P. The circle on which the images formerly lay now becomes a straight line through the centres of the spheres. If the figure (ll) represents a section through the line of centres AB, and if D, D' are the points where the circle of in- tersection cuts the plane of the paper, then, to find the suc- cessive images, draw DA a radius of the first circle, and draw DC, DB, &c., making angles -, , &c. with DA. fe n n ' The points C, B, &c. at which they cut the line of centres will be the positions of the positive images, and the charge of each will be represented by its distances from D. The last of these images will be at the centre of the second circle. To find the negative images draw DP, DQ, &c., making angles 1L , & c . with the line of centres. The intersections of these n n lines with the line of centres will give the positions of the negative images, and the charge of each will be represented by its distance from D. The surface-density at any point of either sphere is the sum of the surface-densities due to the system of images. For instance, the surface-density at any point S of the sphere whose centre is A, is 1 67.] TWO INTERSECTING SPHERES. 241 where A, B, C, &c. are the positive series of images. When 8 is on the circle of intersection the density is zero. To find the total charge on each of the spherical segments, we may find the surface-integral of the induction through that segment due to each of the images. The total charge on the segment whose centre is A due to the image at A whose charge is DA is where is the centre of the circle of intersection. In the same way the charge on the same segment due to the image at B is \ (DB + OB), and so on, lines such as OB measured from to the left being reckoned negative. Hence the total charge on the segment whose centre is A is 0^4- &c.), 167.] The method of electrical images may be applied to any space bounded by plane or spherical surfaces all of which cut one another in angles which are submultiples of two right angles. In order that such a system of spherical surfaces may exist, every solid angle of the figure must be trihedral, and two of its angles must be right angles, and the third either a right angle or a submultiple of two right angles. Hence the cases in which the number of images is finite are (1) A single spherical surface or a plane. (2) Two planes, a sphere and a plane, or two spheres intersecting at an angle - 99 (3) These two surfaces with a third, which may be either plane or spherical, cutting both orthogonally. (4) These three surfaces with a fourth cutting the first two orthogonally and the third at an angle -, . Of these four surfaces one at least must be spherical. We have already examined the first and second cases. In the first case we have a single image. In the second case we have 2nl images arranged in two series in a circle which passes through the influencing point and is orthogonal to both surfaces. VOL. I. R 242 ELECTRIC IMAGES. [168. In the third case we have, besides these images, their images with respect to the third surface, that is, 4# 1 images in all besides the influencing point. In the fourth case we first draw through the influencing point a circle orthogonal to the first two surfaces, and determine on it the positions and magnitudes of the n negative images and the n 1 positive images. Then through each of these 2n points, including the influencing point, we draw a circle orthogonal to the third and fourth surfaces, and determine on it two series of images, n' in each series. We shall obtain in this way, besides the influencing point, 2nn' 1 positive and 2nn' negative images. These 4nn' points are the intersections of n circles with Mother circles, and these circles belong to the two systems of lines of curvature of a cyclide. If each of these points is charged with the proper quantity of electricity, the surface whose potential is zero will consist of n -f n' spheres, forming two series of which the successive spheres of the 7T first set intersect at angles - , and those of the second set at angles , while every sphere of the first set is orthogonal to every sphere of the second set. Case of Two Spheres cutting Orthogonally. See Fig. IV at the end of this volume. 168.] Let A and B, Fig. 12, be the centres of two spheres cutting each other orthogonally in D and .Z/, and let the straight line DI/ cut the line of centres in C. Then C is the image of A with respect to the sphere B, and also the image of B with respect to the sphere whose centre is A. If AD = a, BD = ft then AB= x/a 2 + /3 2 , and if we place at A, B, C quantities of electricity equal to a, ft, and f respectively, then both Fig. 12. spheres will be equipotential surfaces whose potential is unity. We may therefore determine from this system the distribution of electricity in the following cases : 1 68.] TWO SPHERES CUTTING ORTHOGONALLY. 243 (1) On the conductor PDQI? formed of the larger segments of both spheres. Its potential is 1, and its charge is a + /3 -- -~ - =///)+ 7? 7) _/?/} Va 2 +/3 2 This quantity therefore measures the capacity of such a figure when free from the inductive action of other bodies. The density at any point P of the sphere whose centre is A, and the density at any point Q of the sphere whose centre is .B, are respectively ikC'-t&f) and At the points of intersection, D, D' , the density is zero. If one of the spheres is very much larger than the other, the density at the vertex of the smaller sphere is ultimately three times that at the vertex of the larger sphere. (2) The lens P / DQ f J/ formed by the two smaller segments of the spheres, charged with a quantity of electricity = and acted on by points A and B, charged with quantities a and /3, is also at potential unity, and the density at any point is expressed by the same formulae. (3) The meniscus DPD'Q' formed by the difference of the segments charged with a quantity a, and acted on by points B and C, charged respectively with quantities /3 and - , is also in equilibrium at potential unity. (4) The other meniscus QDP / J/ under the action of A and C. We may also deduce the distribution of electricity on the following internal surfaces. The hollow lens P'DQ'D under the influence of the internal electrified point C at the centre of the circle DJ/. The hollow meniscus under the influence of a point at the centre of the concave surface. The hollow formed of the two larger segments of both spheres under the influence of the three points A, B, C. But, instead of working out the solutions of these cases, we shall apply the principle of electrical images to determine the density of the electricity induced at the point P of the external surface of the conductor PDQD' by the action of a point at charged with unit of electricity. R 2 244 ELECTRIC IMAGES. [l68. Let OA = , OB = b, OP = r, BP = p, AD = a, BD = ft ^5 = x/a 2 +/3 2 . Invert the system with respect to a sphere of radius unity and centre 0. The two spheres will remain spheres, cutting each other ortho- gonally, and having their centres in the same radii with A and B. If we indicate by accented letters the quantities corresponding to the inverted system, -_ If, in the inverted system, the potential of the surface is unity, then the density at the point P 7 is If, in the original system, the density at P is o-, then and the potential is -. By placing at a negative charge of electricity equal to unity, the potential will become zero over the surface, and the density at P will be 1 g 2 -a 2 x __ /3 3 " 3 V ' ** * 47T a This gives the distribution of electricity on one of the spherical surfaces due to a charge placed at 0. The distribution on the other spherical surface may be found by exchanging a and b, a and ft and putting q or AQ instead of p. To find the total charge induced on the conductor by the elec- trified point at 0, let us examine the inverted system. In the inverted system we have a charge a at A', and /3' at .Z?', and a negative charge f at a point C' in the line A'B', such that A'C' : C'B' : : a 2 : /3' 2 . If OA'= a', OB'= b', OC' = c' t we find c 2 = 169.] FOUR SPHERES CUTTING ORTHOGONALLY. Inverting this system the charges become 245 a _a a a and a'/3' 7 a/3 Hence the whole charge on the conductor due to a unit of negative electricity at is a/3 ._ Distribution of Electricity on Three Spherical Surfaces which Intersect at Eight Angles. 169.] Let the radii of the spheres be a, /3, y, then BC = x/jS^T?, C^ = Vy* + tf, AB = Let PQR, Fig. 1 3, be the feet of the perpendiculars from ABC on the opposite sides of the tri- angle, and let be the inter- section of perpendiculars. Then P is the image of B in the sphere y, and also the image of C in the sphere /3. Also is the image of P in the sphere a. Let charges a, )3, and y be placed at A, B, and C. Then the charge to be placed at Pis Fig. 13. V W + 7 Also ^ P = sidered as the image of P, is so that the charge at 0, con- r / In the same way we may find the system of images which are 246 ELECTRIC IMAGES. [170. electrically equivalent to four spherical surfaces at potential unity intersecting at right angles. If the radius of the fourth sphere is b, and if we make the charge at the centre of this sphere = 8, then the charge at the intersection of the line of centres of any two spheres, say a and ft, with their plane of intersection, is 1 1 The charge at the intersection of the plane of any three centres ABC with the perpendicular from D is 1 /T~ ~T~ ~T" V 1? + /3 2 H " > 2 and the charge at the intersection of the four perpendiculars is 1 VI 1 1 1 T + ^2 + -2 + ^2 a 2 /3 2 v 2 5 2 /3 System of Four Spheres Intersecting at Right Angles under the Action of an Electrified Point. 170.] Let the four spheres be A, B, C, D, and let the electrified point be 0. Draw four spheres A 19 JS 19 C lt J) 19 of which any one, A 19 passes through and cuts three of the spheres, in this case B, C, and D, at right angles. Draw six spheres (ab), (ac), (ad), (be), (bd), (cd), of which each passes through and through the circle of intersection of two of the original spheres. The three spheres B 1 , C 19 D 1 will intersect in another point besides 0. Let this point be called J? 9 and let B', C', and I/ be the intersections of C it D l9 A, of D if A 19 B L , and of A 19 JS 19 C^ re- spectively. Any two of these spheres, A 19 JB lt will intersect one of the six (cd) in a point ( + - - , + - ) charges ^ 2 a 2 p 2 y' + e, the minus charges being at the points which have 1 or 3 negative coordinates. Then it is obvious the coordinate planes are at potential zero. Now let us invert with regard to any point and we have the case of three spheres cutting orthogonally under the influence of an electrified point. If we invert with regard to one of the electrified points, we find the solution for the case of a con- ductor in the form of three spheres of radii a, /3, y cutting ortho- gonally and freely charged. If to the above system of electrified points we superadd their images in a sphere with its centre at the origin we see that, in addition to the three coordinate planes, the surface of the sphere forms also a part of the surface of zero potential.] Two Spheres not Intersecting. 171.] When a space is bounded by two spherical surfaces which do not intersect, the successive images of an influencing point within this space form two infinite series, all of which lie beyond the spherical surfaces, and therefore fulfil the condition of the applicability of the method of electrical images. Any two non-intersecting spheres may be inverted into two concentric spheres by assuming as the point of inversion either of the two common inverse points of the pair of spheres. '248 ELECTKIC IMAGES. We shall begin, therefore, with the case of two uninsulated concentric spherical surfaces, subject to the induction of an elec- trified point placed between them. Let the radius of the first be b, and that of the second be^, and let the distance of the influencing point from the centre be r = be". Then all the successive images will be on the same radius as the influencing point. Let Q , Fig. 14, be the image of P in the first sphere, P 3 that of o i n the second sphere, Q 1 that of P l in the first sphere, and so on ; then and OP s .Oq 8 . l also OQ Q = be'*, \ & c . Hence OP 8 = be( u+2 *\ OQ 8 = be-( u+28 ^, If the charge of P is denoted by P, then Fig. 14. Next, let Qi be the image of P in the second sphere, P/ that of >/ in the first. &c., OP= OQ;= Of these images all the P's are positive, and all the Q's negative, all the P"s and Q's belong to the first sphere, and all the P's and Q"s to the second. The images within the first sphere form a converging series, the sum of which is -P This therefore is the quantity of electricity on the first or interior sphere. The images outside the second sphere form a diverging series, but the surface-integral of each with respect to the spherical surface is zero. The charge of electricity on the exterior spherical surface is therefore -iW P 172.] TWO SPHERES NOT INTERSECTING. 249 If we substitute for these expressions their values in terms of OA, OB, and OP, we find ^OA PS charge on ^ = -P__, -T.OB AP charge on =-P _ . If we suppose the radii of the spheres to become infinite, the case becomes that of a point placed between two parallel planes A and B. In this case these expressions become charge on A = P AP charge on B= P-r^ 172.] In order to pass from this case to that of any two spheres not intersecting each other, we begin by finding the two com- mon inverse points } (7 through which all circles pass that are orthogonal to both spheres. Then, if we invert the system with respect to either of these points, the spheres become concentric, as in the first case. Fig 15 If we take the point in Fig. 1 5 as centre of inversion, this point will be situated in Fig. 14 somewhere between the two spherical surfaces. Now in Art. 1 7 1 we solved the case where an electrified point is placed between two concentric conductors at zero potential. By inversion of that case with regard to the point we shall therefore deduce the distributions on two spherical conductors at potential zero, exterior to one another, induced by an electrified point in their neighbourhood. In Art. 173 it will be shewn how the results thus obtained may be employed in finding the distributions on two spherical charged conductors subject to their mutual influence only. The radius OAPB in Fig. 1 4 on which the successive images lie becomes in Fig. 15 an arc of a circle through and (7, and the ratio of O'P to OP is equal to Ce u where C is a numerical quantity. 250 ELECTEIC IMAGES. [172 , a A Ifweput 0= = log^p> a==1 oZ' ^ = g ~O then /3 a = or, % + a = 0. All the successive images of P will lie on the arc OAPBO'. The position of the image of P in A is (J) where That of Q in J? is P l where Similarly 6(P t ) = In the same way if the successive images of P in B, A, , &c. are Q ', P/, &', &c., 6(Ps) = 2st3-, 0(Q/) = 2)8 0-f2st3-. To find the charge of any image P 8 we observe that in the inverted figure its charge is P/ V OF' In the original figure we must multiply this by OP S . Hence the charge of P 8 in the dipolar figure is /OP..Orp. ^V OP.&P If we make f = VOP.C/P, and call ^ the parameter of the point P, then we may write P At p s ' or the charge of any image is proportional to its parameter. If we make use of the curvilinear coordinates 6 and (/>, such that x + A/ \y-\-k where 2k is the distance 00', then Jc sinh Jc sin $ cosh cos ' ~ cosh 6 cos ' (x+k coth 0) 2 +/ = k* cosech 2 0, 73'] TWO SPHERES NOT INTERSECTING. 251 _ vcoshtf cos Since the charge of each image is proportional to its parameter, , and is to be taken positively or negatively according as it is of the form P or Q, we find P - P Vcoshfl- -t a ~~ ' ' V cosh (0+ 25OT-) cos Pvcoshtf cost/) V cosh (2 a 2 sv?) cos< p, P \/cosh cos $ \/cosh (0 2 We have now obtained the positions and charges of the two infinite series of images. We have next to determine the total charge on the sphere A by finding the sum of all the images within it which are of the form Q or P'. We may write this P v cosh 6 cos $ ^ s =i i //> Vcosh(0 Pvcoshfl ,-o / . / ,, Vcosh(2a In the same way the total induced charge on B is _ *r+t = Pvcoshtf ^* :1 \/cosh (0 + 2 six] cos _ ^-#=00 1 P vcosh 6 cos(f> ^ S =Q / w \ 173.] We shall apply these results to the determination of the * In these expressions we must remember that 2cosh0 = e e + e~ e , 2sinh0 = e e -e~ 6 , and the other functions of 9 are derived from these by the same definitions as the corresponding trigonometrical functions. The method of applying dipolar coordinates to this case was given by Thomson in Liouville's Journal for 1847. See Thomson's reprint of Electrical Papers, 211, 212. In the text I have made use of the investigation of Prof. Betti, Nuovo Cimento, vol. xx, for the analytical method, but I have retained the idea of electrical images as used by Thomson in his original investigation, Phil. Mag., 1853. 252 ELECTRIC IMAGES. [173. coefficients of capacity and induction of two spheres whose radii are a and I, and the distance between whose centres is c. Let the sphere A be at potential unity, and the sphere at potential zero. Then the successive images of a charge a placed at the centre of the sphere A will be those of the actual distribution of electricity. All the images will lie on the axis between the poles and the centres of the spheres, and it will be observed that of the four systems of images determined in Art. 1 72, only the first and fourth exist in this case. If we put k = 2c k ' k then sinh a = > sinh 3 T a o The values of 6 and $ for the centre of the sphere A are 6 = 2 a, < = 0. Hence in the equations we must substitute a or k -j = for P, sinh a 2 a for 6 and for $, remembering that P itself forms part of the charge of A. We thus find for the coefficient of capacity of A #=oo 1 "^#=( = * 2.s=( h(?CT a) for the coefficient of induction of A on B or of on A =00 1 We may, in like manner, by supposing B at potential unity and A at potential zero, determine the value of q bb . We shall find, with our present notation, To calculate these quantities in terms of a and #, the radii of the spheres, and of c the distance between their centres, we observe that if __ K= Va* + b*+c*2b 2 c*-2(?a' 2 -2a 2 b 2 , we may write . , K . , . . . K smha = --- , smh/3 = j- . Binhar = , 2ac 2bc ' 2ab - - cosh a = - > cosh 8=. - = - > cosh or = - - 7 2ca 2cb 2ab 1 74-] TWO ELECTRIFIED SPHERES. 253 and make use of sinh (a + /3) = sinh a cosh (3 + cosh a sinh /3, cosh (a + (3) == cosh a cosh ft -f sinh a sinh /3. By this process or by the direct calculation of the successive images as shewn in Sir W. Thomson's paper, we find = ab - 174.] We have then the following equations to determine the charges E a and E b of the two spheres when electrified to potentials V a and T b respectively, If we put q aa q bb q ab 2 = = - and p aa = faff, p ab = faff, p bb = faff, whence PaaPnPab = & ; then the equations to determine the potentials in terms of the charges are V a = p aa fi a -\-p ab E b , and^? aa , p ab , andj% are the coefficients of potential. The total energy of the system is, by Art. 85, The repulsion between the spheres is therefore, by Arts. 92, 93, where c is the distance between the centres of the spheres. Of these two expressions for the repulsion, the first, which expresses it in terms of the potentials of the spheres and the 254: ELECTRIC IMAGES. [174. variations of the coefficients of capacity and induction, is the most convenient for calculation. We have therefore to differentiate the ^'s with respect to c. These quantities are expressed as functions of k, a, /3, and smhtn- dk cosh a cosh /3 we find dc sinh w da sinh a cosh dc k sinh OT d{$ _ cosh a sinh (3 dc k sinh CT 2 c(2c 2 -2b 2 -a 2 ) ~'' ~ ~ dc " c 2 c 2 (c 2 -a 2 -b 2 ) aH* {(5c 2 -a 2 -b 2 )(c 2 -a 2 -b 2 )-a 2 d 2 } where s is positive for the sphere A and negative for the sphere B. The charge of each image, when the potential of the spheres is unity, is numerically equal to its distance from the point of contact, and is always negative. There will also be a series of positive images whose distances from the point of contact measured in the direction of the centre of 0, are of the form When s is zero, or a positive integer, the image is in the sphere A. When s is a negative integer the image is in the sphere B. The charge of each image is measured by its distance from the origin and is always positive. The total charge of the sphere A is therefore o> 1 ab -=oo J. 256 ELECTRIC IMAGES. [ 1 75< Each of these series is infinite, but if we combine them in the form ._ ^*=> * 4&1 g(a + 6 the series becomes converging. In the same way we find for the charge of the sphere B, oo cib ab *=-oo 1 The expression for E a is obviously equal to in which form the result in this case was given by Poisson. It may also be shewn (Legendre Traite des Fonctions Elliptiqnes, ii, 438) that the above series for E a is equal to b ab a + b' ) a H- b where y = -57712..., and V(x) = ^- logT(l +#). rf^z? The values of # have been tabulated by Gauss (Werke, Band iii, pp. 161-162.) If we denote for an instant b -=- (a + b) by a?, we find for the difference of the charges E a and E b , d . ab -=- log sm TT^ x Tib cot tf + d a + b When the spheres are equal the charge of each for potential unity 18 ru=> 1 = alog e 2 = -693147180. When the sphere A is very small compared with the sphere the charge on A is a 2 ^-*=o> 1 ^ = T 2*=i ~2~ approximately ; o $ or E a = ~ I 7 7-] SPHERICAL BOWL. 257 The charge on B is nearly the same as if A were removed, or The mean density on each sphere is found by dividing the charge by the surface. In this way we get 47ra 2 = 246' Hence, if a very small sphere is made to touch a very large one, the mean density on the small sphere is equal to that on the large o sphere multiplied by , or 1.644936. Application of Electrical Inversion to the case of a Spherical Bowl. 176.] One of the most remarkable illustrations of the power of Sir W. Thomson's method of Electrical Images is furnished by his investigation of the distribution of electricity on a portion of a spherical surface bounded by a small circle. The results of this investigation, without proof, were communicated to M. Liouville and published in his Journal in 1847. The complete investigation is given in the reprint of Thomson's Electrical Papers, Article XV. I am not aware that a solution of the problem of the distribution of electricity on a finite portion of any curved surface has been given by any other mathematician. As I wish to explain the method rather than to verify the calculation, I shall not enter at length into either the geometry or the integration, but refer my readers to Thomson's work. Distribution of Electricity on an Ellipsoid. 177.] It is shewn by a well-known method* that the attraction of a shell bounded by two similar and similarly situated and concentric ellipsoids is such that there is no resultant attraction on any point within the shell. If we suppose the thickness of the shell to diminish indefinitely while its density increases, we ultimately arrive at the conception of a surface- density varying as the perpendicular from the centre on the tangent plane, and since the resultant attraction of this superficial distribution on any * Thomson and Tait's Natural Philosophy, 520, or Art. 150 of this book. VOL. I. S 258 ELECTRIC IMAGES. [178. point within the ellipsoid is zero, electricity, if so distributed on the surface, will be in equilibrium. Hence, the surface-density at any point of an ellipsoid undis- turbed by external influence varies as the distance of the tangent plane from the centre. Distribution of Electricity on a Disk. By making two of the axes of the ellipsoid equal, and making the third vanish, we arrive at the case of a circular disk, and at an expression for the surface-density at any point P of such a disk when electrified to the potential V and left undisturbed by external influence. If a be the surface-density on one side of the disk, and if KPL be a chord drawn through the point P, then 7 Application of the Principle of Electric Inversion. 178.] Take any point Q as the centre of inversion, and let R be the radius of the sphere of inversion. Then the plane of the disk becomes a spherical surface passing through Q, and the disk itself becomes a portion of the spherical surface bounded by a circle. We shall call this portion of the surface the bowl. If S' is the disk electrified to potential 7' and free from external influence, then its electrical image S will be a spherical segment at potential zero, and electrified by the influence of a quantity 7'H of electricity placed at Q. We have therefore by the process of inversion obtained the solu- tion of the problem of the distribution of electricity on a bowl or a plane disk when under the influence of an electrified point in the surface of the sphere or plane produced. Influence of an Electrified Point placed on the unoccupied part of the Spherical Surface. The form of the solution, as deduced by the principles already given and by the geometry of inversion, is as follows : If C is the central point or pole of the spherical bowl 8 9 and if a is the distance from C to any point in the edge of the segment, then, if a quantity q of electricity is placed at a point Q in the surface of the sphere produced, and if the bowl S is maintained at potential zero, the density cr at any point P of the bowl will be l8o.] SPHERICAL BOWL. 259 CQ, CP, and QP being the straight lines joining the points, C> Q, and P. It is remarkable that this expression is independent of the radius of the spherical surface of which the bowl is a part. It is therefore applicable without alteration to the case of a plane disk. Influence of any Number of Electrified Points. Now let us consider the sphere as divided into two parts, one of which, the spherical segment on which we have determined the electric distribution, we shall call the bowl, and the other the remainder, or unoccupied part of the sphere on which the in- fluencing point Q is placed. If any number of influencing points are placed on the remainder of the sphere, the electricity induced by these on any point of the bowl may be obtained by the summation of the densities induced by each separately. 179.] Let the whole of the remaining surface of the sphere be uniformly electrified, the surface-density being p, then the density at any point of the bowl may be obtained by ordinary integration over the surface thus electrified. We shall thus obtain the solution of the case in which the bowl is at potential zero, and electrified by the influence of the remaining portion of the spherical surface rigidly electrified with density p. Now let the whole system be insulated and placed within a sphere of diameter/, and let this sphere be uniformly and rigidly electrified so that its surface-density is p f . There will be no resultant force within this sphere, and therefore the distribution of electricity on the bowl will be unaltered, but the potential of all points within the sphere will be increased by a quantity V where y = 2-npf. Hence the potential at every point of the bowl will now be V. Now let us suppose that this sphere is concentric with the sphere of which the bowl forms a part, and that its radius exceeds that of the latter sphere by an infinitely small quantity. We have now the case of the bowl maintained at potential V and influenced by the remainder of the sphere rigidly electrified with superficial density p + p'. 180.] We have now only to suppose p + />'= 0, and we get the case of the bowl maintained at potential 7 and free from external influence. S 2 260 ELECTKIC IMAGES. [l8l. If cr is the density on either surface of the bowl at a given point when the bowl is at potential zero, and is influenced by the rest of the sphere electrified to density p, then, when the bowl is main- tained at potential V 9 we must increase the density on the outside of the bowl by p', the density on the supposed enveloping sphere. The result of this investigation is that if f is the diameter of the sphere, a the chord of the radius of the bowl, and r the chord of the distance of P from the pole of the bowl, then the surface- density or on the inside of the bowl is f 2 -a* _j /f 2 -a i^jtan /S/ ^ 1 / ./ <* J t / -/ ' " (T = 27T 2 / ( and the surface-density on the outside of the bowl at the same point is y In the calculation of this result no operation is employed more abstruse than ordinary integration over part of a spherical surface. To complete the theory of the electrification of a spherical bowl we only require the geometry of the inversion of spherical surfaces. 181.] Let it be required to find the surface-density induced at any point of the bowl by a quantity q of electricity placed at a point Q, not now in the spherical surface produced. Invert the bowl with respect to Q, the radius of the sphere of inversion being R. The bowl 8 will be inverted into its image 8' and the point P will have P' for its image. We have now to determine the density dx dy dy dx _ A !_ L_ o dx* * dy* ~ ' dx* r dy* ' Hence both functions satisfy Laplace's equation. M, *. (2) da dx da dp dy dx da dx Ta ^ dx dy If x and y are rectangular coordinates, and if ds 1 is the intercept of the curve (/3 = constant) between the curves a and a + da, and ds z the intercept of a between the curves p and /3 -f dp, then d*i _ ds * .. J IA\ da~d0--R' and the curves intersect at right angles. If we suppose the potential F= V^ + lca, where k is some con- stant, then Twill satisfy Laplace's equation, and the curves (a) will be equipotential curves. The curves (/3) will be lines of force, and 264 CONJUGATE FUNCTIONS. [184. the surface-integral of R over unit-length of a cylindrical surface whose projection on the plane of xy is the curve AB will be Jc(ft B /3^), where ft A and ft B are the values of ft at the extremities of the curve. If one series of curves corresponding to values of a in arithmetical progression be drawn on the plane, and another series corresponding to a series of values of ft having the same common difference, then the two series of curves will everywhere intersect at right angles, and, if the common difference is small enough, the elements into which the plane is divided will be ultimately little squares, whose sides, in different parts of the field, are in different directions and of different magnitudes, being inversely proportional to R. If two or more of the equipotential lines (a) are closed curves enclosing a continuous space between them, we may take these for the surfaces of conductors at potentials (^o + ^i)* (^o + ^ a 2)> & c - respectively. The quantity of electricity upon any one of these be- Jc tween the lines offeree ft and /3 2 will be (ftzft\)' The number of equipotential lines between two conductors will therefore indicate their difference of potential, and the number of lines of force which emerge from a conductor will indicate the quantity of electricity upon it. We must next state some of the most important theorems relating to conjugate functions, and in proving them we may use either the equations (l), containing the differential coefficients, or the original definition, which makes use of imaginary symbols. 184.] THEOKEM I. Ifx and y' are conjugate functions with respect to x and y> and if x" and y" are also conjugate functions with respect to x and y> then the functions x' -f x" and y' +y" will be conjugate functions with respect to x and y. dx' dy' . dx" dy" For - = -f- , and -=- = -f- ; dx dy dx dy therefore dx dy dy' dx" df Also T- = -T-J and T-= -- 7-9 dy dx dy dx dx' therefore dy dx or x + x" and y r -\-y" are conjugate with respect to x and y. 185.] GKAPHIC METHOD. 