A Treatise on Electricity and Magnetism/Volume 1/Preliminary

by James Clerk Maxwell

ELECTRICITY AND MAGNETISM.

ELECTRICITY AND MAGNETISM.

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PRELIMINARY.

ON THE MEASUREMENT OF QUANTITIES.

1.] Every expression of a Quantity consists of two factors or components. One of these is the name of a certain known quantity of the same kind as the quantity to be expressed, which is taken as a standard of reference. The other component is the number of times the standard is to be taken in order to make up the required quantity. The standard quantity is technically called the Unit, and the number is called the Numerical Value of the quantity.

There must be as many different units as there are different kinds of quantities to be measured, but in all dynamical sciences it is possible to define these units in terms of the three fundamental units of Length, Time, and Mass. Thus the units of area and of volume are defined respectively as the square and the cube whose sides are the unit of length.

Sometimes, however, we find several units of the same kind founded on independent considerations. Thus the gallon, or the volume of ten pounds of water, is used as a unit of capacity as well as the cubic foot. The gallon may be a convenient measure in some cases, but it is not a systematic one, since its numerical relation to the cubic foot is not a round integral number.

2.] In framing a mathematical system we suppose the fundamental units of length, time, and mass to be given, and deduce all the derivative units from these by the simplest attainable definitions.

The formulae at which we arrive must be such that a person of any nation, by substituting for the different symbols the numerical value of the quantities as measured by his own national units, would arrive at a true result.

Hence, in all scientific studies it is of the greatest importance to employ units belonging to a properly defined system, and to know the relations of these units to the fundamental units, so that we may be able at once to transform our results from one system to another.

This is most conveniently done by ascertaining the dimensions of every unit in terms of the three fundamental units. When a given unit varies as the $n$th power of one of these units, it is said to be of $n$ dimensions as regards that unit.

For instance, the scientific unit of volume is always the cube whose side is the unit of length. If the unit of length varies, the unit of volume will vary as its third power, and the unit of volume is said to be of three dimensions with respect to the unit of length.

A knowledge of the dimensions of units furnishes a test which ought to be applied to the equations resulting from any lengthened investigation. The dimensions of every term of such an equation, with respect to each of the three fundamental units, must be the same. If not, the equation is absurd, and contains some error, as its interpretation would be different according to the arbitrary system of units which we adopt^[1].

The Three Fundamental Units.

3.] (1) Length. The standard of length for scientific purposes in this country is one foot, which is the third part of the standard yard preserved in the Exchequer Chambers.

In France, and other countries which have adopted the metric system, it is the mètre. The mètre is theoretically the ten millionth part of the length of a meridian of the earth measured from the pole to the equator; but practically it is the length of a standard preserved in Paris, which was constructed by Borda to correspond, when at the temperature of melting ice, with the value of the preceding length as measured by Delambre. The mètre has not been altered to correspond with new and more accurate measurements of the earth, but the arc of the meridian is estimated in terms of the original mètre.

In astronomy the mean distance of the earth from the sun is sometimes taken as a unit of length.

In the present state of science the most universal standard of length which we could assume would be the wave length in vacuum of a particular kind of light, emitted by some widely diffused substance such as sodium, which has well-defined lines in its spectrum. Such a standard would be independent of any changes in the dimensions of the earth, and should be adopted by those who expect their writings to be more permanent than that body.

In treating of the dimensions of units we shall call the unit of length $[L]$. If $l$ is the numerical value of a length, it is understood to be expressed in terms of the concrete unit $[L]$, so that the actual length would be fully expressed by $l[L]$.

4.] (2) Time. The standard unit of time in all civilized countries is deduced from the time of rotation of the earth about its axis. The sidereal day, or the true period of rotation of the earth, can be ascertained with great exactness by the ordinary observations of astronomers; and the mean solar day can be deduced from this by our knowledge of the length of the year.

The unit of time adopted in all physical researches is one second of mean solar time.

In astronomy a year is sometimes used as a unit of time. A more universal unit of time might be found by taking the periodic time of vibration of the particular kind of light whose wave length is the unit of length.

We shall call the concrete unit of time $[T]$, and the numerical measure of time $t$.

5.] (3) Mass. The standard unit of mass is in this country the avoirdupois pound preserved in the Exchequer Chambers. The grain, which is often used as a unit, is defined to be the $7000$th part of this pound.

In the metrical system it is the gramme, which is theoretically the mass of a cubic centimètre of distilled water at standard temperature and pressure, but practically it is the thousandth part of a standard kilogramme preserved in Paris.

The accuracy with which the masses of bodies can be compared by weighing is far greater than that hitherto attained in the measurement of lengths, so that all masses ought, if possible, to be compared directly with the standard, and not deduced from experiments on water.

In descriptive astronomy the mass of the sun or that of the earth is sometimes taken as a unit, but in the dynamical theory of astronomy the unit of mass is deduced from the units of time and length, combined with the fact of universal gravitation. The astronomical unit of mass is that mass which attracts another body placed at the unit of distance so as to produce in that body the unit of acceleration.In framing a universal system of units we may either deduce the unit of mass in this way from those of length and time already defined, and this we can do to a rough approximation in the present state of science; or, if we expect^[2] soon to be able to determine the mass of a single molecule of a standard substance, we may wait for this determination before fixing a universal standard of mass.

We shall denote the concrete unit of mass by the symbol $[M]$ in treating of the dimensions of other units. The unit of mass will be taken as one of the three fundamental units. When, as in the French system, a particular substance, water, is taken as a standard of density, then the unit of mass is no longer independent, but varies as the unit of volume, or as $[L^{3}]$.