265 Graphic Representation of a Function which is the Sum of Two Given Functions. Let a function (a) of x and y be graphically represented by a series of curves in the plane of xy^ each of these curves corre- sponding to a value of a which belongs to a series of such values increasing by a common difference, 8. Let any other function, /3, of x and y be represented in the same way by a series of curves corresponding to a series of values of ft having the same common difference as those of a. Then to represent the function a + ft in the same way, we must draw a series of curves through the intersections of the two former series, from the intersection of the curves (a) and () to that of the curves (a + 8) and (ft 8), then through the intersection of (a + 2 5) and (/3 28), and so on. At each of these points the function will have the same value, namely a + /3. The next curve must be drawn through the points of intersection of (a) and ((3 + 8), of (a + 8) and (/3), of (a + 2 8) and (/3 8), and so on. The function belonging to this curve will be a -f ft -f 8. In this way, when the series of curves (a) and the series (/3) are drawn, the series (a + /3) may be constructed. These three series of curves may be drawn on separate pieces of transparent paper, and when the first and second have been properly superposed, the third may be drawn. The combination of conjugate functions by addition in this way enables us to draw figures of many interesting cases with very little trouble when we know how to draw the simpler cases of which they are compounded. We have, however, a far more powerful method of transformation of solutions, depending on the following theorem. 185.] THEOREM II. If x" and y" are conjugate functions with respect to the variables x' and y\ and if x' and y' are conjugate functions with respect to x and y, then x" and y" will be con- jugate functions with respect to x and y. dx" dx" dx' dx" dy' ~~~~ ~ dy" dy dy" dx' dy dy dx' dy 266 CONJUGATE FUNCTIONS. [185. dx" _ dx" daf dx" dy f dy dx' dy dy' dy _ _ dy' dx dx' dx dx ' and these are the conditions that x" and y" should be conjugate functions of x and y. This may also be shewn from the original definition of conjugate functions. For x" -\- \/ \y" is a function of x f + */ ly, and #'+ \/ \y is a function of #+ \/ ly. Hence, #"+\/ \y" is a function of #-f \/ \y. In the same way we may shew that if x' and y are conjugate functions of x and y, then a? and y are conjugate functions of x and y'. This theorem may be interpreted graphically as follows : Let x ', y' be taken as rectangular coordinates, and let the curves corresponding to values of x" and of y" taken in regular arithmetical series be drawn on paper. A double system of curves will thus be drawn cutting the paper into little squares. Let the paper be also ruled with horizontal and vertical lines at equal intervals, and let these lines be marked with the corresponding values of x f and y f . Next, let another piece of paper be taken in which x and y are made rectangular coordinates and a double system of curves #', y' is drawn, each curve being marked with the corresponding value of x' or y '. This system of curvilinear coordinates will correspond, point for point, to the rectilinear system of coordinates x', y' on the first piece of paper. Hence, if we take any number of points on the curve x" on the first paper, and note the values of x' and y f at these points, and mark the corresponding points on the second paper, we shall find a number of points on the transformed curve x" . If we do the same for all the curves x" ', y" on the first paper, we shall obtain on the second paper a double series of curves as", y" of a different form, but having the same property of cutting the paper into little squares. i86.] THEOREMS. 267 186.] THEOHEM III. If V is any function of x' and /, and if x' and y' are conjugate functions of x and y, then dx 2 r di the integration being between the same limits. For -7 = -=-> -= | Y~f ~T- dx dx dx dy dx dx' dy dx'dy' dx dx dy' 2 dx dx' dx 2 dy' dx* ' and d -. . daf_dtf_ d 2 Fdy'* dx'dy dy dy dy' 2 dy f dx dy 2 4 dy' Adding the last two equations, and remembering- the conditions of conjugate functions (l), we find dx Hence ,u- r , u/ ~ r \ (d dy' dx' dy'\ = ('dx 72 d/*' \dx ~dy "" ~dy ~dx' r f? 2 Fx 77 rr f d*r d^ f dx'd y ' dx'd v \ + -) dxd y = ( + (- dfdy> =(d^ + d> y - If F is a potential, then, by Poisson's equation and we may write the result or the quantity of electricity in corresponding portions of two sys- tems is the same if the coordinates of one system are conjugate functions of those of the other. 268 CONJUGATE FUNCTIONS. [187. Additional Theorems on Conjugate Functions. 187.] THEOREM IV. If x l and y lt and also x. 2 and y^ are eon- jugate functions of x and y, then, if X=x l x 2 y l y^ and Y = x l y 2 + x 2 y l , X and Y will be conjugate functions of x and y. For X+ V^lT = fo + V^T^) (# 2 + V^T y 2 ). THEOREM V. If $ be a solution of the equation _ dx* d ~ 7 '^ n -n ^ fd$ d(h \ , , dx and if 2 It = log ( -~ + -~ ) > and = tan- 1 > \dx dy ' d dy R and will le conjugate functions of x and y. For H and are conjugate functions of ~- and , and these , d. fdf^are conjugate functions of x and y. EXAMPLE I. Inversion. 188.] As an example of the general method of transformation let us take the case of inversion in two dimensions. If is a fixed point in a plane, and OA a fixed direction, and if r = OP = aef>, and = AOP, and if #, y are the rectangular coordinates of P with respect to 0, tf^tan- 1 ^, ) / 5 x x = ae? cos 0, y ae? sin 0, ) p and are conjugate functions of x and y. If p'= np and 0'= n0, p' and 6' will be conjugate functions of p and 0. In the case in which n = 1 we have a 2 = -, and 0'=-0, (6) which is the case of ordinary inversion combined with turning the figure 1 80 round OA. &_ Inversion in Two Dimensions. In this case if ; and / represent the distances of corresponding points from 0, e and ef the total electrification of a body, S and S' superficial elements, V and V solid elements, o- and = 2 J Znog- r7r ; (8) 4r* and if the circle is a section of a hollow conducting cylinder, the surface- density at any point Q is - y Fi S> V- 27TO Invert the system with respect to a point O y making AO = mb, and a 2 = (m 2 then we have a charge at A equal to that at A, where AA'=. m The density at Q' i g AQ' 2 and the potential at any point P' within the circle is 4/ = = 2 ^ (log 5 -log AP), = 2E (log OP'- log AP* - log w). (9) This is equivalent to a combination of a charge E at ^', and a charge E at 0, which is the image of A, with respect to the circle. The imaginary charge at is equal and opposite to that at^'. If the point P' is defined by its polar coordinates referred to the centre of the circle, and if we put p = logr log 3, and p = log A A' logd, then AP"= be^ AA'= be"> t and the potential at the point (p, 6) is = E log (e- 2 <> 2 6?~po e? cos 6 -f e 2 ?) AO = (10) (11) This is the potential at the point (p, 6) due to a charge E, placed at the point (p 0; 0), with the condition that when p = 0, $ = 0. 270 CONJUGATE FUNCTIONS. [190. In this case p and 6 are the conjugate functions in equations (5) : p is the logarithm of the ratio of the radius vector of a point to the radius of the circle, and 6 is an angle. The centre is the only singular point in this system of coordinates, and the line-integral of / -=- ds round a closed curve is zero or 2 TT, according as the closed curve excludes or includes the centre. EXAMPLE III. Neumann's Transformation of this Case*. 190.] Now let a and ft be any conjugate functions of x and y^ such that the curves (a) are equipotential curves, and the curves (/3) are lines of force due to a system consisting of a charge of half a unit at the origin, and an electrified system disposed in any manner at a certain distance from the origin. Let us suppose that the curve for which the potential is a is a closed curve, such that no part of the electrified system except the half-unit at the origin lies within this curve. Then all the curves (a) between this curve and the origin will be closed curves surrounding the origin, and all the curves (/8) will meet in the origin, and will cut the curves (a) orthogonally. The coordinates of any point within the curve (a ) will be deter- mined by the values of a and ft at that point, and if the point travels round one of the curves (a) in the positive direction, the value of ft will increase by 2 ir for each complete circuit. If we now suppose the curve (a ) to be the section of the inner surface of a hollow cylinder of any form maintained at potential zero under the influence of a charge of linear density ~E on a line of which the origin is the projection, then we may leave the external electrified system out of consideration, and we have for the potential at any point (a) within the curve 4> = 2^(a-a ), (12) and for the quantity of electricity on any part of the curve a between the points corresponding to ft 1 and /3 2 , If in this way, or in any other, we have determined the dis- tribution of potential for the case of a given curve of section when the charge is placed at a given point taken as origin, we may pass to the case in which the charge is placed at any other point by an application of the general method of transformation. * See Crelle's Journal, 1861. NEUMANN'S TRANSFORMATION. 271 Let the values of a and ft for the point at which the charge is placed be 04 and 1$ then substituting in equation (ll) a a for p, and 00J for 6, we find for the potential at any point whose co- ordinates are a and 0, $ = Slog (I 2tf a+a i- 2a ocos(0 1 ) + e 2 (+ a i-2o)) -Elog(l-2e^co*(0-p 1 ) + e*(*-^)-2E(a 1 --a ). (14) This expression for the potential becomes zero when a = a , and is finite and continuous within the curve a except at the point (a l5 0j), at which point the second term becomes infinite, and in its immediate neighbourhood is ultimately equal to 2.#log/, where / is the distance from that point. We have therefore obtained the means of deducing the solution of Green's problem for a charge at any point within a closed curve when the solution for a charge at any other point is known. The charge induced upon an element of the curve a between the points and 0-M0 by a charge E placed at the point (a l5 X ) is, with the notation of Art. 183, n , where ds l is measured inwards and a is to be put equal to a after differentiation. This becomes, by (4) of Art. 183, _ -go _ " 2^ 1 - 2 the potential at the point (a l5 X ) within the closed curve, there being no electri- fication within the curve, 1 f" (l-e^-^)Fdft 9 ~ 27T./0 l-2e'i-''o>cos/3-/3 1 + e 2 (''i<>> 272 CONJUGATE FUNCTIONS. [191. EXAMPLE IV. Distribution of Electricity near an Edge of a Conductor formed by Two Plane Faces. 191.] In the case of an infinite plane face of a conductor charged with electricity to the surface-density <7 , we find for the potential at a distance y from the plane where C is the value of the potential of the conductor itself. Assume a straight line in the plane as a polar axis, and transform into polar coordinates, and we find for the potential 7= Cl-no-QaePsinO, and for the quantity of electricity on a parallelogram of breadth unity, and length ae? measured from the axis E = if # 2 = e~* cos fa y^ e~^ sin fa (2) # 2 and y 2 will be conjugate functions. Hence, if 2a> = as 1 + a: 2 = (e+ + e-+)co8fa 1y = ft+ft = (e+e~+) sinfa (3) x and y will also be conjugate functions of < and $. In this case the points for which is constant lie in the ellipse whose axes are e*-\- er* and e$e~$. VOL. I. T 274 CONJUGATE FUNCTIONS. [ I 93- The points for which ^ is constant lie in the hyperbola whose axes are 2 cos \jr and 2 sin ty. On the axis of a, between x=. I and #= + 1 , $ = 0, \j/ as cos- 1 ^-. (4) On the axis of #, beyond these limits on either side, we have # > 1, \jf = 0, = log (#+ \A? 2 1), (5) X< 1, \/r = 7T, = log ( >/# 2 1 5?). Hence, if is the potential function, and \j/ the function of flow, we have the case of electricity flowing from the positive to the negative side of the axis of x through the space between the points 1 and 4- 1 , the parts of the axis beyond these limits being impervious to electricity. Since, in this case, the axis of y is a line of flow, we may suppose it also impervious to electricity. We may also consider the ellipses to be sections of the equi- potential surfaces due to an indefinitely long flat conductor of breadth 2, charged with half a unit of electricity per unit of length. If we make \j/ the potential function, and < the function of flow, the case becomes that of an infinite plane from which a strip of breadth 2 has been cut away and the plane on one side charged to potential IT while the other remains at zero. These cases may be considered as particular cases of the quadric surfaces treated of in Chapter X. The forms of the curves are given in Fig. X. EXAMPLE VI. Fig. XI. 193.] Let us next consider of and y as functions of x and ^, where of I log Va? +/, / = I tan- l *-, (6) x x and y will be also conjugate functions of < and \jf. The curves resulting from the transformation of Fig. X with respect to these new coordinates are given in Fig. XI. If x' and y' are rectangular coordinates, then the properties of the axis of x in the first figure will belong to a series of lines parallel to x in the second figure for which y ' = bri 'TT, where n' is any integer. The positive values of x on these lines will correspond to values of x greater than unity, for which, as we have already seen, *' ,~^ v 6 +v e b _ iJ. 7 I93-] PARTICULAR CASE OF CONJUGATE FUNCTIONS. 275 The negative values of a?' on the same lines will correspond to values of x less than unity, for which, as we have seen, a/ < = 0, \l/ = cos" 1 ^ = cos~ l e b . (8) The properties of the axis of y in the first figure will belong to a series of lines in the second figure parallel to #', for which /= a* ('+!). (9) The value of \j/ along these lines is \j/ = * (n + ) for all points both positive and negative, and / ^ tw_ \ * = log(y+ Vy+1) = \og\e b + V e b + i/, ( 10 ) [The curves for which < and \j/ are constant may be traced directly from the equations As the figure repeats itself for intervals of iib in the values o it will be sufficient to trace the lines for one such interval. Now there will be two cases, according as < or ^ changes sign with y'. Let us suppose that so changes sign. Then any curve for which \fr is constant will be symmetrical about the axis of #', cutting that axis orthogonally at some point on its negative side. If we begin with this point for which < = 0, and gradually in- crease <, the curve will bend round from being at first orthogonal to being, for large values of $, at length parallel to the axis of of. The positive side of the axis of ai is one of the system, viz. ty is there zero, and when/= + ITT^, ^ = JTT. The lines for which \l/ has constant values ranging from to JTT form therefore a system of curves embracing the positive side of the axes of x'. The curves for which has constant values cut the system \^ orthogonally, the values of ranging from +00 to -co. For any one of the curves drawn above the axis of x the value of $ is positive, along the negative side of the axis of x the value is zero, and for any curve below the axis of af the value is negative. We have seen that the system \js is symmetrical about the axis of #; let PQR be any curve cutting that system orthogonally and terminating in P and R in the lines /= + \-nb, the point Q being in the axis of %'. Then the curve PQR is symmetrical about the axis of #', but if c be the value of along PQ, the value of < along QR will be c. This discontinuity in the value of $ will be accounted T a 276 CONJUGATE FUNCTIONS. [194. for by an electrical distribution in the case which will be discussed in Art. 195. If we next suppose that \j/ and not $ changes sign with y', the values of

= we have the negative side of the axis of af, and when < = oo we have a line at an infinite distance perpendicular to the axis of af '. Along any line PQR between these two the value of $ is constant throughout its entire length and positive. Any value \jf now experiences an abrupt change at the point where the curve along which it is constant crosses the negative side of the axis of #', the sign of ^ changing there. The sig- nificance of this discontinuity will appear in Art. 197. The lines we have shewn how to trace are drawn in Fig. XI if we limit ourselves to two-thirds of that diagram, cutting off the uppermost third.] 194.] If we consider as the potential function, and \js as the function of flow, we may consider the case to be that of an in- definitely long strip of metal of breadth 116 with a non-conducting division extending from the origin indefinitely in the positive direction, and thus dividing the positive part of the strip into two separate channels. We may suppose this division to be a narrow slit in the sheet of metal. If a current of electricity is made to flow along one of these divisions and back again along the other, the entrance and exit of the current being at an indefinite distance on the positive side of the origin, the distribution of potential and of current will be given by the functions and x//- respectively. If, on the other hand, we make \js the potential, and $ the function of flow, then the case will be that of a current in the general direction of y', flowing through a sheet in which a number of non-conducting divisions are placed parallel to #', extending from the axis of y r to an indefinite distance in the negative direction. 195.] We may also apply the results to two important cases in statical electricity. (1) Let a conductor in the form of a plane sheet, bounded by a straight edge but otherwise unlimited, be placed in the plane of xz on the positive side of the origin, and let two infinite conducting planes be placed parallel to it and at distances %irb on either side. Then, if \]/ is the potential function, its value is for the middle conductor and J TT for the two planes. Let us consider the quantity of electricity on a part of the middle EDGE OF AN ELECTRIFIED PLATE. 277 conductor, extending to a distance 1 in the direction of *, and from the origin to #'= a. The electricity on the part of this strip extending from x{ to # 2 ' is (< 2 <^j). Hence from the origin to x'= a the amount is 477 If a is large compared with I, this becomes 1 - *" = 477 477^ , v Hence the quantity of electricity on the plane bounded by the straight edge is greater than it would have been if the electricity had been uniformly distributed over it with the same density that it has at a distance from the boundary, and it is equal to the quantity of electricity having the same uniform surface-density, but extending to a breadth equal to I \og e 2 beyond the actual boundary of the plate. This imaginary uniform distribution is indicated by the dotted straight lines in Fig. XI. The vertical lines represent lines of force, and the horizontal lines equipotential surfaces, on the hypo- thesis that the density is uniform over both planes, produced to infinity in all directions. 196.] Electrical condensers are sometimes formed of a plate placed midway between two parallel plates extending considerably beyond the intermediate one on all sides. If the radius of curvature of the boundary of the intermediate plate is great compared with the distance between the plates, we may treat the boundary as approximately a straight line, and calculate the capacity of the condenser by supposing the intermediate plate to have its area extended by a strip of uniform breadth round its boundary, and assuming the surface-density on the extended plate the same as it is in the parts not near the boundary. Thus, if S be the actual area of the plate, L its circumference and B the distance between the large plates, we have a = i-B, (13) 77 278 CONJUGATE FUNCTIONS. [196. and the breadth of the additional strip is so that the extended area is (15) The capacity of the middle plate is Correction for the Thickness of the Plate. Since the middle plate is generally of a thickness which cannot be neglected in comparison with the distance between the plates, we may obtain a better representation of the facts of the case by supposing the section of the intermediate plate to correspond with the curve ^ = \//. The plate will be of nearly uniform thickness, /3 = 2b\}f', at a distance from the boundary, but will be rounded near the edge. The position of the actual edge of the plate is found by putting /= 0, whence #' = j \ oge cos ^ ^ 7 j The value of $ at this edge is 0, and at a point for which #'= a it is a + t>log e 2 ~~b~ Hence, altogether, the quantity of electricity on the plate is the same as if a strip of breadth -# /i W/3\ (log. 2 + log, cos g) , i.e. ^log e (2cos||), (18) had been added to the plate, the density being assumed to be every- where the same as it is at a distance from the boundary. Density near the Edge. The surface-density at any point of the plate is / f_ \/e b : 47rB 1 9 7.] DENSITY NEAR THE EDGE. 279 The quantity within brackets rapidly approaches unity as x' increases, so that at a distance from the boundary equal to n times the breadth of the strip a, the actual density is greater than the normal density by about 2M+1 of the normal density. In like manner we may calculate the density on the infinite planes When %'= 0, the density is 2~* of the normal density. At n times the breadth of the strip on the positive side, the density is less than the normal density by about - n+1 At n times the breadth of the strip on the negative side, the density is about of the normal density. 2 These results indicate the degree of accuracy to be expected in applying this method to plates of limited extent, or in which irregularities may exist not very far from the boundary. The same distribution would exist in the case of an infinite series of similar plates at equal distances, the potentials of these plates being alternately -f Fand V. In this case we must take the distance between the plates equal to B. 197.] (2) The second case we shall consider is that of an infinite series of planes parallel to xz at distances B = vb, and all cut off by the plane of yz, so that they extend only on the negative side of this plane. If we make the potential function, we may regard these planes as conductors at potential zero. Let us consider the curves for which $ is constant. When y ' mrb> that is, in the prolongation of each of the planes, we have %' = b log \ (e+ + *-*) (21) when y-= (n+^)bir, that is, in the intermediate positions af= a log i (*--*). (22) Hence, when < is large, the curve for which (/> is constant is an undulating line whose mean distance from the axis of y is approximately a = b ( log e 2), (23) and the amplitude of the undulations on either side of this line is 280 CONJUGATE FUNCTIONS. When $ is large this becomes be* 2 *, so that the curve approaches to the form of a straight line parallel to the axis of y at a distance a from that axis on the positive side. If we suppose a plane for which #'= #, kept at a constant potential while the system of parallel planes is kept at a different potential, then, since b

, and may be considered as approximately plane. If D is the depth of these undulations from the crest to the trough of each wave, then we find for the corresponding value of $, JD 0=ilog^. (26) e b -l The value of of at the crest of the wave is & log *(** + l+e B and for a slit of infinite depth, putting D oo, the correction is .* (31) To find the surface-density on the series of parallel plates we must find = 0. We find 47T dx l = . (32) i *_ \/e -l The average density on the plane plate at distance A from the edges of the series of plates is o 1 = = Hence, at a distance from 4776 the edge of one of the plates equal to na the surface-density is of this average density. 200.] Let us next attempt to deduce from these results the distribution of electricity in the figure formed by rotating the plane of the figure about the axis /= R. In this case, Poisson's equation will assume the form dv , . ~ Let us assume F=<, the function given in Art. 193, and de- 282 CONJUGATE FUNCTIONS. [2OO. termine the value of p from this equation. We know that the first two terms disappear, and therefore J_ _l _ d(f) '' If we suppose that, in addition to the surface-density already investigated, there is a distribution of electricity in space according to the law just stated, the distribution of potential will be repre- sented by the curves in Fig. XI. Now from this figure it is manifest that -p is generally very / small except near the boundaries of the plates, so that the new distribution may be approximately represented by what actually exists, namely a certain superficial distribution near the edges of the plates. If therefore we integrate ijpdafd/ between the limits /= and /=-, and from #'= oo to x = +oc, we shall find the whole additional charge on one side of the plates due to the curvature. . d(f> d\lf Since -:, = -- f, > we have dy dx (35) Integrating with respect to y f , we find 1 12R + B. 2R + B -p- ^g - r-jjr- (36) This is half the total quantity of electricity which we must suppose distributed in space near the edge of one of the cylindric plates per unit of circumference. Since it is only close to the edge of the plate that the density is sensible, we may suppose it all condensed on the surface of the plate without altering sensibly its action on the opposed plane surface, and in calculating the attraction between that surface and the cylindric surface we may suppose this electricity to belong to the cylindric surface. 200.] CIRCULAR GROOVES. 283 If there had been no curvature the superficial charge on the positive surface of the plate per unit of length would have been Hence, if we add to it the whole of the above distribution, this TO charge must be multiplied by the factor (l + 4 -^-) to get the total charge on the positive side. * In the case of a disk of radius R placed midway between two infinite parallel plates at a distance -5, we find for the capacity of the disk R 2 log 1 2 ,. -p-H-2 * e R+\B. (38) Jj 7T * [In Art. 200, in estimating the total space distribution we might perhaps more correctly take for it the integral ffp 2 IT (R + y') dx'dy', which gives, per unit circum- 1 7? ference of the edge of radius R, -= , thus leading to the same correction as in the text. The case of the disk may be treated in like manner as follows : Let the figure of Art. 195 revolve round a line perpendicular to the plates and at a distance + R from the edge of the middle one. That edge will therefore envelope a circle, which will be the edge of the disk. As in Art. 200, we begin with Poisson'a equation, which in this case will be d-V d*V 1 dV We now assume that F = ^, the potential function of Art. 195. We must therefore suppose electricity to exist in the region between the plates whose volume density p is 47r Rx dx The total amount is p.2*(R-x')dx'dy'. 4/ v %/ Now if R is large in comparison with the distance between the plates this result will be seen, on an examination of the potential lines in Fig. XI, to be sensibly the same as B /*2 fao dti I I - dx'dy'; that is, ^vB. JO J-co^ The total surface distribution if we include both sides of the disk is If, therefore, the volume distribution between the plates be supposed to be concen- trated on the disk the expression for the capacity, the difference of the potentials of the plates and disk being f, becomes T) a result differing from that in the text by j nearly,] 284: CONJUGATE FUNCTIONS. [20 1. Theory of Thomson's Guard-ring. 201.] In some of Sir W. Thomson's electrometers, a large plane surface is kept at one potential, and at a distance a from this surface is placed a plane disk of radius R surrounded by a large plane plate called a Guard-ring with a circular aperture of radius Hf concentric with the disk. This disk and plate are kept at potential zero. The interval between the disk and the guard-plate may be regarded as a circular groove of infinite depth, and of breadth R' R) which we denote by B. The charge on the disk due to unit potential of the large disk, R 2 supposing the density uniform, would be - 4 a. The charge on one side of a straight groove of breadth B and length L 2irR, and of infinite depth, may be estimated by the number of lines of force emanating from the large disk and falling upon the side of the groove. Referring to Art. 197 and footnote we see that the charge will therefore be . RB i.e. Jj 7 t A + a since in this case $=1,0 = 0, and therefore b = A + a'. But since the groove is not straight, but has a radius of curvature TD 7?, this must be multiplied by the factor (l + J -=-) The whole charge on the disk is therefore R 2 RB , B^ _R 2 + R' 2 R f2 -R 2 a' , , ~U~ ~8A A + a'' The value of a' cannot be greater than ^ , = 0.22.Z? nearly. 7T If ^5 is small compared with either A or R this expression will give a sufficiently good approximation to the charge on the disk due to unity of difference of potential. The ratio of A to R may have any value, but the radii of the large disk and of the guard-ring must exceed R by several multiples of A. 202 -] A CASE OF TWO PLANES. 285 EXAMPLE VII. Fie-. XII 202.] Helmholtz, in his memoir on discontinuous fluid motion *, has pointed out the application of several formulae in which the coordinates are expressed as functions of the potential and its conjugate function. One of these may be applied to the case of an electrified plate of finite size placed parallel to an infinite plane surface connected with the earth. Since x^A<^ and y l =A\\r, and also x 2 =Ae* cos ^ and y% = A e* sin \//-, are conjugate functions of $ and x/r, the functions formed by adding X L to x 2 and y x to y 2 will be also conjugate. Hence, if x = A^ + Ae^cos\l/, y = A \fr + A efi sin \j/. then x and y will be conjugate with respect to $ and ^-, and < and \f/ will be conjugate with respect to x and y. Now let x and y be rectangular coordinates, and let kty be the potential, then kfy will be conjugate to /fcx/f, k being any constant. Let us put \l/ = TT, then y = ATI, x = A (0 ^). If varies from oo to 0, and then from to +00, x varies from -co to A and from A to oo. Hence the equipotential surface, for which \j/ = TT, is a plane parallel to a? at a distance b = i7 A from the origin, and extending from oo to x = A. Let us consider a portion of this plane, extending from x = (A + a) to x = A and from z = to z = c, let us suppose its distance from the plane of xz to be y = b = A ir, and its potential to be F= k^r = kit. The charge of electricity on the portion of the plane considered is found by ascertaining the values of at its extremities. We have therefore to determine $ from the equation < will have a negative value <^> 1 and a positive value $ 2 ; at the edge of the plane, where x = A, = 0. Hence the charge on the one side is ckfa-*- 4?:, and that on the other side is c/<-H 477. Konigl. Akad. der WissenscJiaften, zu Berlin, April 23, 1868. 286 CONJUGATE FUNCTIONS. [203. Both these charges are positive and their sum is 47T If we suppose that a is large compared with A, -- -T-1 + &C. A ^ 2 = log {z + 1 +log Oj + x + &c O j- If we neglect the exponential terms in fa we shall find that the charge on the negative surface exceeds that which it would have if the superficial density had been uniform and equal to that at a distance from the boundary, by a quantity equal to the charge on a strip of breadth A = - with the uniform superficial density. The total capacity of the part of the plane considered is The total charge is CV, and the attraction towards the infinite plane, whose equation is y = and potential x/r = 0, is dC ac, 2 ^ db i + 10 P The equipotential lines and lines of force are given in Fig. XII. EXAMPLE VIII. Theory of a Grating of Parallel Wires. Fig. XIII. 203.] In many electrical instruments a wire grating is used to prevent certain parts of the apparatus from being electrified by induction. We know that if a conductor be entirely surrounded by a metallic vessel at the same potential with itself, no electricity can be induced on the surface of the conductor by any electrified body outside the vessel. The conductor, however, when completely surrounded by metal, cannot be seen, and therefore, in certain cases, an aperture is left which is covered with a grating of fine wire. Let us investigate the effect of this grating in diminishing the effect of electrical induction. We shall suppose the grating to consist of a series of parallel wires in one plane and at equal intervals, the diameter of the wires being small compared with the 204.] INDUCTION THHOUGH A GRATING. 287 distance between them, while the nearest portions of the electrified bodies on the one side and of the protected conductor on the other are at distances from the plane of the screen, which are considerable compared with the distance between consecutive wires. 204.] The potential at a distance / from the axis of a straight wire of infinite length charged with a quantity of electricity A per unit of length is F = - 2 A log / + (7. (l) We may express this in terms of polar coordinates referred to an axis whose distance from the wire is unity, in which case we must make / 2 = 1 - 2 r cos + r 2 , (2) and if we suppose that the axis of reference is also charged with the linear density A', we find V Alog(l 2/COS0 + / 2 ) 2A'logr + C. (3) If we now make H r = "!, = ^, (4) then, by the theory of conjugate functions, *)_ F= Alog (l 2e * cos- - + e a ) 2A'log* + 0, (5) where x and y are rectangular coordinates, will be the value of the potential due to an infinite series of fine wires parallel to z in the plane of xz, and passing through points in the axis of x for which so is a multiple of a. Each of these wires is charged with a linear density A. The term involving A' indicates an electrification, producing a constant force in the direction ofy. a The forms of the equipotential surfaces and lines of force when A'= are given in Fig. XIII. The equipotential surfaces near the wires are nearly cylinders, so that we may consider the solution approximately true, even when the wires are cylinders of a diameter which is finite but small compared with the distance between them. The equipotential surfaces at a distance from the wires become more and more nearly planes parallel to that of the grating. If in the equation we make y = 6 lt a quantity large compared with , we find approximately, ^ = - ~p (A + A') + C nearly. (6) If we next make y b 2 , where # 2 is a positive quantity large compared with a, we find approximately, 288 CONJUGATE FUNCTIONS. [205. / + C nearly. (7) If c is the radius of the wires of the grating, c being small compared with a, we may find the potential of the grating itself by supposing that the surface of the wire coincides with the equi- potential surface which cuts the plane of xz at a distance c from the axis of z. To find the potential of the grating we therefore put x = c, and y 0, whence V= 2 A log 2 sin + C. (8) 205.] We have now obtained expressions representing the elec- trical state of a system consisting of a grating of wires whose diameter is small compared with the distance between them, and two plane conducting surfaces, one on each side of the grating, and at distances which are great compared with the distance between the wires. The surface-density o^ on the first plane is got from the equa- tion(6) That on the second plane cr 2 from the equation (?) 4^=^ = 1^'. (10 ) db 2 a If we now write a . vc = - 2 - lo Se(2sm-), and eliminate X and A/ from the equations (6), (7), (8), (9), (10), we find ^4^ +^).F 1 (l + 4)-F' i -rJ, (12) v^ + --r 1 +r 1 (i + -ri. (is) When the wires are infinitely thin, a becomes infinite, and the terms in which it is the denominator disappear, so that the case is reduced to that of two parallel planes without a grating in- terposed. If trie grating is in metallic communication with one of the planes, say the first, V F I} and the right-hand side of the equation for oi becomes F l F 2 . Hence the density o-j induced on the first plane when the grating is interposed is to that which would have been induced on it if the grating were removed, the second plane being maintained at the same potential, as 1 to 1 H 206.] METHOD OF APPROXIMATION. 289 We should have found the same value for the effect of the grating in diminishing the electrical influence of the first surface on the second, if we had supposed the grating connected with the second surface. This is evident since d l and b 2 enter into the expression in the same way. It is also a direct result of the theorem of Art. 88. The induction of the one electrified plane on the other through the grating is the same as if the grating were removed, and the distance between the planes increased from b l + 2 to If the two planes are kept at potential zero, and the grating electrified to a given potential, the quantity of electricity on the grating will be to that which would be induced on a plane of equal area placed in the same position as Ma : M 2 + a (i + ^)- This investigation is approximate only when b l and b 2 are large compared with a, and when a is large compared with c. The quantity a is a line which may be of any magnitude. It becomes infinite when c is indefinitely diminished. If we suppose c = \ a there will be no apertures between the wires of the grating, and therefore there will be no induction through it. We ought therefore to have for this case a = 0. The formula (ll), however, gives in this case a = log 2, =-0-110, 2 7T which is evidently erroneous, as the induction can never be altered in sign by means of the grating. It is easy, however, to proceed to a higher degree of approximation in the case of a grating of cylindrical wires. I shall merely indicate the steps of this process. Method of Approximation. 206.] Since the wires are cylindrical, and since the distribution of electricity on each is symmetrical with respect to the diameter parallel to ^, the proper expansion of the potential is of the form 7= C' logr-f2<7 ~ ~ U - If we write p 2 = ~ and we find +pq) = Hence 4 pgY) + r. ((1 It appears from these equations that the quantity pU+qV con- tinually diminishes, so that whatever be the initial state of elec- trification the receivers are ultimately oppositely electrified, so that the potentials of A and B are in the ratio of p to q. On the other hand, the quantity pUq~P continually increases, so that, however little joC/may exceed or fall short of ^Fat first, the difference will be increased in a geometrical ratio in each 211.] THE RECIPROCAL ELECTROPHORUS. 297 revolution till the electromotive forces become so great that the insulation of the apparatus is overcome. Instruments of this kind may be used for various purposes. For producing a copious supply of electricity at a high potential, as is done by means of Mr. Varley's large machine. For adjusting the charge of a condenser, as in the case of {Thomson's electrometer, the charge of which can be increased or diminished by a few turns of a very small machine of this kind, which is then called a Replenishes For multiplying small differences of potential. The inductors may be charged at first to an exceedingly small potential, as, for instance, that due to a thermo-electric pair, then, by turning the machine, the difference of potentials may be continually multiplied till it becomes capable of measurement by an ordinary electrometer. By determining by experiment the ratio of increase of this difference due to each turn of the machine, the original electromotive force with which the inductors were charged may be deduced from the number of turns and the final electrification. In most of these instruments the carriers are made to revolve about an axis and to come into the proper positions with respect to the inductors by turning an axle. The connexions are made by means of springs so placed that the carriers come in contact with them at the proper instants. 211.] Sir W. Thomson *, however, has constructed a machine for multiplying electrical charges in which the carriers are drops of water falling out of the inside of an inductor into an insulated receiver. The receiver is thus continually supplied with electricity of opposite sign to that of the inductor. If the inductor is electrified positively, the receiver will receive a continually increasing charge of negative electricity. The water is made to escape from the receiver by means of a funnel, the nozzle of which is almost surrounded by the metal of the receiver. The drops falling from this nozzle are therefore nearly free from electrification. Another inductor and receiver of the same construction are arranged so that the inductor of the one system is in connexion with the receiver of the other. The rate of increase of charge of the receivers is thus no longer constant, but increases in a geometrical progression with the time, the charges of the two receivers being of opposite signs. This increase goes on till the falling drops are so diverted from their course by * Proc. E. S., June 20, 1867. 298 ELECTEOSTATIC INSTRUMENTS. [212. the electrical action that they fall outside of the receiver or even strike the inductor. In this instrument the energy of the electrification is drawn from that of the falling drops. 212.] Several other electrical machines have been constructed in which the principle of electric induction is employed. Of these the most remarkable is that of Holtz, in which the carrier is a glass plate varnished with gum-lac and the inductors are pieces of pasteboard. Sparks are prevented from passing between the parts of the apparatus by means of two glass plates, one on each side of the revolving carrier plate. This machine is found to be very effective, and not to be much affected by the state of the atmo- sphere. The principle is the same as in the revolving doubler and the instruments developed out of the same idea, but as the carrier is an insulating plate and the inductors are imperfect conductors, the complete explanation of the action is more difficult than in the case where the carriers are good conductors of known form and are charged and discharged at definite points. 213.] In the electrical machines already described sparks occur whenever the carrier comes in contact with a conductor at a different potential from its own. Now we have shewn that whenever this occurs there is a loss of energy, and therefore the whole work employed in turning the machine is not con- verted into electrification in an available form, but part is spent in producing the heat and noise of electric sparks. I have therefore thought it desirable to shew how an electrical machine may be constructed which is not subject to this loss of efficiency. I do not propose it as a useful form of machine, but as an example of the method by which the contrivance called in heat-engines a regenerator may be applied to an electrical machine to prevent loss of work. In the figure let A, B, e are connected with the earth. Let us suppose that when the carrier P is in the middle of A the coefficient of induction between P and A is A. The capacity of P in this position is greater than J, since it is not completely surrounded by the receiver A. Let it be A +a. Then if the potential of P is U, and that of A, V, the charge on P will be (A + a) UA7. Now let P be in contact with the spring a when in the middle of the receiver A, then the potential of P is F, the same as that of A, and its charge is therefore aV. If P now leaves the spring a it carries with it the charge aV. As P leaves A its potential diminishes, and it diminishes still more when it comes within the influence of C'> which is negatively electrified. If when P comes within C' its coefficient of induction on C' is C', and its capacity is (? + c', then, if U is the potential of P the charge on P is If C'V'=aV, then at this point U the potential of P will be reduced to zero. Let P at this point come in contact with the spring / which is connected with the earth. Since the potential of P is equal to that of the spring there will be no spark at contact. This conductor C", by which the carrier is enabled to be connected to earth without a spark, answers to the contrivance called a regenerator in heat-engines. We shall therefore call it a .Re- generator. Now let P move on, still in contact with the earth-spring /, till it comes into the middle of the inductor , the potential of which is V. If B is the coefficient of induction between P and B at this point, then, since U= the charge on P will be BV. When P moves away from the earth-spring it carries this charge with it. As it moves out of the positive inductor B towards the 300 ELECTROSTATIC INSTRUMENTS. [214. negative receiver A f its potential will be increasingly negative. At the middle of A', if it retained its charge, its potential would be A' + a' and if JB7 is greater than of 7' its numerical value will be greater than that of V . Hence there is some point before P reaches the middle of A' where its potential is V. At this point let it come in contact with the negative receiver-spring a' '. There will be no spark since the two bodies are at the same potential. Let P move on to the middle of A', still in contact with the spring, and therefore at the same potential with A'. During this motion it communicates a negative charge to A '. At the middle of A' it leaves the spring and carries away a charge a'V towards the positive regenerator C, where its potential is reduced to zero and it touches the earth- spring e. It then slides along the earth-spring into the negative inductor _Z?', during which motion it acquires a positive charge BV which it finally communicates to the positive receiver A, and the cycle of operations is repeated. During this cycle the positive receiver has lost a charge #Fand gained a charge B'7'. Hence the total gain of positive electricity is BV'-aV. Similarly the total gain of negative electricity is BY a'V'. By making the inductors so as to be as close to the surface of the carrier as is consistent with insulation, B and B may be made large, and by making the receivers so as nearly to surround the carrier when it is within them, a and a' may be made very small, and then the charges of both the Leyden jars will be increased in every revolution. The conditions to be fulfilled by the regenerators are C'V'=aV, and C7=a'7'. Since a and a are small the regenerators do not require to be either large or very close to the carriers. On Electrometers and Electroscopes. 214.] An electrometer is an instrument by means of which electrical charges or electrical potentials may be measured. In- struments by means of which the existence of electric charges or of differences of potential may be indicated, but which are not capable of affording numerical measures, are called Electroscopes. An electroscope if sufficiently sensitive may be used in electrical measurements, provided we can make the measurement depend on 2 1 5.] COULOMB'S TORSION BALANCE. 301 the absence of electrification. For instance, if we have two charged bodies A and B we may use the method described in Chapter I to determine which body has the greater charge. Let the body A be carried by an insulating support into the interior of an insulated closed vessel C. Let C be connected to earth and again insulated. There will then be no external electrification on C. Now let A be removed, and B introduced into the interior of C, and the elec- trification of C tested by an electroscope. If the charge of B is equal to that of A there will be no electrification, but if it is greater or less there will be electrification of the same kind as that of J?, or the opposite kind. Methods of this kind, in which the thing to be observed is the non-existence of some phenomenon, are called null or zero methods. They require only an instrument capable of detecting the existence of the phenomenon. In another class of instruments for the registration of phe- nomena the instruments may be depended upon to give always the same indication for the same value of the quantity to be registered, but the readings of the scale of the instrument are not proportional to the values of the quantity, and the relation between these readings and the corresponding value is unknown, except that the one is some continuous function of the other. Several electrometers depending on the mutual repulsion of parts of the instrument which are similarly electrified are of this class. The use of such instruments is to register phenomena, not to measure them. Instead of the true values of the quantity to be measured, a series of numbers is obtained, which may be used afterwards to determine these values when the scale of the instrument has been properly investigated and tabulated. In a still higher class of instruments the scale readings are proportional to the quantity to be measured, so that all that is required for the complete measurement of the quantity is a know- ledge of the coefficient by which the scale readings must be multiplied to obtain the true value of the quantity. Instruments so constructed that they contain within themselves the means of independently determining the true values of quan- tities are called Absolute Instruments. Coulomb's Torsion Balance. 215.] A great number of the experiments by which Coulomb 302 ELECTROSTATIC INSTRUMENTS. [215. established the fundamental laws of electricity were made by mea- suring the force between two small spheres charged with electricity, one of which was fixed while the other was held in equilibrium by two forces, the electrical action between the spheres, and the torsional elasticity of a glass fibre or metal wire. See Art. 38. The balance of torsion consists of a horizontal arm of gum-lac, suspended by a fine wire or glass fibre,, and carrying at one end a little sphere of elder pith, smoothly gilt. The suspension wire is fastened above to the vertical axis of an arm which can be moved round a horizontal graduated circle, so as to twist the upper end of the wire about its own axis any number of degrees. The whole of this apparatus is enclosed in a case. Another little sphere is so mounted on an insulating stem that it can be charged and introduced into the case through a hole, and brought so that its centre coincides with a definite point in the horizontal circle described by the suspended sphere. The position of the suspended sphere is ascertained by means of a graduated circle engraved on the cylindrical glass case of the instrument. Now suppose both spheres charged, and the suspended sphere in equilibrium in a known position such that the torsion-arm makes an angle with the radius through the centre of the fixed sphere. The distance of the centres is then 2 a sin \ 0, where a is the radius of the torsion-arm, and if F is the force between the spheres the moment of this force about the axis of torsion is Fa cos J 0. Let both spheres be completely discharged, and let the torsion- arm now be in equilibrium at an angle < with the radius through the fixed sphere. Then the angle through which the electrical force twisted the torsion-arm must have been (f>, and if M is the moment of the torsional elasticity of the fibre, we shall have the equation Hence, if we can ascertain JJf, we can determine F, the actual force between the spheres at the distance 2 a sin \0. To find M t the moment of torsion, let /be the moment of inertia of the torsion-arm, and T the time of a double vibration of the arm under the action of the torsional elasticity, then In all electrometers it is of the greatest importance to know what force we are measuring. The force acting on the suspended 2 I 5.] INFLUENCE OF THE CASE. 303 sphere is due partly to the direct action of the fixed sphere, but partly also to the electrification, if any, of the sides of the case. If the case is made of glass it is impossible to determine the electrification of its surface otherwise than by very difficult mea- surements at every point. If, however, either the case is made of metal, or if a metallic case which almost completely encloses the apparatus is placed as a screen between the spheres and the glass case, the electrification of the inside of the metal screen will depend entirely on that of the spheres, and the electrification of the glass case will have no influence on the spheres. In this way we may avoid any indefiniteness due to the action of the case. To illustrate this by an example in which we can calculate all the effects, let us suppose that the case is a sphere of radius , that the centre of motion of the torsion-arm coincides with the centre of the sphere and that its radius is a ; that the charges on the two spheres are E 1 and E, and that the angle between their positions is 6 ; that the fixed sphere is at a distance 1 from the centre, and that r is the distance between the two small spheres. Neglecting for the present the effect of induction on the dis- tribution of electricity on the small spheres, the force between them will be a repulsion _EE l -T' and the moment of this force round a vertical axis through the centre will be The image of E t due to the spherical surface of the case it a point in the same radius at a distance with a charge E l , and the #! i moment of the attraction between E and this image about the axis of suspension is b* . a sm aa, sin ( aa^ A aa\ )i }-?+ VI If 7), the radius of the spherical case, is large compared with a 304 ELECTKOSTATIC INSTRUMENTS. [216. and a l , the distances of the spheres from the centre, we may neglect the second and third terms of the factor in the denominator. The whole moment tending to turn the torsion- arm may then be written Electrometers for the Measurement of Potentials. 216.] In all electrometers the moveable part is a body charged with electricity, and its potential is different from that of certain of the fixed parts round it. When, as in Coulomb's method, an insulated body having a certain charge is used, it is the charge which is the direct object of measurement. We may, however, connect the balls of Coulomb's electrometer, by means of fine wires, with different conductors. The charges of the balls will then depend on the values of the potentials of these conductors and on the potential of the case of the instrument. The charge on each ball will be approximately equal to its radius multiplied by the excess of its potential over that of the case of the instrument, provided the radii of the balls are small compared with their distances from each other and from the sides or opening of the case. Coulomb's form of apparatus, however, is not well adapted for measurements of this kind, owing to the smallness of the force between spheres at the proper distances when the difference of po- tentials is small. A more convenient form is that of the Attracted Disk Electrometer. The first electrometers on this principle were constructed by Sir W. Snow Harris*. They have since been brought to great perfection, both in theory and construction, by Sir W. Thomson f. When two disks at different potentials are brought face to face with a small interval between them there will be a nearly uniform electrification on the opposite faces and very little electrification on the backs of the disks, provided there are no other conductors or electrified bodies in the neighbourhood. The charge on the positive disk will be approximately proportional to its area, and to the difference of potentials of the disks, and inversely as the distance between them. Hence, by making the areas of the disks large * Phil. Trans. 1834. f See an excellent report on Electrometers by Sir W. Thomson. Report of the British Association, Dundee, 1867. 217.] PRINCIPLE OF THE GUARD-RING. 305 and the distance between them small, a small difference of potential may give rise to a measurable force of attraction. The mathematical theory of the distribution of electricity over two disks thus arranged is given at Art. 202, but since it is im- possible to make the case of the apparatus so large that we may suppose the disks insulated in an infinite space, the indications of the instrument in this form are not easily interpreted numerically. 217.] The addition of the guard-ring to the attracted disk is one of the chief improvements which Sir W. Thomson has made on the apparatus. Instead of suspending the whole of one of the disks and determ- ining the force acting upon it, a central portion of the disk is separated from the rest to form the attracted disk, and the outer ring forming the remainder of the disk is fixed. In this way the force is measured only on that part of the disk where it is most regular, and the want of uniformity of the electrification near the COUMTERPOIse Fig. 19. edge is of no importance, as it occurs on the guard-ring and not on the suspended part of the disk. Besides this, by connecting the guard-ring with a metal case surrounding the back of the attracted disk and all its suspending apparatus, the electrification of the back of the disk is rendered VOL. i. x 306 ELECTROSTATIC INSTRUMENTS, [217. impossible, for it is part of the inner surface of a closed hollow conductor all at the same potential. Thomson's Absolute Electrometer therefore consists essentially of two parallel plates at different potentials, one of which is made so that a certain area, no part of which is near the edge of the plate, is moveable under the action of electric force. To fix our ideas we may suppose the attracted disk and guard-ring uppermost. The fixed disk is horizontal, and is mounted on an insulating stem which has a measurable vertical motion given to it by means of a micrometer screw. The guard-ring is at least as large as the fixed disk ; its lower surface is truly plane and parallel to the fixed disk. A delicate balance is erected on the guard-ring to which is suspended a light moveable disk which almost fills the circular aperture in the guard-ring without rubbing against its sides. The lower surface of the suspended disk must be truly plane, and we must have the means of knowing when its plane coincides with that of the lower surface of the guard-ring, so as to form a single plane interrupted only by the narrow interval between the disk and its guard-ring. For this purpose the lower disk is screwed up till it is in contact with the guard-ring, and the suspended disk is allowed to rest upon the lower disk, so that its lower surface is in the same plane as that of the guard-ring. Its position with respect to the guard- ring is then ascertained by means of a system of fiducial marks. Sir W. Thomson generally uses for this purpose a black hair attached to the moveable part. This hair moves up or down just in front of two black dots on a white enamelled ground and is viewed along- with these dots by means of a piano convex lens with the plane side next the eye. If the hair as seen through the lens appears straight and bisects the interval between the black dots it is said to be in its sighted position, and indicates that the sus- pended disk with which it moves is in its proper position as regards height. The horizontality of the suspended disk may be tested by comparing the reflexion of part of any object from its upper surface with that of the remainder of the same object from the upper surface of the guard-ring. The balance is then arranged so that when a known weight is placed on the centre of the suspended disk it is in equilibrium in its sighted position, the whole apparatus being freed from electrification by putting every part in metallic communication. A metal case is placed over the guard-ring so as to enclose the 2 1 7.] THOMSON'S ABSOLUTE ELECTROMETER. 307 balance and suspended disk, sufficient apertures being left to see the fiducial marks. The guard-ring, case, and suspended disk are all in metallic communication with each other, but are insulated from the other parts of the apparatus. Now let it be required to measure the difference of potentials of two conductors. The conductors are put in communication with the upper and lower disks respectively by means of wires, the weight is taken off the suspended disk, and the lower disk is moved up by means of the micrometer screw till the electrical attraction brings the suspended disk down to its sighted position. We then know that the attraction between the disks is equal to the weight which brought the disk to its sighted position. If W be the numerical value of the weight, and g the force of gravity, the force is Wg, and if A is the area of the suspended disk, D the distance between the disks, and V the difference of the potentials of the disks *, - - * Let us denote the radius of the suspended disk by E, and that of the aperture of the guard-ring by R', then the breadth of the annular interval between the disk and the ring will be B = R'R. If the distance between the suspended disk and the large fixed disk is D, and the difference of potentials between these disks is F, then, by the investigation in Art. 201, the quantity of electricity on the suspended disk will be I SD SD where a = B^^-, or a = 0.220635 (R'- -R). If the surface of the guard-ring is not exactly in the plane of the surface of the suspended disk, let us suppose that the distance between the ^ fixed ^ disk and the guard -ring is not D but D + z = I?, then it appears from the investigation in Art. 225 that there will be an additional charge of electricity near the edge of the disk on account of its height z above the general surface of the guard-ring. The whole charge in this case is therefore, approximately, and in the expression for the attraction we must substitute for A, the area of the disk, the corrected quantity (R'*-IP) -~^ + 8 (B + K) (D'- where E = radius of suspended disk, R'= radius of aperture in the guard-ring, D = distance between fixed and suspended disks, D'= distance between fixed disk and guard-ring, a = 0.220635 (X?-K). When a is small compared with D we may neglect the second term, and when D' D is small we may neglect the last term. X 2 308 ELECTROSTATIC INSTRUMENTS. [218. If the suspended disk is circular, of radius R, and if the radius of the aperture of the guard-ring is R'> then A = i* (-ffi 2 + IP), and V= 4 D 218.] Since there is always some uncertainty in determining the micrometer reading corresponding to D = 0, and since any error in the position of the suspended disk is most important when D is small, Sir W. Thomson prefers to make all his measurements depend on differences of the electromotive force T. Thus, if V and V are two potentials, and D and If the corresponding distances, For instance, in order to measure the electromotive force of a galvanic battery, two electrometers are used. By means of a condenser, kept charged if necessary by a re- plenisher, the lower disk of the principal electrometer is maintained at a constant potential. This is tested by connecting the lower disk of the principal electrometer with the lower disk of a secondary electrometer, the suspended disk of which is connected with the earth. The distance between the disks of the secondary elec- trometer and the force required to bring the suspended disk to its sighted position being constant, if we raise the potential of the condenser till the secondary electrometer is in its sighted position, we know that the potential of the lower disk of the principal electrometer exceeds that of the earth by a constant quantity which we may call T. If we now connect the positive electrode of the battery to earth, and connect the suspended disk of the principal electrometer to the negative electrode, the difference of potentials between the disks will be V+ v, if v is the electromotive force of the battery. Let D be the reading of the micrometer in this case, and let I? be the reading when the suspended disk is connected with earth, then In this way a small electromotive force v may be measured by the electrometer with the disks at conveniently measurable distances. When the distance is too small a small change of absolute distance makes a great change in the force, since the force varies inversely as the square of the distance, so that any 2I9-] GAUGE ELECTROMETER. 309 error in the absolute distance introduces a large error in the result unless the distance is large compared with the limits of error of the micrometer screw. The effect of small irregularities of form in the surfaces of the disks and of the interval between them diminish according to the inverse cube and higher inverse powers of the distance, and what- ever be the form of a corrugated surface, the eminences of which just reach a plane surface, the electrical effect at any distance which is considerable compared to the breadth of the corrugations, is the same as that of a plane at a certain small distance behind the plane of the tops of the eminences. See Arts. 197, 198. By means of the auxiliary electrification, tested by the auxiliary electrometer, a proper interval between the disks is secured. The auxiliary electrometer may be of a simpler construction, in which there is no provision for the determination of the force of attraction in absolute measure, since all that is wanted is to secure a constant electrification. Such an electrometer may be called a gauge electrometer. This method of using an auxiliary electrification besides the elec- trification to be measured is called the Heterostatic method of electrometry, in opposition to the Idiostatic method in which the whole effect is produced by the electrification to be measured. In several forms of the attracted disk electrometer, the attracted disk is placed at one end of an arm which is supported by being attached to a platinum wire passing through its centre of gravity and kept stretched by means of a spring. The other end of the arm carries the hair which is brought to a sighted position by altering the distance between the disks, and so adjusting the force of the electric attraction to a constant value. In these electro- meters this force is not in general determined in absolute measure, but is known to be constant, provided the torsional elasticity of the platinum wire does not change. The whole apparatus is placed in a Leyden jar, of which the inner surface is charged and connected with the attracted disk and guard-ring. The other disk is worked by a micrometer screw and is connected first with the earth and then with the conductor whose potential is to be measured. The difference of readings multiplied by a constant to be determined for each electrometer gives the potential required. 219.] The electrometers already described are not self-acting, but require for each observation an adjustment of a micrometer 310 ELECTKOSTATIC INSTRUMENTS. [219. screw, or some other movement which must be made by the observer. They are therefore not fitted to act as self-registering instruments, which must of themselves move into the proper position. This condition is fulfilled by Thomson's Quadrant Electrometer. The electrical principle on which this instrument is founded may be thus explained : A and B are two fixed conductors which may be at the same or at different potentials. C is a moveable conductor at a high potential, which is so placed that part of it is opposite to the surface of A and part opposite to that of B, and that the proportions of these parts are altered as C moves. For this purpose it is most convenient to make C moveable about an axis, and make the opposed surfaces of A, of B, and of C portions of surfaces of revolution about the same axis. In this way the distance between the surface of C and the opposed surfaces of A or of B remains always the same, and the motion of C in the positive direction simply increases the area opposed to B and diminishes the area opposed to A. If the potentials of A and B are equal there will be no force urging C from A to , but if the potential of C differs from that of B more than from that of A, then C will tend to move so as to increase the area of its surface opposed to B. By a suitable arrangement of the apparatus this force may be made nearly constant for different positions of C within certain limits, so that if C is suspended by a torsion fibre, its deflexions will be nearly proportional to the difference of potentials between A and B multiplied by the difference of the potential of C from the mean of those of A and B. C is maintained at a high potential by means of a condenser provided with a replenisher and tested by a gauge electrometer, and A and B are connected with the two conductors the difference of whose potentials is to be measured. The higher the potential of C the more sensitive is the instrument. This electrification of C, being independent of the electrification to be measured, places this electrometer in the heterostatic class. We may apply to this electrometer the general theory of systems of conductors given in Arts. 93, 127. Let A, B, C denote the potentials of the three conductors re- spectively. Let a, b, c be their respective capacities,^ the coefficient of induction between B and ). If a and /3 are the radii of the spheres, then, when a is large compared with /3, the charge on B is to that on A in the ratio of Now let a- be the uniform surface-density on A when B is re- moved, then the charge on A is and therefore the charge on B is v 3 a or, when ft is very small compared with a, the charge on the hemisphere B is equal to three times that due to a surface-density is the solid angle subtended by the edge of either disk at the point. Hence the potential of the whole system will be V 4:iT(rz + a'c&. The forms of the equipotential surfaces and lines of induction are given on the left-hand side of Fig. XX, at the end of Vol. II. Let us trace the form of the surface for which F= 0. This surface is indicated by the dotted line. Putting the distance of any point from the axis of z = r, then, when r is much less than a, and z is small, we find 0> = 27T 27T- + &C. a Hence, for values of r considerably less than a, the equation of the zero equipotential surface is zc = 47ro-z+2'7ro / c 2 Tit/ h&c. ; becomes approximately a lune of the sphere of unit radius whose angle is tan- 1 {z -*- (r a)}, that is, o> is 2 tan" 1 {z -r- (ra)}, so that dV = dz Hence, when dV v'c _ = 0, ' =+ ,'226.] ACCUMULATORS. 319 The equipotential surface V is therefore composed of a disk- like figure of radius r , and nearly uniform thickness Z Q , and of the part of the infinite plane of xy which lies beyond this figure. The surface-integral over the whole disk gives the charge of electricity on it. It may be found, as in the theory of a circular current in Part IV, Art. 704, to be Q = ' The charge on an equal area of the plane surface is 7ro-r 2 , hence the charge on the disk exceeds that on an equal area of the plane in the ratio of z , STTT t 1 + 8 - log -- to unity, where z is the thickness and r the radius of the disk, z being sup- posed small compared with r. On Electric Accumulators and the Measurement of Capacity. 226.] An Accumulator or Condenser is an apparatus consisting of two conducting surfaces separated by an insulating dielectric medium. A Leyden jar is an accumulator in which an inside coating of tinfoil is separated from the outside coating by the glass of which the jar is made. The original Leyden phial was a glass vessel containing water which was separated by the glass from the hand which held it. The outer surface of any insulated conductor may be considered as one of the surfaces of an accumulator, the other being the earth or the walls of the room in which it is placed, and the intervening air being the dielectric medium. The capacity of an accumulator is measured by the quantity of electricity with which the inner surface must be charged to make the difference between the potentials of the surfaces unity. Since every electrical potential is the sum of a number of parts found by dividing each electrical element by its distance from a point, the ratio of a quantity of electricity to a potential must have the dimensions of a line. Hence electrostatic capacity is a linear quantity, or we may measure it in feet or metres without ambiguity. In electrical researches accumulators are used for two principal purposes, for receiving and retaining large quantities of electricity in as small a compass as possible, and for measuring definite quan- tities of electricity by means of the potential to which they raise the accumulator. 320 ELECTROSTATIC INSTRUMENTS. [ 22 7- For the retention of electrical charges nothing has been devised more perfect than the Leyden jar. The principal part of the loss arises from the electricity creeping along the damp uncoated surface of the glass from the one coating to the other. This may be checked in a great degree by artificially drying the air within the jar, and by varnishing the surface of the glass where it is exposed to the atmosphere. In Sir W. Thomson's electroscopes there is a very small percentage of loss from day to day, and I believe that none of this loss can be traced to direct conduction either through air or through glass when the glass is good, but that it arises chiefly from superficial conduction along the various insulating stems and glass surfaces of the instrument. In fact, the same electrician has communicated a charge to sulphuric acid in a large bulb with a long neck, and has then her- metically sealed the neck by fusing it, so that the charge was com- pletely surrounded by glass, and after some years the charge was found still to be retained. It is only, however, when cold, that glass insulates in this way, for the charge escapes at once if the glass is heated to a temperature below 100C. When it is desired to obtain great capacity in small compass, accumulators in which the dielectric is sheet caoutchouc, mica, or paper impregnated with paraffin are convenient. 227.] For accumulators of the second class, intended for the measurement of quantities of electricity, all solid dielectrics must be employed with great caution on account of the property which they possess called Electric Absorption. The only safe dielectric for such accumulators is air, which has this inconvenience, that if any dust or dirt gets into the narrow space between the opposed surfaces, which ought to be occupied only by air, it not only alters the thickness of the stratum of air, but may establish a connexion between the opposed surfaces, in which case the accumulator will not hold a charge. To determine in absolute measure, that is to say in feet or metres, the capacity of an accumulator, we must either first ascertain its form and size, and then solve the problem of the distribution of electricity on its opposed surfaces, or we must compare its capacity with that of another accumulator, for which this problem has been solved. As the problem is a very difficult one, it is best to begin with an accumulator constructed of a form for which the solution is known. 228.] MEASUREMENT OF CAPACITT. 321 Thus the capacity of an insulated sphere in an unlimited space is known to be measured by the radius of the sphere. A sphere suspended in a room was actually used by MM. Kohl- rausch and Weber, as an absolute standard with which they com- pared the capacity of other accumulators. The capacity, however, of a sphere of moderate size is so small when compared with the capacities of the accumulators in common use that the sphere is not a convenient standard measure. Its capacity might be greatly increased by surrounding the- sphere with a hollow concentric spherical surface of somewhat greater radius. The capacity of the inner surface is then a fourth proportional to the thickness of the stratum of air and the radii of the two surfaces. Sir W. Thomson has employed this arrangement as a standard of capacity, but the difficulties of working the surfaces truly spherical, of making them truly concentric, and of measuring their distance and their radii with sufficient accuracy, are considerable. We are therefore led to prefer for an absolute measure of capacity a form in which the opposed surfaces are parallel planes. The accuracy of the surface of the planes can be easily tested, and their distance can be measured by a micrometer screw, and may be made capable of continuous variation, which is a most important property of a measuring instrument. The only difficulty remaining arises from the fact that the planes must necessarily be bounded, and that the distribution of electricity near the boundaries 'of the planes has not been rigidly calculated. It is true that if we make them equal circular disks, whose radius is large compared with the distance between them, we may treat the edges of the disks as if they were straight lines, and calculate the distribution of electricity by the method due to Helmholtz, and described in Art. 202. But it will be noticed that in this case part of the electricity is distributed on the back of each disk, and that in the calculation it has been supposed that there are no conductors in the neighbourhood, which is not and cannot be the case in a small instrument. 228.] We therefore prefer the following arrangement, due to Sir W. Thomson, which we may call the Guard-ring arrangement, by means of which the quantity of electricity on an insulated disk may be exactly determined in terms of its potential. VOL. I. 322 ELECTROSTATIC INSTRUMENTS. [228. a A B & o(^y G a n , Fig. 21. The Guard-ring Accumulator. Bb is a cylindrical vessel of conducting- material of which the outer surface of the upper face is accurately plane. This upper surface consists of two parts, a disk A, and a broad ring BB surrounding the disk, separated from it by a very small interval all round, just sufficient to prevent sparks passing 1 . The upper surface of the disk is accurately in the same plane with that of the guard-ring. The disk is supported by pillars of insulating material GG. C is a metal disk, the under surface of which is accurately plane and parallel to BB. The disk C is considerably larger than A. Its distance from A is adjusted and measured by means of a micrometer screw, which is not given in the figure. This accumulator is used as a measuring instrument as follows : Suppose C to be at potential zero, and the disk A and vessel Bb both at potential V. Then there will be no electrification on the back of the disk because the vessel is nearly closed and is all at the same potential. There will be very little electrification on the edges of the disk because BB is at the same potential with the disk. On the face of the disk the electrification will be nearly uniform, and therefore the whole charge on the disk will be almost exactly represented by its area multiplied by the surface-density on a plane, as given in Art. 124. In fact, we learn from the investigation in Art. 201 that the charge on the disk is j^ 2 + ^ /2 H"*-W a ) \ SA SA A + aY where R is the radius of the disk, R' that of the hole in the guard- ring, A the distance between A and C, and a a quantity which cannot exceed R-R 1 ^ . If the interval between the disk and the guard-ring is small compared with the distance between A and R 2 that of the part from B to C, and R that of the whole from A to (7, then, since ab = R l C, bc = R 2 C, and ac = RC, the potential at B is which determines the potential at B when the potentials at A and C are given. Resistance of a Multiple Conductor. 276.] Let a number of conductors ABZ t ACZ, ADZ be arranged side by side with their extremities in contact with the same two points A and Z. They are then said to be arranged in multiple arc. Let the resistances of these conductors be R l , R 2J R 3 respect* 277-] SPECIFIC RESISTANCE AND CONDUCTIVITY. 369 ively, and tlie currents C lt C 2 , C 3 , and let the resistance of the multiple conductor be R, and the total current C. Then, since the potentials at A and Z are the same for all the conductors, they have the same difference, which we may call E. We then have E=C 1 R 1 = C 2 R 2 = C 3 R 3 = CR, whence =i + i+i- (7) Or, the reciprocal of the resistance of a multiple conductor is the sum of the reciprocals of the component conductors. If we call the reciprocal of the resistance of a conductor the conductivity of the conductor, then we may say that the con- ductivity of a multiple conductor is the sum of the conductivities of the component conductors. Current in any Branch of a Multiple Conductor. From the equations of the preceding article, it appears that if (?! is the current in any branch of the multiple conductor, and R 1 the resistance of that branch, 4=CJr, (8) where C is the total current, and R is the resistance of the multiple conductor as previously determined. Longitudinal Resistance of Conductors of Uniform Section. 277.] Let the resistance of a cube of a given material to a current parallel to one of its edges be />, the side of the cube being unit of length, p is called the ' specific resistance of that material for unit of volume.' Consider next a prismatic conductor of the same material whose length is I, and whose section is unity. This is equivalent to I cubes arranged in series. The resistance of the conductor is there- fore I p. Finally, consider a conductor of length I and uniform section s. This is equivalent to s conductors similar to the last arranged in multiple arc. The resistance of this conductor is therefore ju#. s When we know the resistance of a uniform wire we can determine VOL. i. B b 370 LINEAR ELECTEIC CURRENTS. [ 2 7^. the specific resistance of the material of which it is made if we can measure its length and its section. The sectional area of small wires is most accurately determined by calculation from the length, weight, and specific gravity of the specimen. The determination of the specific gravity is sometimes inconvenient, and in such cases the resistance of a wire of unit length and unit mass is used as the e specific resistance per unit of weight/ If r is this resistance, I the length, and m the mass of a wire, then On the Dimensions of the Quantities involved in these Equations. 278.] The resistance of a conductor is the ratio of the electro- motive force acting on it to the current produced. The conduct- ivity of the conductor is the reciprocal of this quantity, or in other words, the ratio of the current to the electromotive force producing it. Now we know that in the electrostatic system of measurement the ratio of a quantity of electricity to the potential of the con- ductor on which it is spread is the capacity of the conductor, and is measured by a line. If the conductor is a sphere placed in an unlimited field, this line is the radius of the sphere. The ratio of a quantity of electricity to an electromotive force is therefore a line, but the ratio of a quantity of electricity to a current is the time during which the current flows to transmit that quantity. Hence the ratio of a current to an electromotive force is that of a line to a time, or in other words, it is a velocity. The fact that the conductivity of a conductor is expressed in the electrostatic system of measurement by a velocity may be verified by supposing a sphere of radius r charged to potential F", and then connected with the earth by the given conductor. Let the sphere contract, so that as the electricity escapes through the conductor the potential of the sphere is always kept equal to T. Then the charge on the sphere is rV at any instant, and the current is -j- (rF), but, since V is constant, the current is jr V, and the electromotive force through the conductor is V. The conductivity of the conductor is the ratio of the current to the electromotive force, or , that is, the velocity with which the Cvt radius of the sphere must diminish in order to maintain the potential 28o.] SYSTEM OF LINEAR CONDUCTORS. 371 constant when the charge is allowed to pass to earth through the conductor. In the electrostatic system, therefore, the conductivity of a con- ductor is a velocity, and of the dimensions [LT~ 1 ]. The resistance of the conductor is therefore of the dimensions IL-*T\. The specific resistance per unit of volume is of the dimension of [T], and the specific conductivity per unit of volume is of the dimension of [2 7 " 1 ]. The numerical magnitude of these coefficients depends only on the unit of time, which is the same in different countries. The specific resistance per unit of weight is of the dimensions 279.] We shall afterwards find that in the electromagnetic system of measurement the resistance of a conductor is expressed by a velocity, so that in this system the dimensions of the resist- ance of a conductor are [XT 7 " 1 ]. The conductivity of the conductor is of course the reciprocal of this. The specific resistance per unit of volume in this system is of the dimensions [X 2 ! 7 " 1 ], and the specific resistance per unit of weight is of the dimensions [L~ 1 T~ 1 M~\. On Linear Systems of Conductors in general. 280.] The most general case of a linear system is that of n points, A l} A 2 ,...A n} connected together in pairs by \n(nr-\) linear conductors. Let the conductivity (or reciprocal of the re- sistance) of that conductor which connects any pair of points, say A p and A q , be called K pq , and let the current from A p to A q be C pq . Let P p and P q be the electric potentials at the points A p and A q respectively, and let the internal electromotive force, if there be any, along the conductor from A p to A q be E pq . The current from A p to A q is, by Ohm's Law, C tq = K M (P t -P q + K pq ). . (1) Among these quantities we have the following sets of relations : The conductivity of a conductor is the same in either direction, * = *. (a) The electromotive force and the current are directed quantities, so that ^U = -^p> and C pq = -C qp . (3) Let P lt P 2 , ...P n be the potentials at A 19 A 2 , ... A n respectively, and let Q lt Q 2 ,... Q n be the quantities of electricity which enter B b 2 372 LINEAR ELECTRIC CURRENTS. [280. the system in unit of time at each of these points respectively. These are necessarily subject to the condition of ' continuity ' &+&... + &=<), (4) since electricity can neither be indefinitely accumulated nor pro- duced within the system. The condition of ' continuity ' at any point A p is Q p = C pl + C p2 + &c. + O pn . (5) Substituting the values of the currents in terms of equation (l), this becomes E pn ). (6) The symbol K pp does not occur in this equation. Let us therefore give it the value K pp = - (K pl + K p2 + &c. + K pn ) ; (7) that is, let K pp be a quantity equal and opposite to the sum of all the conductivities of the conductors which meet in A p . We may then write the condition of continuity for the point A p , E pn -Q p . (8) By substituting 1 , 2, &c. n for p in this equation we shall obtain n equations of the same kind from which to determine the n potentials P lt P 2 , &c., P n . Since, however, if we add the system of equations (8) the result is identically zero by (3), (4) and (7), there will be only n 1 in- dependent equations. These will be sufficient to determine the differences of the potentials of the points, but not to determine the absolute potential of any. This, however, is not required to calcu- late the currents in the system. If we denote by D the determinant (9) and by D pq , the minor of K pq , we find for the value of I p P n , K qn E qn -Q q )D pq + &c. (10) In the same way the excess of the potential of any other point, say A q , over that of A n may be determined. We may then de- termine the current between A p and A q from equation (l), and so solve the problem completely. 282 a.] SYSTEM OF LINEAR CONDUCTORS. 373 281.] We shall now demonstrate a reciprocal property of any two conductors of the system, answering to the reciprocal property we have already demonstrated for statical electricity in Art. 88. The coefficient of Q q in the expression for P p is -^. That of Q p in the expression for P q is -~ Now D pq differs from D qp only by the substitution of the symbols such as K qp for K pq . But, by equation (2), these two symbols are equal, since the conductivity of a conductor is the same both ways. Hence D pq =-.D qp . (ll) It follows from this that the part of the potential at A p arising from the introduction of a unit current at A q is equal to the part of the potential at A q arising from the introduction of a unit current at A p . We may deduce from this a proposition of a more practical form. Let A, B, C, D be any four points of the system, and let the effect of a current Q, made to enter the system at A and leave it at B, be to make the potential at C exceed that at D by P. Then, if an equal current Q be made to enter the system at C and leave it at D, the potential at A will exceed that at B by the same quantity P. If an electromotive force E be introduced, acting in the conductor from A to B, and if this causes a current C from X to Y, then the same electromotive force E introduced into the conductor from X to Y will cause an equal current C from A to B. The electromotive force E may be that of a voltaic battery intro- duced between the points named, care being taken that the resist- ance of the conductor is the same before and after the introduction of the battery. 282 a.] If an electromotive force E^ act along the conductor A p A q , the current produced along another conductor of the system A A is easily found to be K n K, t E pq (D re +D, t -D rq -D. r ) * D. There will be no current if j) rp + Z) sq rq J) s p = 0. (12) But, by (11), the same equation holds if, when the electromotive force acts along A r A t , there is no current in A p A q . On account of this reciprocal relation the two conductors referred to are said to be conjugate. The theory of conjugate conductors has been investigated by 374 LINEAR ELECTRIC CURRENTS. [282 I. Kirchhoff, who has stated the conditions of a linear system in the following manner, in which the consideration of the potential is avoided. (1) (Condition of 'continuity.') At any point of the system the sum of all the currents which flow towards that point is zero. (2) In any complete circuit formed by the conductors the sum of the electromotive forces taken round the circuit is equal to the sum of the products of the current in each conductor multiplied by the resistance of that conductor. We obtain this result by adding equations of the form (l) for the complete circuit, when the potentials necessarily disappear. *282 .] If the conducting wires form a simple network and if we suppose that a current circulates round each mesh, then the actual current in the wire which forms a thread of each of two neighbouring meshes will be the difference between the two currents circulating in the two meshes, the currents being reckoned positive when they circulate in a direction opposite to the motion of the hands of a watch. It is easy to establish in this case the following proposition : Let x be the current, E the electromotive force, and R the total resistance in any mesh ; let also y, z, ... be currents circulating in neighbouring meshes which have threads in common with that in which x circulates, the resistances of those parts being ,?, ^, . . . ; then Rxsytz&c. = E. To illustrate the use of this rule we will take the arrangement known as Wheatstone's Bridge, adopting the figure and notation of Art. 347. We have then the three following equations repre- senting the application of the rule in the case of the three circuits OBC, OCA, OAB in which the currents #, y> z respectively circulate, viz. (a + p+y),,. _ yy -/3* = JS; y > and I m n respectively at A, B and m = N-r-> n = N^r' (6) dx dy dz 378 CONDUCTION IN THREE DIMENSIONS. [288. ..,( dX d\ dX') y = N lu + v~ + w- r l > I dx dy dz (?) Hence, if y is the component of the current normal to the surface, dX dX ~dy ' ~~ dz j If y = there will be no current through the surface, and the surface may be called a Surface of Flow, because the lines of motion are in the surface. 288.] The equation of a surface of flow is therefore dX dX w ~^ |" v ~~^ dx dy If this equation is true for all values of A, all the surfaces of the family will be surfaces of flow. 289.] Let there be another family of surfaces, whose parameter is A', then, if these are also surfaces of flow, we shall have (3) we may call the coefficients r the coefficients of Longitudinal con- ductivity, and p and q those of Transverse conductivity. The coefficients of resistance are inverse to those of conductivity. This relation may be defined as follows : Let [PQR\ be the determinant of the coefficients of resistance, and [pqr] that of the coefficients of conductivity, then [PQR] [pqr] = 1, (6) [PQR] A = (P 2 P 3 - ft 50 , [pqr] P l = (AA-& r ^ ( 7 ) &c. &c. The other equations may be formed by altering the symbols P, Q, R 3 P) q, r, and the suffixes 1, 2, 3 in cyclical order. Rate of Generation of Heat. 299.] To find the work done by the current in unit of time in overcoming resistance, and so generating heat, we multiply the components of the current by the corresponding components of the electromotive force. We thus obtain the following expressions for W t the quantity of work expended in unit of time : 300.] COEFFICIENTS OF CONDUCTIVITY. 385 (8) uv' ) (9) By a proper choice of axes, either of the two latter equations may be deprived of the terms involving the products of u, v, w or of Jf, Y t Z. The system of axes, however, which reduces W to the form is not in general the same as that which reduces it to the form It is only when the coefficients P lt P 2 , P 3 are equal respectively to Q 19 Q 2 , Q 3 that the two systems of axes coincide. If with Thomson * we write P=S+T, Q = S-1 and p = s +t, q = s t then we have l 2 ,s, and [PQR] r t = R 2 R 9 -S* + T^ (13) [p<^K=-tf, If therefore we cause S lt $ 2 , S 3 to disappear, s 1 will not also dis- appear unless the coefficients T are zero. Condition of Stability. 300.] Since the equilibrium of electricity is stable, the work spent in maintaining the current must always be positive. The conditions that W may be positive are that the three coefficients R lt R 2y R 3 , and the three expressions must all be positive. There are similar conditions for the coefficients of conductivity. * Trans. R. S. Edin., 1853-4, p. 165. VOL. I. C C 386 EESISTANCE AND CONDUCTIVITY. [3OI. Equation of Continuity in a Homogeneous Medium. 301.] If we express the components of the electromotive force as the derivatives of the potential V t the equation of continuity du dv dw becomes in a homogeneous medium d 2 7 1 Jx 2 2 dy 2 3 dz 2 1 dy dz If the medium is not homogeneous there will be terms arising from the variation of the coefficients of conductivity in passing from one point to another. This equation corresponds to Laplace's equation in an isotropic medium. 302.] If we put LA* ft I >*/>'> I O O O O 4* O 2 ___ /% a 2 ^^ M a a f 1 7 I / o / -i Tn I o -p u o-i Oo Oq"~ 1 "i 'O 09 ' Q oo , I 1 / I J i Z O ' L 6 6 J.I 4O'\y where [W]^ = r 2 r 3 s-f, (19) and so on, the system A, B will be inverse to the system r, s, and if we make A : x 2 +A 2 y* + A 3 z 2 +2B l yz+2B 2 zx+2 B 3 xy = [AS] p 2 , (20) we shall find that r =T~- 477 p is a solution of the equation. In the case in which the coefficients T are zero, the coefficients A and B become identical with R and S. When T exists this is not the case. In the case therefore of electricity flowing out from a centre in an infinite, homogeneous, but not isotropic, medium, the equipotential surfaces are ellipsoids, for each of which p is constant. The axes of these ellipsoids are in the directions of the principal axes of con- ductivity, and these do not coincide with the principal axes of resistance unless the system is symmetrical. By a transformation of this equation we may take for the axes of #, y, z the principal axes of conductivity. The coefficients of the forms 8 and B will then be reduced to zero, and each coefficient 33] SKEW SYSTEM. 387 of the form A will be the reciprocal of the corresponding coefficient of the form r. The expression for p will be 303.] The theory of the complete system of equations of resist- ance and of conductivity is that of linear functions of three vari- ables, and it is exemplified in the theory of Strains *, and in other parts of physics. The most appropriate method of treating it is that by which Hamilton and Tait treat a linear and vector function of a vector. We shall not, however, expressly introduce Quaternion notation. The coefficients T^ T 2 , T s may be regarded as the rectangular components of a vector T, the absolute magnitude and direction of which are fixed in the body, and independent of the direction of the axes of reference. The same is true of t lt t z , t 3 , which are the components of another vector t. The vectors T and t do not in general coincide in direction. Let us now take the axis of z so as to coincide with the vector T, and transform the equations of resistance accordingly. They will then have the form v, j | ) (23) Z S 2 u + S 1 It appears from these equations that we may consider the elec- tromotive force as the resultant of two forces, one of them depending only on the coefficients E and 8, and the other depending on T alone. The part depending on R and 8 is related to the current in the same way that the perpendicular on the tangent plane of an ellipsoid is related to the radius vector. The other part, depending on T, is equal to the product of T into the resolved part of the current perpendicular to the axis of T, and its direction is per- pendicular to T and to the current, being always in the direction in which the resolved part of the current would lie if turned 90 in the positive direction round T. If we consider the current and T as vectors, the part of the electromotive force due to T is the vector part of the product, Tx current. The coefficient T may be called the Rotatory coefficient. "We have reason to believe that it does not exist in any known sub- * See Thomson and Tait's Natural PhHo'orfi/, 154. C C 2 388 RESISTANCE AND CONDUCTIVITY. [304. stance. It should be found, if anywhere, in magnets, which have a polarization in one direction, probably due to a rotational phe- nomenon in the substance. 304.] Assuming then that there is no rotatory coefficient, we shall shew how Thomson's Theorem given in Art. 100 may be extended to prove that the heat generated by the currents in the system in a given time is a unique minimum. To simplify the algebraical work let the axes of coordinates be chosen so as to reduce expression (9), and therefore also in this case expression (10), to three terms; and let us consider the general characteristic equation (16) which thus reduces to d 2 F d 2 F d 2 F_ r l J2 ~f~ r 2 ~7Tz "^ T 3 ^2 * ( / Also, let and let 5, c be three functions of #, y, z satisfying the condition da db dc , . -r + T- + T- ( 25 ) dx du dz v ' (26) c = Finally, let the triple-integral tf + R^)dxdydz (27) be extended over spaces bounded as in the enunciation of Art. 100 ; such viz. that ^is constant over certain portions or else the normal component of the vector , #, c is given, the latter condition being accompanied by the further restriction that the integral of this component over the whole bounding surface must be zero : then W will be a minimum when u = 0, v = 0, w = 0. For we have in this case and therefore, by (26), W dF dF =///(-. - ! ///("s + 'f+S'"**'- < 28 ' 305.] EXTENSION OF THOMSON'S THEOREM. 389 du dv dw to + Ty + Tz = ' the third term vanishes by virtue of the conditions at the limits. The first term of (28) is therefore the unique minimum value of W. 305.] As this proposition is of great importance in the theory of electricity, it may be useful to present the following proof of the most general case in a form free from analytical operations. Let us consider the propagation of electricity through a conductor of any form, homogeneous or heterogeneous. Then we know that (1) If we draw a line along the path and in the direction of the electric current, the line must pass from places of high potential to places of low potential. (2) If the potential at every point of the system be altered in a given uniform ratio, the currents will be altered in the same ratio, according to Ohm's Law. (3) If a certain distribution of potential gives rise to a certain distribution of currents, and a second distribution of potential gives rise to a second distribution of currents, then a third distribution in which the potential is the sum or difference of those in the first and second will give rise to a third distribution of currents, such that the total current passing through a given finite surface in the third case is the sum or difference of the currents passing through it in the first and second cases. For, by Ohm's Law, the additional current due to an alteration of potentials is independent of the original current due to the original distribution of potentials. (4) If the potential is constant over the whole of a closed surface, and if there are no electrodes or intrinsic electromotive forces within it, then there will be no currents within the closed surface, and the potential at any point within it will be equal to that at the surface. If there are currents within the closed surface they must either be closed curves, or they must begin and end either within the closed surface or at the surface itself. But since the current must pass from places of high to places of low potential, it cannot flow in a closed curve. Since there are no electrodes within the surface the current cannot begin or end within the closed surface, and since the potential at all points of the surface is the same, there can be no current along lines passing from one point of the surface to another. 390 RESISTANCE AND CONDUCTIVITY. [306. Hence there are no currents within the surface, and therefore there can be no difference of potential, as such a difference would produce currents, and therefore the potential within the closed surface is everywhere the same as at the surface. (5) If there is no electric current through any part of a closed surface, and no electrodes or intrinsic electromotive forces within the surface, there will be no currents within the surface, and the potential will be uniform. We have seen that the currents cannot form closed curves, or begin or terminate within the surface, and since by the hypothesis they do not pass through the surface, there can be no currents, and therefore the potential is constant. (6) If the potential is uniform over part of a closed surface, and if there is no current through the remainder of the surface, the potential within the surface will be uniform for the same reasons. (7) If over part of the surface of a body the potential of every point is known, and if over the rest of the surface of the body the current passing through the surface at each point is known, then only one distribution of potentials at points within the body can exist. For if there were two different values of the potential at any point within the body, let these be V\ in the first case and V 2 in the second case, and let us imagine a third case in which the potential of every point of the body is the excess of potential in the first case over that in the second. Then on that part of the surface for which the potential is known the potential in the third case will be zero, and on that part of the surface through which the currents are known the currents in the third case will be zero, so that by (6) the potential everywhere within the surface will be zero, or there is no excess of V^ over T 2i or the reverse. Hence there is only one possible distribution of potentials. This proposition is true whether the solid be bounded by one closed surface or by several. On the Approximate Calculation of the Resistance of a Conductor of a given Form. 306.] The conductor here considered has its surface divided into three portions. Over one of these portions the potential is main- tained at a constant value. Over a second portion the potential has a constant value different from the first. The whole of th^remainder of the surface is impervious to electricity. We may suppose the 306.] RESISTANCE OF A WIRE OF VARIABLE SECTION. 391 conditions of the first and second portions to be fulfilled by applying- to the conductor two electrodes of perfectly conducting material, and that of the remainder of the surface by coating it with per- fectly non-conducting material. Under these circumstances the current in every part of the conductor is simply proportional to the difference between the potentials of the electrodes. Calling this difference the electro- motive force, the total current from the one electrode to the other is the product of the electromotive force by the conductivity of the conductor as a whole, and the resistance of the conductor is the reciprocal of the conductivity. It is only when a conductor is approximately in the circumstances above defined that it can be said to have a definite resistance, or conductivity as a whole. A resistance coil, consisting of a thin wire terminating in large masses of copper, approximately satisfies these conditions, for the potential in the massive electrodes is nearly constant, and any differences of potential in different points of the same electrode may be neglected in comparison with the difference of the potentials of the two electrodes. A very useful method of calculating the resistance of such con- ductors has been given, so far as I know, for the first time, by Lord Rayleigh, in a paper on the Theory of Resonance *. It is founded on the following considerations. If the specific resistance of any portion of the conductor be changed, that of the remainder being unchanged, the resistance of the whole conductor will be increased if that of the portion is increased, and diminished if that of the portion be diminished. This principle may be regarded as self-evident, but it may easily be shewn that the value of the expression for the resistance of a system of conductors between two points selected as electrodes, increases as the resistance of each member of the system in- creases. It follows from this that if a surface of any form be described in the substance of the conductor, and if we further suppose this surface to be an infinitely thin sheet of a perfectly conducting substance, the resistance of the conductor as a whole will be diminished unless the surface is one of the equipotential surfaces in the natural state of the conductor, in which case no effect will be produced by making it a perfect conductor, as it is already in electrical equilibrium. * Phil. Trans., 1871, p. 77. See Art. 102. 392 RESISTANCE AND CONDUCTIVITY. [306. If therefore we draw within the conductor a series of surfaces, the first of which coincides with the first electrode, and the last with the second, while the intermediate surfaces are bounded by the non-conducting surface and do not intersect each other, and if we suppose each of these surfaces to be an infinitely thin sheet of perfectly conducting matter, we shall have obtained a system the resistance of which is certainly not greater than that of the original conductor, and is equal to it only when the surfaces we have chosen are the natural equipotential surfaces. To calculate the resistance of the artificial system is an operation of much less difficulty than the original problem. For the resist- ance of the whole is the sum of the resistances of all the strata contained between the consecutive surfaces, and the resistance of each stratum can be found thus : Let dS be an element of the surface of the stratum, v the thick- ness of the stratum perpendicular to the element, p the specific resistance, E the difference of potential of the perfectly conducting surfaces, and dC the current through dS, then dC=E dS. (1) P v and the whole current through the stratum is the integration being extended over the whole stratum bounded by the non-conducting surface of the conductor. Hence the conductivity of the stratum is <"> and the resistance of the stratum is the reciprocal of this quantity. If the stratum be that bounded by the two surfaces for which the function F has the values F and F+ dF respectively, then and the resistance of the stratum is dF -VFdS p To find the resistance of the whole artificial conductor, we have only to integrate with respect to F, and we find 307.] RESISTANCE OF A WIRE OF VARIABLE SECTION. 393 -VFdS P The resistance E of the conductor in its natural state is greater than the value thus obtained, unless all the surfaces we have chosen are the natural equipotential surfaces. Also, since the true value of R is the absolute maximum of the values of R l which can thus be obtained, a small deviation of the chosen surfaces from the true equipotential surfaces will produce an error of R which is com- paratively small. This method of determining a lower limit of the value of the resistance is evidently perfectly general, and may be applied to conductors of any form, even when p, the specific resistance, varies in any manner within the conductor. The most familiar example is the ordinary method of determining the resistance of a straight wire of variable section. In this case the surfaces chosen are planes perpendicular to the axis of the wire, the strata have parallel faces, and the resistance of a stratum of section S and thickness ds is and that of the whole wire of length s is . <) where S is the transverse section and is a function of s. This method in the case of wires whose section varies slowly with the length gives a result very near the truth, but it is really only a lower limit, for the true resistance is always greater than this, except in the case where the section is perfectly uniform. 307.] To find the higher limit of the resistance, let us suppose a surface drawn in the conductor to be rendered impermeable to electricity. The effect of this must be to increase the resistance of the conductor unless the surface is one of the natural surfaces of flow. By means of two systems of surfaces we can form a set of tubes which will completely regulate the flow, and the effect, if there is any, of this system of impermeable surfaces must be to increase the resistance above its natural value. The resistance of each of the tubes may be calculated by the method already given for a fine wire, and the resistance of the whole conductor is the reciprocal of the sum of the reciprocals of the resistances of all the tubes. The resistance thus found is greater 394 RESISTANCE AND CONDUCTIVITY. [307. than the natural resistance, except when the tubes follow the natural lines of flow. In the case already considered, where the conductor is in the form of an elongated solid of revolution, let us measure x along the axis, and let the radius of the section at any point be b. Let one set of impermeable surfaces be the planes through the axis for each of which

and \l/, we had assumed that the flow through each tube is proportional to d\l/ d$, we should have obtained as the value of the resistance under this additional constraint (17) which is evidently greater than the former value, as it ought to be, on account of the additional constraint. In Lord Rayleigh's paper this is the supposition made, and the superior limit of the resistance there given has the value (17), which is a little greater than that which we have obtained in (16). 308.] We shall now apply the same method to find the correction which must be applied to the length of a cylindrical conductor of radius a when its extremity is placed in metallic contact with a massive electrode, which we may suppose of a different metal. For the lower limit of the resistance we shall suppose that an infinitely thin disk of perfectly conducting matter is placed between the end of the cylinder and the massive electrode, so as to bring the end of the cylinder to one and the same potential throughout. The potential within the cylinder will then be a function of its length only, and if we suppose the surface of the electrode where the cylinder meets it to be approximately plane, and all its dimen- sions to be large compared with the diameter of the cylinder, the distribution of potential will be that due to a conductor in the form of a disk placed in an infinite medium. See Arts. 151, 177. If E is the difference of the potential of the disk from that of the distant parts of the electrode, C the current issuing from the 396 RESISTANCE AND CONDUCTIVITY. [309. surface of the disk into the electrode, and p' the specific resistance of the electrode ; then if Q is the amount of electricity on the disk, which we assume distributed as in Art. 151, we have p'C= 1.47r<2 = 2v~, by Art. 151, 7T 2 = 4J0. (18) Hence, if the length of the wire from a given point to the electrode is Z-, and its specific resistance p, the resistance from that point to any point of the electrode not near the junction is and this may be written ~ ira 2 v p 4 ' ' where the second term within brackets is a quantity which must be added to the length of the cylinder or wire in calculating its resistance, and this is certainly too small a correction. To understand the nature of the outstanding error we may observe, that whereas we have supposed the flow in the wire up to the disk to be uniform throughout the section, the flow from the disk to the electrode is not uniform, but is at any point in- versely proportional to the minimum chord through that point. In the actual case the flow through the disk will not be uniform, but it will not vary so much from point to point as in this supposed case. The potential of the disk in the actual case will not be uniform, but will diminish from the middle to the edge. 309.] We shall next determine a quantity greater than the true resistance by constraining the flow through the disk to be uniform at every point. We may suppose electromotive forces introduced for this purpose acting perpendicular to the surface of the disk. The resistance within the wire will be the same as before, but in the electrode the rate of generation of heat will be the surface- integral of the product of the flow into the potential. The rate of flow at any point is :r, and the potential is the same as that of ira 2 an electrified surface whose surface-density is or, where C p f ^ 27ro-=-^, (20) TI a* p' being the specific resistance. 309.] CORRECTION FOR THE ENDS OP THE WIRE. 397 We have therefore to determine the potential energy of the electrification of the disk with the uniform surface-density a-. * The potential at the edge of a disk of uniform density o- is easily found to be 4 z respectively. Let us take the case of the surface which separates a medium having these coefficients of conduction from an isotropic medium having a coefficient of conduction equal to r. Let X', Y', Z' be the values of X, Y } Z in the isotropic medium, then we have at the surface r=r, (4) or Xdx + Tdy + Zdz = X'dx + Tdy + Z'dz, (5) when Idx + mdy + ndz = 0. (6) This condition gives J'=JT+47r # = 7T -> w = j-j-> (13) k dx k ay k dz and if v is the normal drawn at any point of the surface of separa- tion from the first medium towards the second, the condition of continuity is 1 d7^ J_ d7 2 k dv ~~ $ 2 dv If 1 and 02, are the angles which the lines of flow in the first and second media respectively make with the normal to the surface of separation, then the tangents to these lines of flow are in the same plane with the normal and on opposite sides of it, and /&! tan 1 = 2 tan 2 . (15) This may be called the law of refraction of lines of flow. 311.] As an example of the conditions which must be fulfilled when electricity crosses the surface of separation of two media, let us suppose the surface spherical and of radius a, the specific resistance being ^ within and Jc 2 without the surface. Let the potential, both within and without the surface, be ex- panded in solid harmonics, and let the part which depends on the surface harmonic S t be (1) (2) within and without the sphere respectively. At the surface of separation where r = a we must have Fi=F2 and i iS. (3) &! dr k% dr From these conditions we get the equations These equations are sufficient, when we know two of the four quantities A 19 A. 2t J$ lt -Z? 2 > ^ deduce the other two. Let us suppose A l and J3 L known, then we find the following expressions for A 2 and B 2 , 4 _^ 1 (f _ l - ^(2 Ml) SPHERICAL SHELL. 401 In this way we can find the conditions which each term of the harmonic expansion of the potential must satisfy for any number of strata bounded by concentric spherical surfaces. 312.] Let us suppose the radius of the first spherical surface to be a l3 and let there be a second spherical surface of radius a 2 greater than a l9 beyond which the specific resistance is 3 . If there are no sources or sinks of electricity within these spheres there will be no infinite values of F", and we shall have B l = 0. We then find for A 3 and j5 3 , the coefficients for the outer medium, - (6) The value of the potential in the outer medium depends partly on the external sources of electricity, which produce currents in- dependently of the existence of the sphere of heterogeneous matter within, and partly on the disturbance caused by the introduction of the heterogeneous sphere. The first part must depend on solid harmonics of positive degrees only, because it cannot have infinite values within the sphere. The second part must depend on harmonics of negative degrees, because it must vanish at an infinite distance from the centre of the sphere. Hence the potential due to the external electromotive forces must be expanded in a series of solid harmonics of positive degree. Let A be the coefficient of one of these, of the form Then we can find A 19 the corresponding coefficient for the inner sphere by equation (6), and from this deduce A 2 , -# 2 > an ^ -^3- Of these jB 3 represents the effect on the potential in the outer medium due to the introduction of the heterogeneous spheres. Let us now suppose 3 = & lt so that the case is that of a hollow shell for which k = k% , separating an inner from an outer portion of the same medium for which k = k. If we put VOL. I. D d 402* CONDUCTION IN HETEROGENEOUS MEDIA. then A = & The difference between A B the undisturbed coefficient, and A l its value in the hollow within the spherical shell, is A 3 . (8) Since this quantity is always positive whatever be the values f k-L and /fc 2 , it follows that, whether the spherical shell conducts better or worse than the rest of the medium, the electrical action in the space occupied by the shell is less than it would otherwise be. If the shell is a better conductor than the rest of the medium it tends to equalize the potential all round the inner sphere. If it is a worse conductor, it tends to prevent the electrical currents from reaching 1 the inner sphere at all. The case of a solid sphere may be deduced from this by making 0j_ = 0, or it may be worked out independently. 313.] The most important term in the harmonic expansion is that in which i = 1, for which (9) The case of a solid sphere of resistance 2 may be deduced from this by making a v = 0. We then have (10) a, 3 A.. It is easy to shew from the general expressions that the value of .Z? 3 in the case of a hollow sphere having a nucleus of resistance #1, surrounded by a shell of resistance ^ 2 , is the same as that of a uniform solid sphere of the radius of the outer surface, and of resistance K, where , ., = 8 "' 3 14-] MEDIUM CONTAINING SMALL SPHERES. 403 314.] If there are n spheres of radius x and resistance & lt placed in a medium whose resistance is 2> at such distances from each other that their effects in disturbing- the course of the current may be taken as independent of each other, then if these spheres are all contained within a sphere of radius 2 , the potential at a great distance from the centre of this sphere will be of the form. 7= (Ar + n3-^)coB6 t (12) where the value of is The ratio of the volume of the n small spheres to that of the sphere which contains them is The value of the potential at a great distance from the sphere may therefore be written (15) Now if the whole sphere of radius a 2 had been made of a material of specific resistance K, we should have had That the one expression should be equivalent to the other, 2^ + flg + X*!-^) lf / 17 \ = 2> This, therefore, is the specific resistance of a compound medium consisting of a substance of specific resistance k. 2 > in which are disseminated small spheres of specific resistance k lt the ratio of the volume of all the small spheres to that of the whole being p. In order that the action of these spheres may not produce effects depending on their interference, their radii must be small compared with their distances, and therefore^? must be a small fraction. This result may be obtained in other ways, but that here given involves only the repetition of the result already obtained for a single sphere. When the distance between the spheres is not great compared ^ _ ^ with their radii, and when j - ~ is considerable, then other terms enter into the result, which we shall not now consider. In consequence of these terms certain systems of arrangement of D d 2, 404 CONDUCTION IN HETEROGENEOUS MEDIA. [315. the spheres cause the resistance of the compound medium to be different in different directions. Application of the Principle of Images. 315.] Let us take as an example the case of two media separated by a plane surface, and let us suppose that there is a source 8 of electricity at a distance a from the plane surface in the first medium, the quantity of electricity flowing from the source in unit of time being 8. If the first medium had been infinitely extended the current at any point P would have been in the direction SP, and the TTf O 7 potential at P would have been where E = - and r L = SP. In the actual case the conditions may be satisfied by taking a point 7, the image of 8 in the second medium, such that IS is normal to the plane of separation and is bisected by it. Let r 2 be the distance of any point from 7, then at the surface of separation '!-*, (1) dv dv Let the potential V at any point in the first medium be that due to a quantity of electricity E placed at S 9 together with an imaginary quantity E 2 at 7, and let the potential F 2 at any point of the second medium be that due to an imaginary quantity E l at 8, then if E K K FT = f- and Fo = (3) r T r the superficial condition V^ = F 2 gives TlJ - 777 777 / A \ and the condition A 1 dv k% dv gives j- (EEz) =-^E lt (6) whence E = 7-^5 E 2 = -^ T^' (^) The potential in the first medium is therefore the same as would be produced in air by a charge E placed at 8, and a charge U 2 at 7 on the electrostatic theory, and the potential in the second medium is the same as that which would be produced in air by a charge E l at S. 3 1 7-] STRATUM WITH PARALLEL SIDES. 405 The current at any point of the first medium is the same as would have been produced by the source S together with a source 2 ~ * S ftj -f- A^ placed at / if the first medium had been infinite, and the current at any point of the second medium is the same as would have been 2& 8 produced by a source 2 placed at/S if the second medium had (#1 -t K t ) been infinite. We have thus a complete theory of electrical images in the case of two media separated by a plane boundary. Whatever be the nature of the electromotive forces in the first medium, the potential they produce in the. first medium may be found by combining their direct effect with the effect of their image. If we suppose the second medium a perfect conductor, then k. 2 = 0, and the image at / is equal and opposite to the source at 8. This is the case of electric images, as in Thomson's theory in electrostatics. If we suppose the second medium a perfect insulator, then 2 = oo, and the image at /is equal to the source at S and of the same sign. This is the case of images in hydrokinetics when the fluid is bounded by a rigid plane surface. 316.] The method of inversion, which is of so much use in electrostatics when the. bounding surface is supposed to be that of a perfect conductor, is not applicable to the more general case of the surface separating two conductors of unequal electric resist- ance. The method of inversion in two dimensions is, however, applicable, as well as the more general method of transformation in two dimensions given in Art. 190*. Conduction through a Plate separating Two Media. 317.] Let us next consider the effect of a plate of thickness AB of a medium whose resist- ance is / 2 , and separating \. two media whose resist- ances are k^ and / 3 , in ~T~ ~f~ J~ altering the potential due to a source S in the first medium. The potential will be Fi S- 24 - * See Kirchhoff, Pogg. Ann. Ixiv. 497, and Ixvii. 344 ; Quincke, Pogg. xcvii. 382; and Smith, Proc. R. S. din., 1869-70, p. 79. 406 CONDUCTION IN HETEROGENEOUS MEDIA. equal to that due to a system of charges placed in air at certain points along the normal to the plate through S. Make AI= SA, 5/i = SB, A^ = T L A, BI 2 = ^ B, AJ 2 = I 2 A, &c. ; then we have two series of points at distances from each other equal to twice the thickness of the plate. 318.] The potential in the first medium at any point P is equal to HILL W + P7 + ?t + Pt +&c " ^ that at a point P' in the second w r // // 4 and that at a point P" in the third where /, I', &c. represent the imaginary charges placed at the points /, &c., and the accents denote that the potential is to be taken within the plate. Then, by the last Article, for the surface through A we have, (11) t #2 + ATj A?2 r K\ For the surface through B we find ^3 "l~ ^2 2 Similarly for the surface through ^4 again, and for the surface through 5, we find for the potential in the first medium, ' "- 1 -. (15) 3 1 9-] STRATIFIED CONDUCTORS. 407 For the potential in the third medium we find (16) If the first medium is the same as the third, then ^ = / 3 and p = p', and the potential on the other side of the plate will be If the plate is a very much better conductor than the rest of the medium, p is very nearly equal to 1. If the plate is a nearly perfect insulator, p is nearly equal to 1, and if the plate differs little in conducting power from the rest of the medium, p is a small quantity positive or negative. The theory of this case was first stated by Green in his ' Theory of Magnetic Induction' (Essay, p. 65). His result, however, is correct only when p is nearly equal to 1 *. The quantity g which he uses is connected with p by the equations 2p _ \-kt *g _ *! * 8 = 3 p *! + 2V p ' 2+ff *!+V If we put p = - , we shall have a solution of the problem of 1 + 27TK the magnetic induction excited by a magnetic pole in an infinite plate whose coefficient of magnetization is K. On Stratified Conductors. 319.] Let a conductor be composed of alternate strata of thick- ness c and c' of two substances whose coefficients of conductivity are different. Required the coefficients of resistance and" conduc- tivity of the compound conductor. Let the plane of the strata be normal to Z. Let every symbol relating to the strata of the second kind be accented, and let every symbol relating to the compound conductor be marked with a bar thus, T. Then T= X = X', (c + c')u = cu 4- cV, T= Y = 7', We must first determine u, u , v, v' 9 Z and Z' in terms of J, Tand w from the equations of resistance, Art. 297, or those * See Sir W. Thomson's 'Note on Induced Magnetism in a Plate,' Camb. and Dub. Math. Journ., Nov. 1845, or Reprint, art. ix. 156. 408 CONDUCTION. IN: HETEROGENEOUS MEDIA. [320-. of conductivity, Art. 298. If we put D for the determinant of the coefficients of resistance, we find Similar equations with, the symbols accented give the values of u , if and Zf . Having 1 found u, v and w in terms of X, Y and Z, we may write down the equations of conductivity of the stratified c c' If we make k = and h'= -,, we find h + h> h+h' cps + c'pj kJi(q,q, #3 = _ eg* + c'q. 3 ' M'(p 2 -p z ') fe- c + c' (h+h')(c + c f ) c + c' 320.] If neither of the two substances of which the strata are formed has the rotatory property of Art. 303, the value of any P or p will be equal to that of its corresponding Q or q. From this it follows that in the stratified conductor also Pi = ?i Pz = 2* Ps = ?3> or there is no rotatory property developed by stratification, unless it exists in the materials. 321.] If we now suppose that there is no rotatory property, and also that the axes of x, y and z- are the principal axes, then the p and q coefficients vanish, and _ c I o If we begin with both substances isotropic, but of different 322.] STRATIFIED CONDUCTORS. 409 conductivities, then the result of stratification will be to make the resistance greatest in the direction of a normal to the strata, and the resistance in all directions in the plane of the strata will be equal. 322.] Take an isotropic substance of conductivity r, cut it into exceedingly thin slices of thickness , and place them alternately with slices of a substance whose conductivity is and k^c in that of z. The conductivities of the conductor so formed in the directions of x, y y and z are to be found by three applications in order of the results of Art. 321. We thereby obtain _ (L+ (1 -f- The accuracy of this investigation depends upon the three dimen- sions of the parallelepipeds being of different orders of magnitude, so that we may neglect the conditions to be fulfilled at their edges and angles. If we make 1} 2 an ^ ^3 eac ^ 3r+5s : If r = 0, that is, if the medium of which the parallelepipeds are made is a perfect insulator, then 410 CONDUCTION IN HETEROGENEOUS MEDIA. [323. If r oo, that is, if the parallelepipeds are perfect conductors, r i = T s > r 2 == f *> r 3 = 2s. In every case, provided ^ 2 / 3 , it may be shewn that r lt r 2 and r 3 are in ascending order of magnitude, so that the greatest conductivity is in the direction of the longest dimensions of the parallelepipeds, and the greatest resistance in the direction of their shortest dimensions. 323.] In a rectangular parallelepiped of a conducting solid, let there be a conducting channel made from one angle to the opposite, the channel being a wire covered with insulating material, and let the lateral dimensions of the channel be so small that the conductivity of the solid is not affected except on account of the current conveyed along the wire. Let the dimensions of the parallelepiped in the directions of the coordinate axes be a, b } c, and let the conductivity of the channel, extending from the origin to the point (a be), be abcK. The electromotive force acting between the extremities of the channel is aX+bY+ cZ, and if C' be the current along the channel C'=Kabc(aX+bY+cZ). The current across the face be of the parallelepiped is 6 en, and this is made up of that due to the conductivity of the solid and of that due to the conductivity of the channel, or leu = bc or u = (/-! + Ka 2 ) X+ ( j 3 + Ka b) Y+ (q. 2 + Kca) Z. In the same way we may find the values of v and w. The coefficients of conductivity as altered by the effect of the channel will be Pt + Kbc, jp^ + Kca, q l + Kb c, 2 -f Kca, In these expressions, the additions to the values of p l9 &c., due to the effect of the channel, are equal to the additions to the values of q 1 , &c. Hence the values of p^ and q t cannot be rendered unequal by the introduction of linear channels into every element of volume of the solid, and therefore the rotatory property of Art. 303, if it does not exist previously in a solid, cannot be introduced by such means. 324.] COMPOSITE CONDUCTOR. 411 3.24.] To construct a framework of linear conductors which shall have any given coefficients of conductivity forming a symmetrical system. Let the space be divided into equal small cubes, of which let the figure represent one. Let the coordinates of the points 0, L, M, N t and their potentials be as follows : x y z Potential Lt 000 X+Y+Z L 1 1 X 1/101 Y N 1 I Z Let these four points be connected by six conductors, OL, OM, ON, MN, NL, LM, of which the conductivities are respectively A, 3, C, P, Q, X, The electromotive forces along these conductors will be Y+Z, Z+X, J+7, 7-Z, Z-X, X-7, and the currents A(7+Z), 3(Z + X) 9 C(X+7), P(Y-Z], Q(Z-X), R(X-7). Of these currents, those which convey electricity in the positive direction of x are those along LH, LN, OM and ON, and the quantity conveyed is / 71 , fv \ f) i f}\ ~y i [ /~i T?\V j_ / 7? _ d\ P Similarly v = (CR)X +(C+A w=(3-Q)X +(A-P)Y whence we find by comparison with the equations of conduction, Art. 298, 4 A = T., + n /i + 2,, 4P = ro + r, r, 20,, CHAPTER X. CONDUCTION IN DIELECTRICS. 325.] WE have seen that when electromotive force acts on a dielectric medium it produces in it a state which we have called electric polarization, and which we have described as consisting of electric displacement within the medium in a direction which, in isotropic media, coincides with that, of the electromotive force, combined with a superficial charge on every element of volume into which we may suppose the dielectric divided, which is negative on the side towards which the force acts, and positive on the side from which it acts. When electromotive force acts on a conducting, medium it also produces what is called an electric current. Now dielectric media, with very few, if any, exceptions, are also more or less imperfect conductors, and many media which are not good insulators exhibit phenomena of dielectric induction. Hence we are led to study the state of a medium in which induction and conduction are going on at the same time. For simplicity we shall suppose the medium isotropic at every point, but not necessarily homogeneous at different points. In this case, the equation of Poisson becomes,- by Art. 83, d / rrdY\ d ,^dV\ d / ^dV\ -r (K-r) + -7-(A.T-J + -T-(K-7-) + 4'np=0, (l) ax ^ ax' dy ^ dy/ dz v &&' where K is the ( specific inductive capacity.' The f equation of continuity' of electric currents becomes dx^r dos dy\r dy* dz.^r dz' dt where r is the specific resistance referred to unit of volume. When K or r is discontinuous, these equations must be trans- formed into those appropriate to surfaces of discontinuity. 326.] THEORY OF A CONDENSER. 413 In a strictly homogeneous medium r and K are both constant, so that we find P dp whence p = Ce * r '; (4) Kr -- or, if we put T = , p = Ce T . (5) This result shews that under the action of any external electric forces on a homogeneous medium, the interior of which is originally charged in any manner with electricity, the internal charges will die away at a rate which does not depend on the external forces, so that at length there will be no charge of electricity within the medium, after which no external forces can either produce or maintain a charge in any internal portion of the medium, pro- vided the relation between electromotive force, electric polarization and conduction remains the same. When disruptive discharge occurs these relations cease to be true, and internal charge may be produced. On Conduction through a Condenser. 326.] Let C be the capacity of a condenser, R its resistance, and E the electromotive force which acts on it, that is, the difference of potentials of the surfaces of the metallic electrodes. Then the quantity of electricity on the side from which the electromotive force acts will be CU, and the current through the substance of the condenser in the direction of the electromotive force will be If the electrification is supposed to be produced by an electro- motive force E acting in a circuit of which the condenser forms part, and if -~ represents the current in that circuit, then do W = *L + C - (6) Let a battery of electromotive force E Q and resistance r t be introduced into this circuit, then Hence, at any time t l9 ' where Z\=-^-. (8) 414 CONDUCTION IN DIELECTRICS. [3 2 7 Next, let the circuit r t be broken for a time t 2 , -h. E( = E 2 ) = E^ T Z where T 2 = CR. (9) Finally, let the surfaces of the condenser be connected by means of a wire whose resistance is r s for a time t z) E( = Z t ) = E t *~* where T 3 = g^ . (10) If 3 is the total discharge through this wire in the time t 3 , In this way we may find the discharge through a wire which is made to connect the surfaces of a condenser after being charged for a time tf l5 and then insulated for a time t 2 . If the time of charging is sufficient, as it generally is, to develope the whole charge, and if the time of discharge is sufficient for a complete discharge, the discharge is 327.] In a condenser of this kind, first charged in any way, next discharged through a wire of small resistance, and then insulated, no new electrification will appear. In most actual condensers, however, we find that after discharge and insulation a new charge is gradually developed, of the same kind as the original charge, but inferior in intensity. This is called the residual charge. To account for it we must admit that the constitution of the dielectric medium is different from that which we have just described. We shall find, however, that a medium formed of a conglomeration of small pieces of different simple media would possess this property. Theory of a Composite Dielectric. 328.] We shall suppose, for the sake of simplicity, that the dielectric consists of a number of plane strata of different materials and of area unity, and that the electric forces act in the direction of the normal to the strata. Let # 15 a z , &c. be the thicknesses of the different strata. X 1) X 2 , &c. the resultant electrical forces within the strata. p l9 _p 2 , &c. the currents due to conduction through the strata. f\>fi-> & c> the electric displacements. u lt 2 , &c. the total currents, due partly to conduction and partly to variation of displacement. STKATIFIEI) DIELECTRIC. 415 r lt r 2 , &c. the specific resistances referred to unit of volume. K lt K^ &c. the specific inductive capacities. k^, k z , &c. the reciprocals of the specific inductive capacities. E the electromotive force due to a voltaic battery, placed in the part of the circuit leading from the last stratum towards the first, which we shall suppose good conductors. Q the total quantity of electricity which has passed through this part of the circuit up to the time t. E Q the resistance of the battery with its connecting wires. o- 12 the surface-density of electricity on the surface which separates the first and second strata. Then in the first stratum, we have, by Ohm's Law, I l = r l p 1 . (1) By the theory of electrical displacement, X lL = ^k l f l . (2) By the definition of the total current, ' (3) with similar equations for the other strata, in each of which the quantities have the suffix belonging to that stratum. To determine the surface-density on any stratum, we have an equation of the form ^ = / 2 /j, (4) and to determine its variation we have By differentiating (4) with respect to t, and equating the result to (5), we obtain 1 -* () or, by taking account of (3), ,_ = u 2 =. &c. = u. (7) That is, the total current u is the same in all the strata, and is equal to the current through the wire and battery. We have also, in virtue of equations (l) and (2), 1 1 dX l (9) from which we may find X l by the inverse operation on u, " 416 CONDUCTION IN DIELECTRICS. [329. The total electromotive force E is ^=0 1 J 1 + 2 I 2 + &c., (10) r ~ II an equation between E, the external electromotive force, and u, the external current. If the ratio of r to k is the same in all the strata, the equation reduces itself to JT . (12) which is the case we have already examined, and in which, as we found, no phenomenon of residual charge can take place. If there are n substances having different ratios of r to Jc, the general equation (11), when cleared of inverse operations, will be a linear differential equation, of the nth order with respect to E and of the (n l)th order with respect to u f t being the independent variable. From the form of the equation it is evident that the order of the different strata is indifferent, so that if there are several strata of the same substance we may suppose them united into one without altering the phenomena. 329.] Let us now suppose that at first f-^f^ &c. are all zero, and that an electromotive force E is suddenly made to act, and let us find its instantaneous effect. Integrating (8) with respect to t, we find q =Judt = ^-j'x i dt+ _-L-.r l + const. (13) Now, since X : is always in this case finite, / X-^dt must be in- sensible when t is insensible, and therefore, since X x is originally zero, the instantaneous effect will be . (14) Hence, by equation (10), ^=477 (Vl + V 2 + &O & (15) and if C be the electric capacity of the system as measured in this instantaneous way, 329.] ELECTRIC 'ABSORPTION/ 417 This is the same result that we should have obtained if we had neglected the conductivity of the strata. Let us next suppose that the electromotive force E is continued uniform for an indefinitely long time, or till a uniform current of conduction equal to p is established through the system. We have then X 1 = r^p^ etc., and therefore by (10), # = fa i + r 2 a 2 + &c.)j3. ( 1 7) If It be the total resistance of the system, E R = = r x x + r 2 2 + &c. (18) In this state we have by (2), so that ^=(>- (19) If we now suddenly connect the extreme strata by means of a conductor of small resistance, E will be suddenly changed from its original value 2$ to zero, and a quantity Q of electricity will pass through the conductor. To determine Q we observe that if X be the new value of X 1 , then by (13), X/ = X l + 4 TT^ Q. (20) Hence, by (10), putting E = 0, = 1 X 1 + &c. + 47r(a 1 1 + 2 2 + &c.)Q, (21) or = E +Q. (22) Hence Q = CE Q where C is the capacity, as given by equation (IB). The instantaneous discharge is therefore equal to the in- stantaneous charge. Let us next suppose the connexion broken immediately after this discharge. We shall then have u 0, so that by equation (8), 47T*! , Zi = re ^ , (23) where X' is the initial value after the discharge. Hence, at any time t, The value of E at any time is therefore VOL. I. EC 418 CONDUCTION IN DIELECTEICS. [330. and the instantaneous discharge after any time t is EG. This is called the residual discharge. If the ratio of r to li is the same for all the strata, the value of E will be reduced to zero. If, however, this ratio is not the same, let the terms be arranged according to the values of this ratio in descending order of magnitude. The sum of all the coefficients is evidently zero, so that when t = 0, E = 0. The coefficients are also in descending order of magnitude, and so are the exponential terms when t is positive. Hence, when t is positive, E will be positive, so that the residual discharge is always of the same sign as the primary discharge. When t is indefinitely great all the terms disappear unless any of the strata are perfect insulators, in which case r-^ is infinite for that stratum, and R is infinite for the whole system, and the final value of E is not zero but E = E^l-iTia^C). (25) Hence, when some, but not all, of the strata are perfect insulators, a residual discharge may be permanently preserved in the system. 330.] We shall next determine the total discharge through a wire of resistance RQ kept permanently in connexion with the extreme strata of the system, supposing the system first charged by means of a long-continued application of the electromotive force E. At any instant we have E = a l r 1 p l + a 2 r 2 p 2 + 8cc.+R Q u = 0, (26) and also, by (3), U =p 1 +-~j. (27) Hence (R + A>) = i *i f^ + 2 '2 ^ + &c. (28) Integrating with respect to t in order to find Q, we get (R + )$ = a, r, (/I'-/,) + a, r, (//-/ 2 ) + &c., (29) where/! is the initial, and// the final value of/i- In this case//= 0, and by (2) and (20) / t = Hence (S + ^Q = + + &c. -E.CR, (30) where the summation is extended to all quantities of this form belonging to every pair of strata. 33 1 -] RESIDUAL DISCHARGE. 419 It appears from this that Q is always negative, that is to say, in the opposite direction to that of the current employed in charging the system. This investigation shews that a dielectric composed of strata of different kinds may exhibit the phenomena known as electric absorption and residual discharge, although none of the substances of which it is made exhibit these phenomena when alone. An investigation of the cases in which the materials are arranged otherwise than in strata would lead to similar results, though the calculations would be more complicated, so that we may conclude that the phenomena of electric absorption may be ex- pected in the case of substances composed of parts of different kinds, even though these individual parts should be microscopically small. It by no means follows that every substance which exhibits this phenomenon is so composed, for it may indicate a new kind of electric polarization of which a homogeneous substance may be capable, and this in some cases may perhaps resemble electro- chemical polarization much more than dielectric polarization. The object of the investigation is merely to point out the true mathematical character of the so-called electric absorption, and to shew how fundamentally it differs from the phenomena of heat which seem at first sight analogous. 331.] If we take a thick plate of any substance and heat it on one side, so as to produce a flow of heat through it, and if we then suddenly cool the heated side to the same temperature as the other, and leave the plate to itself, the heated side of the plate will again become hotter than the other by conduction from within. Now an electrical phenomenon exactly analogous to this can be produced, and actually occurs in telegraph cables, but its mathe- matical laws, though exactly agreeing with those of heat, differ entirely from those of the stratified condenser. In the case of heat there is true absorption of the heat into the substance with the result of making it hot. To produce a truly analogous phenomenon in electricity is impossible, but we may imitate it in the following way in the form of a lecture- room experiment. Let A l , A 2 , &c. be the inner conducting surfaces of a series of condensers, of which i? , -Z? 15 2t &c. are the outer surfaces. Let A lt A 2 , &c. be connected in series by connexions of rcsist- 420 CONDUCTION IN DIELECTRICS. ance E t and let a current be passed along this series from left to right. Let us first suppose the plates _Z? , JB lt J5 2 , each insulated and free from charge. Then the total quantity of electricity on each of the plates JB must remain zero, and since the electricity on the plates A is in each case equal and opposite to that of the opposed Fig. 26. surface they will not be electrified, and no alteration of the current will be observed. But let the plates IB be all connected together, or let each be connected with the earth. Then, since the potential of A l is positive, while that of the plates B is zero, A 1 will be positively electrified and Z? x negatively. If P 19 P 2 , &c. are the potentials of the plates A l} A 2 , &c. } and C the capacity of each, and if we suppose that a quantity of electricity equal to Q Q passes through the wire on the left, Qi through the connexion R lt and so on, then the quantity which exists on the plate A-L is Q Q 13 and we have .-!= 43- Similarly &-&= G A> and so on. But by Ohm's Law we have If we suppose the values of C the same for each plate, and those of R the same for each wire, we shall have a series of equations of the form 332.] THEORY OF ELECTKIC CABLES. 421 If there are n quantities of electricity to be determined, and if either the total electromotive force, or some other equivalent con- ditions be given, the differential equation for determining 1 any one of them will be linear and of the nih order. By an apparatus arranged in this way, Mr. Varley succeeded in imitating the electrical action of a cable 12,000 miles long. When an electromotive force is made to act along the wire on the left hand, the electricity which flows into the system is at first principally occupied in charging the different condensers beginning with A lt and only a very small fraction of the current appears at the right hand till a considerable time has elapsed. If galvano- meters be placed in circuit at R-^ R 2 , &c. they will be affected by the current one after another, the interval between the times of equal indications being greater as we proceed to-the right. 332.] In the case of a telegraph cable the conducting wire is separated from conductors outside by a cylindrical sheath of gutta- percha, or other insulating material* Each portion, of the cable thus becomes a condenser, the outer surface of which is always at potential zero. Hence, in a given portion of the cable, the quantity of free electricity at the surface of the conducting- wire is equal to the product of the potential into the capacity of the portion of the cable considered as a condenser. If a 13 a 2 are the outer and inner radii of the insulating sheath, and if K is its specific dielectric capacity, the capacity of unit of length of the cable is, by Art. 126, Let v be the potential at any point of the wire, which we may consider as the same at every part of the same section. Let Q be the total quantity of electricity which has passed through that section since the beginning of the current. Then the quantity which at the time t exists between sections at x and at is . dO v d or - and this is, by what we have said, equal to cvbx. 422 CONDUCTION IN DIELECTRICS. [333. Hence " = ~ 5T ( 2 ) Again, the electromotive force at any section is -j-, and by Ohm's Law, dv . dQ ~s=*ir ( 3 ) where k is the resistance of unit of length of the conductor, and - is the strength of the current. Eliminating Q between (2) and (3), we find c ^ = ^. (4) This is the partial differential equation which must be solved in order to obtain the potential at any instant at any point of the cable. It is identical with that which Fourier gives to determine the temperature at any point of a stratum through which heat is flowing in a direction normal to the stratum. In the case of heat c represents the capacity of unit of volume, or -what Fourier denotes by CD, and k represents the reciprocal of the conductivity. If the sheath is not a perfect insulator, and if k is the resist- ance of unit of length of the sheath to conduction through it in a radial direction, then if p L is the specific resistance of the insulating material, it is easy to shew that The equation (2) will no longer be true, for the electricity is expended not only in charging the wire to the extent represented by cv, but in escaping at a rate represented by . Hence the rate of expenditure of electricity will be I whence, by comparison with (3), we get dv d 2 v k . . and this is the equation of conduction of heat in a rod or ring as given by Fourier *. 333.] If we had supposed that a body when raised to a high potential becomes electrified throughout its substance as if elec- tricity were compressed into it, we should have arrived at equa- tions of this very form. It is remarkable that Ohm himself, * Theorie de la Ckaleur, Art. 105. 334-] HYDROSTATICAL ILLUSTRATION. 423 -A - - -o - misled by the analogy between electricity and heat, entertained an opinion of this kind, and was thus, by means of an erroneous opinion, led to employ the equations of Fourier to express the true laws of conduction of electricity through a long wire, long before the real reason of the appropriateness of these equations had been suspected. Mechanical Illustration of the Properties of a Dielectric. 334.] Five tubes of equal sectional area A, B, C, D and P are arranged in circuit as in the figure. A, B, C and D are vertical and equal, jr f* P *5^\ and P is horizontal. The lower halves of A, B, C, D are filled with mercury, their upper halves and the horizontal tube P are filled with water. A tube with a stopcock Q con- nects the lower part of A and B with that of C and _Z>, and a piston P is made to slide in the horizontal tube. Let us begin by supposing that the level of the mercury in the four tubes is the same, and that it is indicated by A^ B Q , <7 , D , that the piston is at P , and that the stopcock Q is shut. Now let the piston be moved from P to P lt a distance a. Then, since the sections of all the tubes are equal, the level of the mercury in A and C will rise a distance a, or to A l and C lt and the mercury in B and D will sink an equal distance #, or to B l and D . The difference of pressure on the two sides of the piston will be represented by 4 a. This arrangement may serve to represent the state of a dielectric acted on by an electromotive force 4 a. The excess of water in the tube D may be taken to represent a positive charge of electricity on one side of the dielectric, and the excess of mercury in the tube A may represent the negative charge on the other side. The excess of pressure in the tube P on the side of the piston next D will then represent the excess of potential on the positive side of the dielectric. ( ^1 1^ ~\ - C - 1 -v ~ B o~ v -v -*, : Q Fig. 27. 424 CONDUCTION IN DIELECTRICS, [334- If the piston is free to move it will move back to P and be in equilibrium there. This represents the complete discharge of the dielectric. During the discharge there is a reversed motion of the liquids throughout the whole tube, and this represents that change of electric displacement which we have supposed to take place in a dielectric. I have supposed every part of the system of tubes filled with incompressible liquids, in order to represent the property of all electric displacement that there is no real accumulation of elec- tricity at any place. Let us now consider the effect of opening the stopcock Q while the piston P is at P t . The level of A and D L will remain unchanged, but that of B and C will become the same, and will coincide with JB and C . The opening of the stopcock Q corresponds to the existence of a part of the dielectric which has a slight conducting power, but which does not extend through the whole dielectric so as to form an open channel. The charges on the opposite sides of the dielectric remain in- sulated, but their difference of potential diminishes. In fact, the difference of pressure on the two sides of the pi&ton sinks from \a to 2 a during the passage of the fluid through Q. If we now shut the stopcock Q and allow the piston P to move freely, it will come to equilibrium at a point P 2 , and the discharge will be apparently only half of the charge. The level of the mercury in A and B will be \a above its original level, and the level in the tubes C and D will be \a below its original level. This is indicated by the levels A 2 , _Z? 2 , C 2 , A- If the piston is now fixed and the stopcock opened, mercury will flow from to C till the level in the two tubes, is again at JB Q and C . There will then be a difference of pressure == a on the two sides of the piston P. If the stopcock is then closed and the piston P left free to move, it will again come to equilibrium at a point P B , half way between P 2 and P . This corresponds to the residual charge which is observed when a charged dielectric is first dis- charged and then left to itself. It gradually recovers part of its charge, and if this is again discharged a third charge is formed, the successive charges diminishing in quantity. In the case of the illustrative experiment each charge is half of the preceding, and the 334-] HYDROSTATICAL ILLUSTRATION. 425 discharges, which are J, J, &c. of the original charge, form a series whose sum is equal to the original charge. If, instead of opening and closing the stopcock, we had allowed it to remain nearly, but not quite, closed during the whole experiment, we should have had a case resembling that of the electrification of a dielectric which is a perfect insulator and yet exhibits the pheno- menon called ' electric absorption.' To represent the case in which there is true conduction through the dielectric we must either make the piston leaky, or we must establish a communication, between the top of the tube A and the top of the tube D. In this way we may construct a mechanical illustration of the properties of a dielectric of any kind, in which the two electricities are represented by two real fluids, and the electric potential is represented by fluid pressure. Charge and discharge are repre- sented by the motion of the piston P, and electromotive force by the resultant force on the piston. CHAPTEK XT. THE MEASUREMENT OF ELECTRIC RESISTANCE. 335.] IN the present state of electrical science, the determination of the electric resistance of a conductor may be considered as the cardinal operation in electricity, in the same sense that the deter- mination of weight is the cardinal operation in chemistry. The reason of this is that the determination in absolute measure of other electrical magnitudes, such as quantities of electricity, electromotive forces, currents, &c., requires in each case a com- plicated series of operations, involving generally observations of time, measurements of distances, and determinations of moments of inertia, and these operations, or at least some of them, must be repeated for every new determination, because it is impossible to preserve a unit of electricity, or of electromotive force, or of current, in an unchangeable state, so as to be available for direct comparison. But when the electric resistance of a properly shaped conductor of a properly chosen material has been once determined, it is found that it always remains the same for the same temperature, so that the conductor may be used as a standard of resistance, with which that of other conductors can be compared, and the comparison of two resistances is an operation which admits of extreme accuracy. When the unit of electrical resistance has been fixed on, material copies of this unit, in the form of ' Resistance Coils, 5 are prepared for the use of electricians, so that in every part of the world electrical resistances may be expressed in terms of the same unit. These unit resistance coils are at present the only examples of material electric standards which can be preserved, copied, and used for the purpose of measurement. Measures of electrical capacity, which are also of great importance, are still defective, on account of the disturbing influence of electric absorption. 336.] The unit -of resistance may be an entirely arbitrary one, as in the case of Jacobi's Etalon, which was a certain copper wire of 22.4932 grammes weight, 7.61975 metres length, and 0.667 339-] STANDARDS OF RESISTANCE. 427 millimetres diameter. Copies of this have been made by Leyser of Leipsig, and are to be found in different places. According to another method the unit may be defined as the resistance of a portion of a definite substance of definite dimensions. Thus, Siemens' unit is defined as the Tesistance of a column of mercury of one metre long-, and one square millimetre section, at the temperature 0C. 337.] Finally, the unit may be defined with reference to the electrostatic or the electromagnetic system of units. In practice the electromagnetic system is used in all telegraphic operations, and therefore the only systematic units actually in use are those of this system. In the electromagnetic system, as we shall shew at the proper place, a resistance is a quantity homogeneous with a velocity, and may therefore be expressed as a velocity. See Art. 628. 338.] The first actual measurements on this system were made by Weber, who employed as his unit one millimetre per second. Sir W. Thomson afterwards used one foot per second as a unit, but a large number of electricians have now agreed to use the unit of the British Association, which professes to represent a resistance which, expressed *as a velocity, is ten millions of metres per second. The magnitude of this unit is -more convenient than that of Weber's unit, which is too small. It is sometimes referred to as the B.A. unit, but in order to connect it with the name of the discoverer of the iaws of resistance, it is called the Ohm. 339.] To recollect its value in absolute measure it is useful to know that ten millions of metres is professedly the distance from the pole to the equator, measured along the meridian of Paris. A body, therefore, which in one second travels along a meridian from the pole to the equator would have a velocity which, on the electromagnetic system, is professedly represented by an Ohm. I say professedly, because, if more accurate researches should prove that the Ohm, as constructed from the British Association's material standards, is not really represented by this velocity, elec- tricians would not alter their standards, but would apply a cor- rection. In the same way the metre is professedly one ten-millionth of a certain quadrantal arc, but though this is found not to be exactly true, the length of the metre has not been altered, but the dimensions of the earth are expressed by a less simple number. According to the system of the British Association, the absolute value of the unit is originally chosen so as to represent as nearly 428 MEASUREMENT OF RESISTANCE. [340. as possible a quantity derived from the electromagnetic absolute system. 340.] When a material unit representing this abstract quantity has been made, other standards are constructed by copying this unit, a process capable of extreme accuracy of much greater accuracy than, for instance, the copying of foot-rules from a standard foot. These copies, made of the most permanent materials, are dis- tributed over all parts of the world, so that it is not likely that any difficulty will be found in obtaining copies of them if the original standards should be lost. But such units as that of Siemens can without very great labour be reconstructed with considerable accuracy, so that as the relation of the Ohm to Siemens unit is known, the Ohm can be reproduced even without having a standard to copy, though the labour is much greater and the accuracy much less than by the method of copying. Finally, the Ohm may be reproduced by the electromagnetic method by which it was originally determined. This method, which is considerably more laborious than the determination of a foot from the seconds pendulum, is probably inferior in accuracy to that last mentioned. On the other hand, the determination of the electromagnetic unit in terms of the Ohm with an amount of accuracy corresponding to the progress of electrical science, is a most important physical research and well worthy of being repeated. The actual resistance coils constructed to represent the Ohm were made of an alloy of two parts of silver and one of pla- tinum in the form of wires. from .5 milli- metres to .8 millimetres diameter, and from one to two metres in length. These wires were soldered to stout copper electrodes. The wire itself was covered with two layers of silk, imbedded in solid paraffin, and. enclosed in a thin brass case, so that it can be easily brought to a temperature at which its resistance is accurately one Ohm. This temperature is marked on the insulating support of the coil. (See Fig. 28.) Fig. 28. 34 1 -] RESISTANCE COILS. 429 On the Forms of Resistance Coils. 341 .] A Resistance Coil is a conductor capable of being easily placed in the voltaic circuit, so as to introduce into the circuit a known resistance. The electrodes or ends of the coil must be such that no appre- ciable error may arise from the mode of making the connexions. For resistances of considerable magnitude it is sufficient that the electrodes should be made of stout copper wire or rod well amal- gamated with mercury at the ends, and that the ends should be made to press on flat amalgamated copper surfaces placed in mercury cups. For very great resistances it is sufficient that the electrodes should be thick pieces of brass, and that the connexions should be made by inserting a wedge of brass or copper into the interval between them. This method is found very convenient. The resistance coil itself consists of a wire well covered with silk, the ends of which are soldered permanently to the elec- trodes. The coil must be so arranged that its temperature may be easily observed. For this purpose the wire is coiled on a tube and covered with another tube, so that it may be placed in a vessel of water, and that the water may have access to the inside and the outside of the coil. To avoid the electromagnetic effects of the current in the coil the wire is first doubled back on itself and then coiled on the tube, so that at every part of the coil there are equal and opposite currents in the adjacent parts of the wire. When it is desired to keep two coils at the same temperature the wires are sometimes placed side by side and coiled up together. This method is especially useful when it is more important to secure equality of resistance than to know the absolute value of the resistance, as in the case of the equal arms of Wheatstone's Bridge, (Art. 347). When measurements of resistance were first attempted, a resist- ance coil, consisting of an uncovered wire coiled in a spiral groove round a cylinder of insulating material, was much used. It was called a Rheostat. The accuracy with which it was found possible to compare resistances was soon found to be inconsistent with the use of any instrument in which the contacts are not more perfect than can be obtained in the rheostat. The rheostat, however, is 430 MEASUREMENT OF RESISTANCE, [342. still used for adjusting the resistance where accurate measurement is not required. Resistance coils are generally made of those metals whose resist- ance is greatest and which vary least with temperature. German silver fulfils these conditions very well, but some specimens are found to change their properties during the lapse of years. Hence, for standard coils, several pure metals, and also an alloy of platinum and silver, have been employed, and the relative resistance of these during several years has been, found constant up to the limits of modern accuracy. 342.] For very great resistances, such as several millions of Ohms, the wire must be either very long or very thin, and the construction of the coil is expensive and difficult. Hence tellurium and selenium have been proposed as materials for constructing standards of great resistance. A very ingenious and easy method of construction has been lately proposed by Phillips *. On a piece of ebonite or ground glass a fine pencil-line is drawn. The ends of this filament of plumbago are connected to metallic electrodes, and the whole is then covered with insulating varnish. If it should be found that the resistance of such a pencil-line remains constant, this will be the best method of obtaining a resistance of several millions of Ohms. 343.] There are various arrangements by which resistance coils may be easily introduced into a circuit. For instance, a series of coils of which the resistances are 1, 2, 4, 8, 16, &c., arranged according to the powers of 2, may be placed in a box in series. The electrodes consist of stout brass plates, so arranged on the outside of the box that by inserting a brass plug or wedge between * Phil Mag., July, 1870. 344-] RESISTANCE BOXES. 431 two of them as a shunt, the resistance of the corresponding coil may be put out of the circuit. This arrangement was introduced by Siemens. Each interval between the electrodes is marked with the resist- ance of the corresponding coil, so that if we wish to make the resistance box equal to 107 we express 107 in the binary scale as 64 + 32 + 8 + 2 + 1 or 1101011. We then take the plugs out of the holes corresponding to 64, 32, 8, 2 and 1, and leave the plugs in 16 and 4. This method, founded on the binary scale, is that in which the smallest number of separate coils is needed, and it is also that which can be most readily tested. For if we have another coil equal to 1 we can test the equality of 1 and l', then that of 1 + 1' and 2, then that of 1 + 1' + 2 and 4, and so on. The only disadvantage of the arrangement is that it requires a familiarity with the binary scale of notation, which is not generally possessed by those accustomed to express every number in the decimal scale. 344.] A box of resistance coils may be arranged in a different way for the purpose of mea- suring conductivities instead of resistances. The coils are placed so that one end of each is connected with a long thick piece of metal which forms one elec- trode of the box, and the other Fig. 30. end is connected with a stout piece of brass plate as in the former case. The other electrode of the box is a long brass plate, such that by inserting brass plugs between it and the electrodes of the coils it may be connected to the first electrode through any given set of coils. . The conductivity of the box is then the sum of the con- ductivities of the coils. In the figure, in which the resistances of the coils are 1, 2, 4, &c., and the plugs are inserted at 2 and 8, the conductivity of the box is 1 + 1- = f , and the resistance of the box is therefore or 1.6. This method of combining resistance coils for the measurement of fractional resistances was introduced by Sir W. Thomson under the name of the method of multiple arcs. See Art. 276. 324 MEASUREMENT OF EESISTANCE. [345- On the Comparison of Resistances. 345.] If E is the electromotive force of a battery, and R the resistance of the battery and its connexions, including the galvan- ometer used in measuring the current, and if the strength of the current is 7 when the battery connexions are closed, and I 13 7 2 when additional resistances r lt r 2 are introduced into the circuit, then, by Ohm's Eliminating U, the electromotive force of the battery, and R the resistance of the battery and its connexions, we get Ohm's formula ^ _ (7-7 T )7 2 This method requires a measurement of the ratios of /, 7 t and 7 2 , and this implies a galvanometer graduated for absolute mea- surements. If the resistances r^ and r 2 are equal, then 2 1 and 7 2 are equal, and we can test the equality of currents by a galvanometer which is not capable of determining their ratios. But this is rather to be taken as an example of a faulty method than as a practical method of determining resistance. The electro- motive force E cannot be maintained rigorously constant, and the internal resistance of the battery is also exceedingly variable, so that any methods in which these are assumed to be even for a short time constant are not to be depended on. 346.] The comparison of resistances can be made with extreme accuracy by either of two methods, in which the result is in- dependent of variations of R and JE. 346.] COMPARISON OP RESISTANCES. 433 The first of these methods depends on the use of the differential galvanometer, an instrument in which there are two coils, the currents in which are independent of each other, so that when the currents are made to flow in opposite directions they act in opposite directions on the needle, and when the ratio of these currents is that of m to n they have no resultant effect on the galvanometer needle. Let I I , 7 2 be the currents through the two coils of the galvan- ometer, then the deflexion of the needle may be written 8 = m! 1 nI 2 . Now let the battery current I be divided between the coils of the galvanometer, and let resistances A and B be introduced into the first and second coils respectively. Let the remainder of the resistance of the coils and their connexions be a and ft respect- ively, and let the resistance of the battery and its connexions between C and D be r, and its electromotive force E. Then we find, by Ohm's Law, for the difference of potentials between C and D, and since 7? + /3 ET -- _- _ where D = The deflexion of the galvanometer needle is therefore and if there is no observable deflexion, then we know that the quantity enclosed in brackets cannot differ from zero by more than a certain small quantity, depending on the power of the battery, the suitableness of the arrangement, the delicacy of the galvano- meter, and the accuracy of the observer. Suppose that B has been adjusted so that there is no apparent deflexion. Now let another conductor A' be substituted for A^ and let A' be adjusted till there is no apparent deflexion. Then evidently to a first approximation A'= A. To ascertain the degree of accuracy of this estimate, let the altered quantities in the second observation be accented, then VOL. i. r f 434 MEASUKEMENT OF EESISTANCE. [346. ~ Hence n(A'-A) = b-~b'. If b and S', instead of being both apparently zero, had been only observed to be equal, then, unless we also could assert that E = W^ the right-hand side of the equation might not be zero. In fact, the method would be a mere modification of that already described. The merit of the method consists in the fact that the thing observed is the absence of any deflexion, or in other words, the method is a Null method, one in which the non-existence of a force is asserted from an observation in which the force, if it had been different from zero by more than a certain small amount, would have produced an observable effect. Null methods are of great value where they can be employed, but they can only be employed where we can cause two equal and opposite quantities of the same kind to enter into the experiment together. In the case before us both b and b' are quantities too small to be observed, and therefore any change in the value of E will not affect the accuracy of the result. The actual degree of accuracy of this method might be ascer- tained by making a number of observations in each of which A' is separately adjusted, and comparing the result of each observation with the mean of the whole series. But by putting A' out of adjustment by a known quantity, as, for instance, by inserting at A or at B an additional resistance equal to a hundredth part of A or of JB, and then observing the resulting deviation of the galvanometer needle, we can estimate the number of degrees corresponding to an error of one per cent. To find the actual degree of precision we must estimate the smallest deflexion which could not escape observation, and compare it with the deflexion due to an error of one per cent. * If the comparison is to be made between A and -8, and if the positions of A and JB are exchanged, then the second equation becomes * This investigation is taken from Weber's treatise on Galvanometry. Gottingen Transactions, x. p. 65. DIFFERENTIAL GALVANOMETER. 435 ) = ~b' y whence (m + n) (BA) = ~ b - ^ 6'. j j If ^ and n, A and .#, a and /3 are approximately equal, then Here 5-5' may be taken to be the smallest observable deflexion of the galvanometer. If the galvanometer wire be made longer and thinner, retaining the same total mass, then n will vary as the length of the wire and a as the square of the length. Hence there will be a minimum value of M + ')^ + + 2r) when If we suppose r, the battery resistance, small compared with A, this gives a=$A; or, the resistance of each coil of the galvanometer should oe one-third of the resistance to lie measured. We then find 2 If we allow the current to flow through one only of the coils of the galvanometer, and if the deflexion thereby produced is A (supposing the deflexion strictly proportional to the deflecting force), then mE 3nE . 1 . A =s -T- -- = - if / = and a = - A. A + a + r 4 A 3 Hence In the differential galvanometer two currents are made to produce equal and opposite effects on the suspended needle. The force with which either current acts on the needle depends not only on the strength of the current, but on the position of the windings of the wire with respect to the needle. Hence, unless the coil is very carefully wound, the ratio of m to n may change when the position of the needle is changed, and therefore it is necessary to determine this ratio by proper methods during each i fa 436 MEASUREMENT OF RESISTANCE, [347. course of experiments if any alteration of the position of the needle is suspected. The other null method, in which Wheatstone's Bridge is used, requires only an ordinary galvanometer, and the observed zero deflexion of the needle is due, not to the opposing action of two currents, but to the non-existence of a current in the wire. Hence we have not merely a null deflexion, but a null current as the phenomenon observed, and no errors can arise from want of regularity or change of any kind in the coils of the galvanometer. The galvanometer is only required to be sensitive enough to detect the existence and direction of a current, without in any way determining its value or comparing its value with that of another current. 347.] Wheatstone's Bridge consists essentially of six conductors connecting four points. An electromotive force E is made to act between two of the points by means of a voltaic battery in- troduced between B and C. The current between the other two points and A is measured by a galvanometer. Under certain circumstances this current becomes zero. The conductors BC and OA are then said to be conjugate to each other, which implies a certain relation between the resistances of the other four conductors, and this relation is made use of in measuring resistances. If the current in OA is zero, the potential at must be equal to that at A. Now when we know the potentials at B and C we can determine those at and A by the rule given in Art. 275, provided there is no current in OA, . ^x _ whence the condition is where I, c, /3, y are the resistances in CA, AS, BO, and OC re- spectively. To determine the degree of accuracy attainable by this method we must ascertain the strength of the current in OA when this condition is not fulfilled exactly. Let A, B, C and be the four points. Let the currents along BC, CA and AB be so, y and z, and the resistances of these WHEATSTONE'S BRIDGE. 437 conductors 0, b and c. Let the currents along OA, OB and OC be r;, and the resistances a, (3 and y. Let an electromotive force E act along BC. Required the current along OA. Let the potentials at the points A t B, C and be denoted by the symbols A, B, C and 0. The equations of conduction are cz = A B, with the equations of continuity = C\ -* = o, x = 0, y = 0. By considering the system as made up of three circuits OBC, OCA and OAB, in which the currents are #, y, 2 respectively, and applying Kirchhoff's rule to each cycle, we eliminate the values of the potentials 0, A, J3, C, and the currents f, r/, and obtain the following equations for x, y and z t yx Hence, if we put az =0, -ay y a -a we find and = - 348.] The value of D may be expressed in the symmetrical form, or, since we suppose the battery in the conductor a and the galvanometer in a, we may put B the battery resistance for a and G the galvanometer resistance for a. We then find If the electromotive force E were made to act along OA, the resistance of OA being still a, and if the galvanometer were placed 438 MEASUREMENT OF RESISTANCE. [349- in EC, the resistance of BC being still #, then the value of D would remain the same, and the current in BC due to the electro- motive force E acting along OA would be equal to the current in OA due to the electromotive force E acting in BC. But if we simply disconnect the battery and the galvanometer, and without altering their respective resistances connect the battery to and A and the galvanometer to B and C, then in the value of D we must exchange the values of B and G. If I/ be the value of D after this exchange, we find D-V = G- Let us suppose that the resistance of the galvanometer is greater than that of the battery. Let us also suppose that in its original position the galvanometer connects the junction of the two conductors of least resistance /3, y with the junction of the two conductors of greatest resistance b, c, or, in other words, we shall suppose that if the quantities b, c, y, /3 are arranged in order of magnitude, t> and c stand together, and y and ft stand together. Hence the quantities & (3 and cy are of the same sign, so that their product is positive, and therefore D fi' is of the same sign as B G. If therefore the galvanometer is made to connect the junction of the two greatest resistances with that of the two least, and if the galvanometer resistance is greater than that of the battery, then the value of D will be less, and the value of the deflexion of the galvanometer greater, than if the connexions are exchanged. The rule therefore for obtaining the greatest galvanometer de- flexion in a given system is as follows : Of the two resistances, that of the battery and that of the galvanometer, connect the greater resistance so as to join the two greatest to the two least of the four other resistances. 349.] We shall suppose that we have to determine the ratio of the resistances of the conductors AB and AC, and that this is to be done by finding a point on the conductor BOC, such that when the points A and are connected by a wire, in ihe course of which a galvanometer is inserted, no sensible deflexion of the galvano- meter needle occurs when the battery is made to act between B and C. The conductor BOG may be supposed to be a wire of uniform resistance divided into equal parts, so that the 'ratio of the resist- ances of BO and OC may be read off at once. 349-] WHEATSTONE'S BRIDGE. 439 Instead of the whole conductor being a uniform wire, we may make the part near of such a wire, and the parts on each side may be coils of any form, the resistance of which is accurately known. We shall now use a different notation instead of the symmetrical notation with which we commenced. Let the whole resistance of BAC be R. Let c = mR and b (lm) R. Let the whole resistance of BOG be S. Let /3 = nS and y = (1 n) S. The value of n is read off directly, and that of m is deduced from it when there is no sensible deviation of the galvanometer. Let the resistance of the battery and its connexions be B, and that of the galvanometer and its connexions G. We find as before 2mn) BRS, and if f is the current in the galvanometer wire t ERS . (=(n-m). In order to obtain the most accurate results we must make the deviation of the needle as great as possible compared with the value of (n m). This may be done by properly choosing the dimensions of the galvanometer and the standard resistance wire. It will be shewn, when we come to Galvanometry, Art. 716, that when the form of a galvanometer wire is changed while its mass remains constant, the deviation of the needle for unit current is proportional to the length, but the resistance increases as the square of the length. Hence the maximum deflexion is shewn to occur when the resistance of the galvanometer wire is equal to the constant resistance of the rest of the circuit. In the present case, if 6 is the deviation, where C is some constant, and G is the galvanometer resistance which varies as the square of the length of the wire. Hence we find that in the value of D t when 8 is a maximum, the part involving G must be made equal to the rest of the expression. If we also put m = n, as is the case if we have made a correct observation, we find the best value of G to be 440 MEASUREMENT OF RESISTANCE. [350. This result is easily obtained by considering the resistance from A to through* the system, remembering that C, being conjugate to AO, has no effect on this resistance. In the same way we. should find that if the total area of the acting surfaces of the battery is given, the most advantageous arrangement of the battery is when Finally, we shall determine the value of 8 such that a given change in the value of n may produce the greatest galvanometer deflexion. By differentiating the expression for f we find BE " If we have a great many determinations of resistance to make in which the actual resistance has nearly the same value, then it may be worth while to prepare a galvanometer and a battery for this purpose. In this case we find that the best arrangement is and if n = i G= \R. On the Use of WheaUtonJs Bridge. 350.] We have already explained the general theory of Wheat- stone's Bridge, we shall now consider some of its applications. Fig. 33. The comparison which can be effected with the greatest exact- ness is that of two equal resistances. 35o.] USE OF WHEATSTONE'S BRIDGE. 441 Let us suppose that {3 is a standard resistance coil, and that we wish to adjust y to be equal in resistance to /3. ' Two other coils, b and c, are prepared which are equal or nearly equal to each other, and the four coils are placed with their electrodes in mercury cups so that the current of the battery is divided between two branches, one consisting of (3 and y and the other of b and c. The coils b and c are connected by a wire PR, as uniform in its resistance as possible, and furnished with a scale of equal parts. The galvanometer wire connects the junction of (3 and y with a point Q of the wire PR, and the point of contact at Q is made to vary till on closing first the battery circuit and then the galvanometer circuit, no deflexion of the galvanometer needle is observed. The coils /3 and y are then made to change places, and a new position is found for Q. If this new position is the same as the old one, then we know that the exchange of /3 and y has produced no change in the proportions of the resistances, and therefore y is rightly adjusted. If Q has to be moved, the direction and amount of the change will indicate the nature and amount of the alteration of the length of the wire of y, which will make its resistance equal to that of /3. If the resistances of the coils b and c, each including part of the wire PR up to its zero reading, are equal to that of b and c divisions of the wire respectively, then, if x is the scale reading of Q in the first case, and y that in the second, e + a? _/3 c+y __ y bx ~ y ' by ~~ /3* > 2 whence = 1 Since y is nearly equal to c + x, and both are great with respect to x or y, we may write this and "When y is adjusted as well as we can, we substitute for b and c other coils of (say) ten times greater resistance. The remaining difference between /3 and y will now produce a ten times greater difference in the position of Q than with the 442 MEASUREMENT OF RESISTANCE. [351. original coils I and c, and in this way we can continually increase the accuracy of the comparison. The adjustment by means of the wire with sliding contact piece is more quickly made than by means of a resistance box, and it is capable of continuous variation. The battery must never be introduced instead of the galvano- meter into the wire with a sliding contact, for the passage of a powerful current at the point of contact would injure the surface of the wire. Hence this arrangement is adapted for the case in which the resistance of the galvanometer is greater than that of the battery. When y, the resistance to be measured, a the resistance of the battery, and a the resistance of the galvanometer, are given, the best values of the other resistances have been shewn by Mr. Oliver Heaviside (Phil. Mag. Feb. 1873) to be c = */aa, a-f y On the Measurement of Small 'Resistances. 351.} When a short and thick conductor is introduced into a circuit its resistance is so small compared with the resistance occasioned by unavoidable faults in the connexions, such as want of contact or imperfect soldering, that no correct value of the resistance can be deduced from experi- ments made in the way described above. The object of such experiments is generally to determine the specific re- sistance of the substance, and it is re- sorted to in cases when the substance cannot be obtained in the form of a long thin wire, or when the resistance to transverse as well as to longitudinal conduction has to be measured. Sir W. Thomson* has described a method applicable to such cases, which we may take as an example of a system of nine conductors. 35i.] THOMSON'S METHOD FOR SMALL RESISTANCES. 443 The most important part of the method consists in measuring the resistance, not of the whole length of the conductor, but of the part between two marks on the conductor at some little dis- tance from its ends. The resistance which we wish to measure is that experienced by a current whose intensity is uniform in any section of the conductor, and which flows in a direction parallel to its axis. Now close to the extremities, when the current is introduced by means of electrodes, either soldered, amalgamated, or simply pressed to the ends of the conductor, there is generally a want of uniformity in the distribution of the current in the conductor. At a short distance from the extremities the current becomes Fig. 35. sensibly uniform. The student may examine for himself the investigation and the diagrams of Art. 193, where a current is introduced into a strip of metal with parallel sides through one of the sides, but soon becomes itself parallel to the sides. The resistances of the conductors between certain marks S, S' and T, T' are to be compared. The conductors are placed in series, and with connexions as perfectly conducting as possible, in a battery circuit of small resist- ance. A wire SVT is made to touch the conductors at S and T, and S' VT' is another wire touching them at S' and T'. The galvanometer wire connects the points ^and V of these wires. The wires SFT and S'V'T are of resistance so great that the resistance due to imperfect connexion at S, T, S' or T' may be neglected in comparison with the resistance of the wire, and 7 t V are taken so that the resistances in the branches of either wire leading to the two conductors are nearly in the ratio of the resist- ances of the two conductors. Calling^ and F the resistances of the conductors SS' and T'T. A and C those of the branches 87 sail VT. 444 MEASUREMENT OF RESISTANCE. [352. Calling P and R those of the branches 8' V and V'T'. Q that of the connecting piece S'T'. that of the battery and its connexions. G that of the galvanometer and its connexions. The symmetry of the system may be understood from the skeleton diagram. Fig. 34. The condition that B the battery and G the galvanometer may be conjugate conductors is, in this case, F_ H_ ( R^ A__J2__ C " A + \C "" A> P+Q + R ~ Now the resistance of the connector Q is as small as we can make it. If it were zero this equation would be reduced to F __ ff ~C~ A ' and the ratio of the resistances of the conductors to be compared would be that of C to A, as in Wheatstone's Bridge in the ordinary form. In the present case the value of Q is small compared with P or with R, so that if we assume the points F, V so that the ratio of R to C is nearly equal to that of P to A, the last term of the equation will vanish, and we shall have F:ff::C:A. The success of this method depends in some degree on the per- fection of the contact between the wires and the tested conductors at S, S', T' and T. In the following method, employed by Messrs. Matthiessen and Hockin*, this condition is dispensed with. Pig. 36. 352.] The conductors to be tested are arranged in the manner * Laboratory. Matthiessen and Hockin on Alloys. 352.] MATTHIESSEN AND HOCKIN's METHOD. 445 already described, with the connexions as well made as possible, and it is required to compare the resistance between the marks SS' on the first conductor with the resistance between the marks T Ton. the second. Two conducting points or sharp edges are fixed in a piece of insulating material so that the distance between them can be accurately measured. This apparatus is laid on the conductor to be tested, and the points of contact with the conductor are then at a known distance SS'. Each of these contact pieces is connected with a mercury cup, into which one electrode of the galvanometer may be plunged. The rest of the apparatus is arranged, as in Wheatstone's Bridge, with resistance coils or boxes A and C, and a wire PR with a sliding contact piece Q, to which the other electrode of the galva- nometer is connected. Now let the galvanometer be connected to S and Q, and let A l and C 1 be so arranged, and the position of Q so determined, that there is no current in the galvanometer wire. Then we know that XS A where XS, PQ, &c. stand for the resistances in these conductors. From this we get XY ' Now let the electrode of the galvanometer be connected to S*, and let resistance be transferred from C to A (by carrying resistance coils from one side to the other) till electric equilibrium of the galvanometer wire can be obtained by placing Q at some point of the wire, say Q 2 . Let the values of C and A be now C 2 and A 2i and let A 2 + C 2 + PR = A + C L + PR = R. Then we have, as before, XS' _ XT' R XY' R In the same way, placing the apparatus on the second conductor at TT' and again transferring resistance, we get, when the electrode is in T\ YT' XY " R 44:6 MEASUREMENT OF RESISTANCE. [S53 and when it is in T, XT XT' R Whence T'T _ A,- XT' We can now deduce the ratio of the resistances SS' and T f T t for When great accuracy is not required we may dispense with the resistance coils A and C 9 and we then find 88' T'T The readings of the position of Q on a wire of a metre in length cannot be depended on to less than a tenth of a millimetre, and the resistance of the wire may vary considerably in different parts owing to inequality of temperature, friction, &c. Hence, when great accuracy is required, coils of considerable resistance are intro- duced at A and C, and the ratios of the resistances of these coils can be determined more accurately than the ratio of the resistances of the parts into which the wire is divided at Q. It will be observed that in this method the accuracy of the determination depends in no degree on the perfection of the con- tacts at 8, y or T, T'. This method may be called the differential method of using Wheatstone's Bridge, since it depends on the comparison of ob- servations separately made. An essential condition of accuracy in this method is that the resistance of the connexions should continue the same during the course of the four observations required to complete the deter- mination. Hence the series of observations ought always to be repeated in order to detect any change in the resistances. On the Comparison of Great Resistances. 353.] When the resistances to be measured are very great, the comparison of the potentials at different points of the system may be made by means of a delicate electrometer, such as the Quadrant Electrometer described in Art. 219. If the conductors whose resistances are to be measured are placed in series, and the same current passed through them by means of a battery of great electromotive force, the difference of the potentials 355-] GREAT RESISTANCES. 447 at the extremities of each conductor will be proportional to the resistance of that conductor. Hence, by connecting the electrodes of the electrometer with the extremities, first of one conductor and then of the other, the ratio of their resistances may be de- termined. This is the most direct method of determining resistances. It involves the use of an electrometer whose readings may be depended on, and we must also have some guarantee that the current remains constant during the experiment. Four conductors of great resistance may also be arranged as in Wheatstone's Bridge, and the bridge itself may consist of the electrodes of an electrometer instead of those of a galvanometer. The advantage of this method is that no permanent current is required to produce the deviation of the electrometer, whereas the galvanometer cannot be deflected unless a current passes through the wire. 354.] When the resistance of a conductor is so great that the current which can be sent through it by any available electromotive force is too small to be directly measured by a galvanometer, a condenser may be used in order to accumulate the electricity for a certain time, and then, by discharging the condenser through a galvanometer, the quantity accumulated may be estimated. This is Messrs. Bright and Clark's method of testing the joints of submarine cables. 355.] But the simplest method of measuring the resistance of such a conductor is to charge a condenser of great capacity and to connect its two surfaces with the electrodes of an electrometer and also with the extremities of the conductor. If E is the dif- ference of potentials as shewn by the electrometer, S the capacity of the condenser, and Q the charge on either surface, R the resist- ance of the conductor and x the current in it, then, by the theory of condensers, Q = SE. By Ohm's Law, E = Ex, and by the definition of a current, *--^. dt Hence -Q=RS^, t_ and Q = Q ef*s, where Q Q is the charge at first when t = 0. 448 MEASUREMENT OF RESISTANCE. [356. Similarly E E Q e RS ' where H is the original reading of the electrometer, and E the same after a time t. From this we find which gives J2 in absolute measure. In this expression a knowledge of the value of the unit of the electrometer scale is not required. If S, the capacity of the condenser, is given in electrostatic measure as a certain number of metres, then R is also given in electrostatic measure as the reciprocal of a velocity. If S is given in electromagnetic measure its dimensions are fZ -j-, and R is a velocity. Since the condenser itself is not a perfect insulator it is necessary to make two experiments. In the first we determine the resistance of the condenser itself, 72 , and in the second, that of the condenser when the conductor is made to connect its surfaces. Let this be R'. Then the resistance, R, of the conductor is given by the equation _L.. jL JL R R' RQ This method has been employed by MM. Siemens. Thomson's * Method for the Determination of the Resistance of the Galvanometer. 356.] An arrangement similar to Wheatstone's Bridge has been Galvanometer Fig. 37. employed with advantage by Sir W. Thomson in determining the * Proc. R. 8., Jan. 19, 1871. 357-] MANCE'S METHOD. 449 resistance of the galvanometer when in actual use. It was sug- gested to Sir W. Thomson by Mance's Method. See Art. 357. Let the battery be placed, as before, between B and C in the figure of Article 347, but let the galvanometer be placed in CA instead of in OA. If Ificy is zero, then the conductor OA is conjugate to BC, and, as there is no current produced in OA by the battery in BC, the strength of the current in any other conductor is independent of the resistance in OA. Hence, if the galvano- meter is placed in CA its deflexion will remain the same whether the resistance of OA is small or great. We therefore observe whether the deflexion of the galvanometer remains the same when and A are joined by a conductor of small resistance, as when this connexion is broken, and if, by properly adjusting the re- sistances of the conductors, we obtain this result, we know that the resistance of the galvanometer is -3F- where c, y, and /3 are resistance coils of known resistance. It will be observed that though this is not a null method, in the sense of there being no current in the galvanometer, it is so in the sense of the fact observed being the negative one, that the deflexion of the galvanometer is not changed when a certain con- tact is made. An observation of this kind is of greater value than an observation of the equality of two different deflexions of the same galvanometer, for in the latter case there is time for alteration in the strength of the battery or the sensitiveness of the galvanometer, whereas when the deflexion remains constant, in spite of certain changes which we can repeat at pleasure, we are sure that the current is quite independent of these changes. The determination of the resistance of the coil of a galvanometer can easily be effected in the ordinary way of using Wheatstone's Bridge by placing another galvanometer in OA. By the method now described the galvanometer itself is employed to measure its own resistance. Mance's* Method of determining the Eesistance of the Battery. 357.] The measurement of the resistance of a battery when in action is of a much higher order of difficulty, since the resistance of the battery is found to change considerably for some time after * Proc. R. S., Jan. 19, 1871. VOL. I. G g 450 MEASUREMENT OF RESISTANCE. [357. the strength of the current through it is changed. In many of the methods commonly used to measure the resistance of a battery such alterations of the strength of the current through it occur in the course of the operations, and therefore the results are rendered doubtful. In Mance's method, which is free from this objection, the battery is placed in BC and the galvanometer in CA. The connexion between and B is then alternately made and broken. Now the deflexion of the galvanometer needle will remain un- altered, however the resistance in OB be changed, provided that OB and AC are conjugate. This may be regarded as a particular case of the result proved in Art. 347, or may be seen directly on the elimination of z and ft from the equations of that article, viz. we then have If y is independent of #, and therefore of /3, we must have a a = cy. The resistance of the battery is thus obtained in terms of c, y, a. When the condition a a = cy is fulfilled, the current through the galvanometer is then Ea Ey -> or ab-\-y(a + To test the sensibility of the method let us suppose that the condition cy = a a is nearly, but not accurately, fulfilled, and that Fig. 38. y is the current through the galvanometer when and B are connected by a conductor of no sensible resistance, and y^ the current when and B are completely disconnected. To find these values we must make /3 equal to and to oo in the general formula for y, and compare the results. 357-] COMPARISON OF ELECTROMOTIVE FORCES. 451 The general value for y is where D denotes the same expression as in Art. 348. Making use of the values of y given above we can then easily shew that the expressions for y Q and y l are approximately c(cy-aa) y* and - y(y+a) From these values we find The resistance, c, of the conductor AB should be equal to a, that of the battery; a and y should be equal and as small as possible; and b should be equal to q + y. Since a galvanometer is most sensitive when its deflexion is small, we should bring the needle nearly to zero by means of fixed magnets before making contact between and B. In this method of measuring the resistance of the battery, the current in the galvanometer is not in any way interfered with during the operation, so that we may ascertain the resistance of the battery for any given strength of current in the galvanometer so as to determine how the strength of the current affects the resistance *. If y is the current in the galvanometer, the actual current through the battery is X Q with the key down and ^ with the key up, where / b ac \ / b \ y the resistance of the battery is cy a = a and the electromotive force of the battery is C , X \ * [In the Philosophical Magazine for 1857, vol. i. pp. 515-525, Mr. Oliver Lodge has pointed out as a defect in Mance's method that as the electromotive force of the battery depends upon the current passing through the battery, the deflexion of the galvanometer needJe cannot be the same in the two cases when the key is down or up, if the equation a a = cy is true. Mr. Lodge describes a modification of Mance's method which he has employed with success.] 452 MEASUREMENT OF RESISTANCE. [358- The method of Art. 356 for finding the resistance of the galva- nometer differs from this only in making and breaking contact between and A instead of between and B, and by exchanging a and 3 we obtain for this case y On the Comparison of Electromotive Forces. 358.] The following method of comparing the electromotive forces of voltaic and thermoelectric arrangements, when no current passes through them, requires only a set of resistance coils and a constant battery. Let the electromotive force E of the battery be greater than that of either of the electromotors to be compared, then, if a sufficient resistance, R 19 be interposed between the points A lt B 1 of the primary circuit EB 1 A 1 E, the electromotive force from B : to A l may be made equal to that of the electromotor E. If the elec- trodes of this electromotor are now connected with the points A l9 B l no current will flow through the electromotor. By placing a galvanometer G in the circuit of the electromotor E l9 and adjusting the resistance between A 1 and B lt till the galvanometer G indicates no current, we obtain the equation where R l is the resistance between A 1 and B 19 and C is the strength of the current in the primary circuit. In the same way, by taking a second electromotor E% and placing its electrodes at A 2 and B 2 , so that no current is indicated by the galvanometer 6r 2 , COMPARISON OF ELECTROMOTIVE FORCES. 453 where 7? 2 is the resistance between A 2 and JS 2 . If the observations of the galvanometers G t and G 2 are simultaneous, the value of C, the current in the primary circuit, is the same in both equations, and we find EI : E 2 i ' RI ' H 2 In this way the electromotive force of two electromotors may be compared. The absolute electromotive force of an electromotor may be measured either electrostatically by means of the electrometer, or electromagnetically by means of an absolute galvanometer. This method, in which, at the time of the comparison, there is no current through either of the electromotors, is a modification of Poggendorff's method, and is due to Mr. Latimer Clark, who has deduced the following values of electromotive forces : Concentrated v u solution of Daniell I. Amalgamated Zinc HSO 4 + 4 aq. Cu SO 4 Copper = 1.079 II. HS0 4 + 12aq. CuS0 4 Copper =0.978 III. HS0 4 +12aq. CuNOa Copper =1.00 JSunsenl. HNO fl Carbon 1.964 II. sp. g. 1.38 Carbon =1.888 Grove HS0 4 + 4 aq. HNO, Platinum = 1.956 A Volt is an electromotive force equal to 100,000,000 units of the centimetre-gramme- second system. CHAPTER XII. ON THE ELECTRIC RESISTANCE OF SUBSTANCES. 359.] THERE are three classes in which we may place different substances in relation to the passage of electricity through them. The first class contains all the metals and their alloys, some sulphurets, and other compounds containing metals, to which we must add carbon in the form of gas-coke, and selenium in the crystalline form. In all these substances conduction takes place without any decomposition, or alteration of the chemical nature of the substance, either in its interior or where the current enters and leaves the body. In all of them the resistance increases as the temperature rises. The second class consists of substances which are called electro- lytes, because the current is associated with a decomposition of the substance into two components which appear at the electrodes. As a rule a substance is an electrolyte only when in the liquid form, though certain colloid substances, such as glass at 100C, which are apparently solid, are electrolytes. It would appear from the experiments of Sir B. C. Brodie that certain gases are capable of electrolysis by a powerful electromotive force. In all substances which conduct by electrolysis the resistance diminishes as the temperature rises, The third class consists of substances the resistance of which is so great that it is only by the most refined methods that the passage of electricity through them can be detected. These are called Dielectrics. To this class belong a considerable number of solid bodies, many of which are electrolytes when melted, some liquids, such as turpentine, naphtha, melted paraffin, &c., and all gases and vapours. Carbon in the form of diamond, and selenium in the amorphous form, belong to this class. The resistance of this class of bodies is enormous compared with that of the metals. It diminishes as the temperature rises. It 360.] RESISTANCE. 455 is difficult, on account of the great resistance of these substances, to determine whether the feeble current which we can force through them is or is not associated with electrolysis. On the Electric Resistance of Metals. 360.] There is no part of electrical research in which more numerous or more accurate experiments have been made than in the determination of the resistance of metals. It is of the utmost importance in the electric telegraph that the metal of which the wires are made should have the smallest attainable resistance. Measurements of resistance must therefore be made before selecting the materials. When any fault occurs in the line, its position is at once ascertained by measurements of resistance, and these mea- surements, in which so many persons are now employed, require the use of resistance coils, made of metal the electrical properties of which have been carefully tested. The electrical properties of metals and their alloys have been studied with great care by MM. Matthiessen, Vogt, and Hockin, and by MM. Siemens, who have done so much to introduce exact electrical measurements into practical work. It appears from the researches of Dr. Matthiessen, that the effect of temperature on the resistance is nearly the same for a considerable number of the pure metals, the resistance at 100 C C being to that at C C in the ratio of 1.414 to 1, or of 100 to 70.7. For pure iron the ratio is 1.645, and for pure thallium 1.458. The resistance of metals tins been observed by Dr. C. W. Siemens* through a much wider range of temperature, extending from the freezing point to 350C, and in certain cases to 1000C. He finds that the resistance increases as the temperature rises, but that the rate of increase diminishes as the temperature rises. The formula, which he finds to agree very closely both with the resistances observed at low temperatures by Dr. Matthiessen and with his own observations through a range of 1000 C C, is r = aT* + /37+y, where T is the absolute temperature reckoned from 273 C C, and a, )3, y are constants. Thus, for Platinum r= 0.039369 T* + 0.00216407 7-0.2413, Copper r= 0.026577 7* + 0.0031443 7-0.22751, Iron.! r = 0.072545 2*4 0.0038133 T- 1.23971. * Proe. R. S., April 27, 1871. 456 RESISTANCE. [361. From data of this kind the temperature of a furnace may be determined by means of an observation of the resistance of a platinum wire placed in the furnace. Dr. Matthiessen found that when two metals are combined to form an alloy, the resistance of the alloy is in most cases greater than that calculated from the resistance of the component metals and their proportions. In the case of alloys of gold and silver, the resistance of the alloy is greater than that of either pure gold or pure silver, and, within certain limiting proportions of the con- stituents, it varies very little with a slight alteration of the pro- portions. For this reason Dr. Matthiessen recommended an alloy of two parts by weight of gold and one of silver as a material for reproducing the unit of resistance. The effect of change of temperature on electric resistance is generally less in alloys than in pure metals. Hence ordinary resistance coils are made of German silver, on account of its great resistance and its small variation with tem- perature. An alloy of silver and platinum is also used for standard coils. 361.] The electric resistance of some metals changes when the metal is annealed; and until a wire has been tested by being repeatedly raised to a high temperature without permanently altering its resistance, it cannot be relied on as a measure of resistance. Some wires alter in resistance in course of time without having been exposed to changes of temperature. Hence it is important to ascertain the specific resistance of mercury, a metal which being fluid has always the same molecular structure, and which can be easily purified by distillation and treatment with nitric acid. Great care has been bestowed in determining the resistance of this metal by W. and C. F. Siemens, who introduced it as a standard. Their researches have been supplemented by those of Matthiessen and Hockin. The specific resistance of mercury was deduced from the observed resistance of a tube of length I containing a weight w of mercury, in the following manner. No glass tube is of exactly equal bore throughout, but if a small quantity of mercury is introduced into the tube and occupies a length A of the tube, the middle point of which is distant x from one end of the tube, then the area s of the section near this point Q will be s = , where C is some constant. A 362.] OF METALS. 457 The weight of mercury which fills the whole tube is w p I sdx pCl, (-} -> J v A y n where n is the number of points, at equal distances along the tube, where A has been measured, and p is the mass of unit of volume. The resistance of the whole tube is where r is the specific resistance per unit of volume. Hence wR = rp 2 (A) 2 (i) ^ , wR n 2 and -J2 pl gives the specific resistance of unit of volume. To find the resistance of unit of length and unit of mass we must multiply this by the density. It appears from the experiments of Matthiessen and Hockin that the resistance of a uniform column of mercury of one metre in length, and weighing one gramme at 0C, is 13.071 Ohms, whence it follows that if the specific gravity of mercury is 13.595, the resistance of a column of one metre in length and one square millimetre in section is 0.96146 Ohms. 362.] In the following table R is the resistance in Ohms of a column one metre long and one gramme weight at 0C, and r is the resistance in centimetres per second of a cube of one centi- metre, according to the experiments of Matthiessen *. Percentage increment of Specific resistance for gravity R r 1C at 20C. Silver ....... 10.50 hard drawn 0.1689 1609 0.377 Copper ...... 8.95 hard drawn 0.1469 1642 0.388 Gold ....... 19.27 hard drawn 0.4150 2154 0.365 Lead ....... H.391 pressed 2.257 19847 0.387 Mercury ..... 13.595 liquid 13.071 96146 0.072 Gold 2, Silver 1 . . 15.218 hard or annealed 1.668 10988 0.065 Selenium at 1 00C Crystalline form 6 x 1 13 1.00 * Phil. Mag., May, 1865. 458 EESTSTANCE. [363. On the Electric Resistance of Electrolytes. 363.] The measurement of the electric resistance of electrolytes is rendered difficult on account of the polarization of the electrodes, which causes the observed difference of potentials of the metallic electrodes to be greater than the electromotive force which actually produces the current. This difficulty can be overcome in various ways. In certain cases we can get rid of polarization by using electrodes of proper material, as, for instance, zinc electrodes in a solution of sulphate of zinc. By making the surface of the electrodes very large com- pared with the section of the part of the electrolyte whose resist- ance is to be measured, and by using only currents of short duration in opposite directions alternately, we can make the measurements before any considerable intensity of polarization has been excited by the passage of the current. Finally, by making two different experiments, in one of which the path of the current through the electrolyte is much longer than in the other, and so adjusting the electromotive force that the actual current, and the time during which it flows, are nearly the same in each case, we can eliminate the effect of polarization altogether. 364.] In the experiments of Dr. Paalzow * the electrodes were in the form of large disks placed in separate flat vessels filled with the electrolyte, and the connexion was made by means of a long siphon filled with the electrolyte and dipping into both vessels. Two such siphons of different lengths were used. The observed resistances of the electrolyte in these siphons being R^ and R 2 , the siphons were next filled with mercury, and their resistances when filled with mercury were found to be R and 22 2 '. The ratio of the resistance of the electrolyte to that of a mass of mercury at 0C of the same form was then found from the formula 7 > r> HI H.-> = F/^- To deduce from the values of p the resistance of a centimetre in length having a section of a square centimetre, we must multiply them by the value of r for mercury at 0C. See Art. 361. * Berlin Monatsbericht, July, 1868. 365-] OF ELECTROLYTES. 459 The results given by Paalzow are as follow : Mixtures of Sulphuric Acid and Wafer. Temp Resistance compared with mercury. H 2 SO 4 15 C C 96950 H 2 SO 4 + 14 IPO 19 C C 14157 H 2 SO 4 + 13H 2 22C 13310 H 2 S0 4 +499H 2 22 C C 184773 Sulphate of Zinc and Water. ZnSO 4 -f- 23 IPO 23 C C 194400 ZnSO 4 + 24H 2 23C 191000 ZnSO 4 +105H 2 O 23C 354000 Sulphate of Copper and Water. CuSO 4 + 45 IPO 22C 202410 CuSO 4 +105H 2 O 22 C C 339341 Sulphate of Magnesium and Water. MgSO 4 -f 34H 2 22C 199180 MgSO 4 +107H 2 O 22C 324600 Hydrochloric Acid and Water. HC1 + 15 IPO 23 C C 13626 HC1 + 500IPO 23C 86679 365.] MM. F. Kohlrausch and W. A. Nippoldt* have de- termined the resistance of mixtures of sulphuric acid and water. They used alternating magneto-electric currents, the electromotive force of which varied from \ to 7 * T of that of a Grove's cell, and by means of a thermoelectric copper-iron pair they reduced the electromotive force to T^^TTO of ^ na ^ f a Grove's cell. They found that Ohm's law was applicable to this electrolyte throughout the range of these electromotive forces. The resistance is a minimum in a mixture containing about one- third of sulphuric acid. The resistance of electrolytes diminishes as the temperature increases. The percentage increment of conductivity for a rise of 1C is given in the following table. * Pogg., Ann. cxxxviii. p. 286, Oct. 1869. 460 RESISTANCE. [ 3 66. Resistance of Mixtures of Sulphuric Acid and Water at 22C in terms of Mercury at 0C. MM. Kohlrausch and Nippoldt. Specific gravity at 185 Percentage ofH 2 SO, Resistance at 22C (Hg-1) Percentage increment of conductivity . for 1C 0.9985 0.0 746300 0.47 1.00 0.2 465100 0.47 1.0504 8.3 34530 0.653 1.0989 14.2 18946 0.646 1.1431 20.2 14990 0.799 1.2045 28.0 13133 1.317 1.2631 35.2 13132 1.259 1.3163 41.5 14286 1.410 1.3547 46.0 15762 1.674 1.3994 50.4 17726 1.582 1.4482 55.2 20796 1.417 1.5026 60.3 25574 1.794 On the Electrical 'Resistance of Dielectrics. 366.] A great number of determinations of the resistance of gutta-percha, and other materials used as insulating media, in the manufacture of telegraphic cables, have been made in order to ascertain the value of these materials as insulators. The tests are generally applied to the material after it has been used to cover the conducting wire, the wire being used as one electrode, and the water of a tank, in which the cable is plunged, as the other. Thus the current is made to pass through a cylin- drical coating of the insulator of great area and small thickness. It is found that when the electromotive force begins to act, the current, as indicated by the galvanometer, is by no means constant. The first effect is of course a transient current of considerable intensity, the total quantity of electricity being that required to charge the surfaces of the insulator with the superficial distribution of electricity corresponding to the electromotive force. This first current therefore is a measure not of the conductivity, but of the capacity of the insulating layer. But even after this current has been allowed to subside the residual current is not constant, and does not indicate the true conductivity of the substance. It is found that the current con- tinues to decrease for at least half an hour, so that a determination 366.] OF ELECTROLYTES. 461 of the resistance deduced from the current will give a greater value if a certain time is allowed to elapse than if taken immediately after applying the battery. Thus, with Hooper's insulating material the apparent resistance at the end of ten minutes was four times, and at the end of nineteen hours twenty-three times that observed at the end of one minute. When the direction of the electromotive force is reversed, the resistance falls as low or lower than at first and then gradually rises. These phenomena seem to be due to a condition of the gutta- percha, which, for want of a better name, we may call polarization, and which we may compare on the one hand with that of a series of Leyden jars charged by cascade, and, on the other, with Hitter's secondary pile, Art. 271. If a number of Leyden jars of great capacity are connected in series by means of conductors of great resistance (such as wet cotton threads in the experiments of M. Gaugain), then an electro- motive force acting on the series will produce a current, as indicated by a galvanometer, which will gradually diminish till the jars are fully charged. The apparent resistance of such a series will increase, and if the dielectric of the jars is a perfect insulator it will increase without limit. If the electromotive force be removed and connexion made between the ends of the series, a reverse current will be observed, the total quantity of which, in the case of perfect insulation, will be the same as that of the direct current. Similar effects are observed in the case of the secondary pile, with the difference that the final insulation is not so good, and that the capacity per unit of surface is immensely greater. In the case of the cable covered with gutta-percha, &c , it is found that after applying the battery for half an hour, and then con- necting the wire with the external electrode, a reverse current takes place, which goes on for some time, and gradually reduces the system to its original state. These phenomena are of the same kind with those indicated by the 'residual discharge' of the Leyden jar, except that the amount of the polarization is much greater in gutta-percha, &c. than in glass. This state of polarization seems to be a directed property of the material, which requires for its production not only electromotive force, but the passage, by displacement or otherwise, of a con- 462 RESISTANCE. [367. siderable quantity of electricity, and this passage requires a con- siderable time. When the polarized state has been set up, there is an internal electromotive force acting 1 in the substance in the reverse direction, which will continue till it has either produced a reversed current equal in total quantity to the first, or till the state of polarization has quietly subsided by means of true con- duction through the substance. The whole theory of what has been called residual discharge, absorption of electricity, electrification, or polarization, deserves a careful investigation, and will probably lead to important dis- coveries relating to the internal structure of bodies. 367.] The resistance of the greater number of dielectrics di- minishes as the temperature rises. Thus the resistance of gutta-percha is about twenty times as great at 0C as at 24 C C. Messrs. Bright and Clark have found that the following formula gives results agreeing with their experiments. If r is the resistance of gutta-percha at temperature T centigrade, then the resistance at temperature T-\- 1 will be E = rx 0.8878*, the number varies between 0.8878 and 0.9. Mr. Hockin has verified the curious fact that it is not until some hours after the gutta-percha has taken its temperature that the resistance reaches its corresponding value. The effect of temperature on the resistance of india-rubber is not so great as on that of gutta-percha. The resistance of gutta-percha increases considerably on the application of pressure. The resistance, in Ohms, of a cubic metre of various specimens of gutta-percha used in different cables is as follows *. Name of Cable. Red Sea , 267 x 10 12 to .362 x 10 12 Malta-Alexandria 1.23 x 10 12 Persian Gulf... 1.80 x 10 12 Second Atlantic 3.42 X 10 12 Hooper's Persian Gulf Core... 74.7 x 10 12 Gutta-percha at 24C 3.53 x 10 12 368.] The following table, calculated from the experiments of * Jenkin's Cantor Lectures. 37-] OF DIELECTRICS. 403 M. Buff, described in Art. 271, shews the resistance of a cubic metre of glass in Ohms at different temperatures. Temperature. Resistance. 200 C C 227000 250 13900 300 1480 350 1035 400 735 369.] Mr. C. F. Varley * has recently investigated the conditions of the current through rarefied gases, and finds that the electro- motive force E is equal to a constant E Q together with a part depending on the current according to Ohm's Law, thus For instance, the electromotive force required to cause the current to begin in a certain tube was that of 323 Daniell's cells, but an electromotive force of 304 cells was just sufficient to maintain the current. The intensity of the current, as measured by the galvanometer, was proportional to the number of cells above 304. Thus for 305 cells the deflexion was 2, for 306 it was 4, for 307 it was 6, and so on up to 380, or 304-1-76 for which the deflexion was 150, or 76 x 1.97. From these experiments it appears that there is a kind of polarization of the electrodes, the electromotive force of which is equal to that of 304 DanielFs cells, and that up to this electro- motive force the battery is occupied in establishing this state of polarization. When the maximum polarization is established, the excess of electromotive force above that of 304 cells is devoted to maintaining the current according to Ohm's Law. The law of the current in a rarefied gas is therefore very similar to the law of the current through an electrolyte in which we have to take account of the polarization of the electrodes. In connexion with this subject we should study Thomson's results, described in Art. 57, in which the electromotive force required to produce a spark in air was found to be proportional not to the distance, but to the distance together with a constant quantity. The electromotive force corresponding to this constant quantity may be regarded as the intensity of polarization of the electrodes. 370.] MM. Wiedemann and Riihlmann have recently f investi- * Proc. R. ., Jan. 12, 1871. t Berichte der Konigl. Sachs. Gesellschaft, Oct. 20, 1871. 464 RESISTANCE OF DIELECTRICS. gated the passage of electricity through gases. The electric current was produced by Holtz's machine, and the discharge took place between spherical electrodes within a metallic vessel containing rarefied gas. The discharge was in general discontinuous, and the interval of time between successive discharges was measured by means of a mirror revolving along with the axis of Holtz's machine. The images of the series of discharges were observed by means of a heliometer with a divided object-glass, which was adjusted till one image of each discharge coincided with the other image of the next discharge. By this method very consistent results were obtained. It was found that the quantity of electricity in each discharge is independent of the strength of the current and of the material of the electrodes, and that it depends on the nature and density of the gas, and on the distance and form of the electrodes. These researches confirm the statement of Faraday * that the electric tension (see Art. 48) required to cause a disruptive discharge to begin at the electrified surface of a conductor is a little less when the electrification is negative than when it is positive, but that when a discharge does take place, much more electricity passes at each discharge when it begins at a positive surface. They also tend to support the hypothesis stated in Art. 57, that the stratum of gas condensed on the surface of the electrode plays an important part in the phenomenon, and they indicate that this condensation is greatest at the positive electrode. * Exp. Res., 1501. VOT..J. ecirt'aty, FIG. I Art H iiit-K c/'Forcc a/ifl Equipotential A = ZO . of lh& I'f Fin. ii. Art 119 Lines offeree and Ecjui potential Surfacrs A =2C =-3 P, Point tif fyuMrutjn,. AP = 2 AB Q, Spherical surface ofZs,rojy0fcriticU'. M, Ibint of Majcimujris/'orce aloru/ fJie OJCLS . Th* dotted Uru- is the Line d Jtorce Y * O.I fAtts For the Delegates of the Clare-ridon Press. -'"i FIG in Art 120 of Fore r find Equipolenlial For tke Delegates of the- Clar&ncLon Press. fr . ..< FIG iv. Art 121 s <>/' Force ft/if/ Hfin'ff >/('/! tifif Surfaces --12 For the Delegates of th& Clarendon * of force and fUpofattux Swfax&s vi u, section/ of a, t ?j>tte>rical Surface m which, tfe juperftcial density u a, harmonic of the first decree . For iht Delegates of Ihe- CLar&ncLon< Press. FIG vi Art J43 Spkcrical Harmonic of the third, order o = y For the Delegates of the Clarendon Press. Fro vn Art 143 Spherical Harmonic r a Cr For the Delegates of the, Clarendon Press 14 DAY USE RETURN TO DESK FROM WHICH BORROWED ASTRONOMY, MATHEMATlcS- STATISTICS LIBRARY This book is due on the last date stamped below, or on the date to which renewed. Renewed books are subject to immediate recall. JAN 4 1998 MAI 2J/963 OCT 1^1965 JUN2 J1211 976 ILL 062002 OAN LTT21-50m-6,'60 (B1321slO)476 General Library University of California Berkeley