If, as in the astronomical system, the unit of mass is defined with respect to its attractive power, the dimensions of $[M]$ are $[L^{3}T^{-2}]$.

For the acceleration due to the attraction of a mass $m$ at a distance $r$ is by the Newtonian Law ${\frac {m}{r^{2}}}$. Suppose this attraction to act for a very small time $t$ on a body originally at rest, and to cause it to describe a space $s$, then by the formula of Galileo,

$s={\frac {1}{2}}ft^{2}={\frac {1}{2}}{\frac {m}{r^{2}}}t^{2};$

whence $m=2{\frac {r^{2}s}{t^{2}}}$. Since $r$ and $s$ are both lengths, and $t$ is a time, this equation cannot be true unless the dimensions of $m$ are $[L^{3}T^{-2}]$. The same can be shewn from any astronomical equation in which the mass of a body appears in some but not in all of the terms ^[3].

Derived Units.

6.] The unit of Velocity is that velocity in which unit of length is described in unit of time. Its dimensions are $[LT^{-1}]$.

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 velocity increases by unity in unit of time. Its dimensions are $[LT^{-2}]$.

The unit of Density is the density of a substance which contains unit of mass in unit of volume. Its dimensions are $[ML^{-3}]$.

The unit of Momentum is the momentum of unit of mass moving with unit of velocity. Its dimensions are $[MLT^{-1}]$.

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 force 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 force of gravity, which varies from place to place, so that statements involving these quantities are not complete without a knowledge of the force of gravity in the places where these statements were found to be true.

The abolition, for all scientific purposes, of this method of measuring forces is mainly due to the introduction of a general system of making observations of magnetic force in countries in which the force 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.

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 expression for the quantity consists of two factors, the unit and the numerical 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 when, if 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 when, if the variables alter continuously, the quantity itself alters continuously.

Thus, if $u$ is a function of $x$, and if, while $x$ passes continuously from $x_{0}$ to $x_{1}$, $u$ passes continuously from $u_{0}$ to $u_{1}$, but when $x$ passes from $x_{1}$ to $x_{2}$, $u$ passes from $u_{1}'$ to $u_{2}$, $u_{1}'$ being different from $u_{1}$, then $u$ is said to have a discontinuity in its variation with respect to $x$ for the value $x=x_{1}$, because it passes abruptly from $u_{1}$ to $u_{1}'$ while $x$ passes continuously through $x_{1}$.

If we consider the differential coefficient of $u$ with respect to $x$ for the value $x=x_{1}$ as the limit of the fraction

${\frac {u_{2}-u_{0}}{x_{2}-x_{0}}}$,

when $x_{2}$ and $x_{0}$ are both made to approach $x_{1}$ without limit, then, if $x_{0}$ and $x_{2}$ are always on opposite sides of $x_{1}$, the ultimate value of the numerator will be $u_{1}'-u_{1}$ and that of the denominator will be zero. If $u$ is a quantity physically continuous, the discontinuity can exist only with respect to the particular variable $x$. We must in this case admit that it has an infinite differential coefficient when $x=x_{1}$. If $u$ is not physically continuous, it cannot be differentiated 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 $x_{0}$ is a very little less than $x_{1}$ and $x_{2}$ a very little greater than $x_{1}$, then $u_{0}$ will be very nearly equal to $u_{1}$ and $u_{2}$ to $u_{1}'$. We may now suppose $u$ to vary in any arbitrary but continuous manner from $u_{0}$ to $u_{2}$ between the limits $x_{0}$ and $x_{2}$. In many physical questions we may begin with a hypothesis of this kind, and then investigate the result when the values of $x_{0}$ and $x_{2}$ are made to approach that of $x_{1}$ and ultimately to reach it. The result will in most cases be independent of the arbitrary manner in which we have supposed $u$ to vary between the limits.

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 $x$, and these will be connected with the values of the other variables by an equation which we may write

$\phi =\phi (x,y,z,\And c.)=0.$

The discontinuity will occur when $\phi =0$. When $\phi$ is positive the function will have the form $F_{2}(x,y,z,\And c.)$. When $\phi$ is negative it will have the form $F_{1}(x,y,z,\And c.)$. There need be no necessary relation between the forms $F_{1}$ and $F_{2}$.

To express this discontinuity in a mathematical form, let one of the variables, say $x$, be expressed as a function of $\phi$ and the other variables, and let $F_{1}$ and $F_{2}$ be expressed as functions of $\phi ,y,z,\And c.$ We may now express the general form of the function by any formula which is sensibly equal to $F_{2}$ when $\phi$ is positive, and to $F_{1}$ when $\phi$ is negative. Such a formula is the following—

$F={\frac {F_{1}+e^{n\phi }F_{2}}{1+e^{n\phi }}}$

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 $\phi$ is positive, and equal to $F_{1}$ when $\phi$ is negative.

Discontinuity of the Derivatives of a Continuous Function.

The first derivatives of a continuous function may be discontinuous. Let the values of the variables for which the discontinuity of the derivatives occurs be connected by the equation

$\phi =\phi (x,y,z\ldots )=0,$

and let $F_{1}$ and $F_{2}$ be expressed in terms of $\phi$ and $n-1$ other variables, say $(y,z...)$.

Then, when $\phi$ is negative, $F_{1}$ is to be taken, and when $\phi$ is positive $F_{2}$ is to be taken, and, since $F$ is itself continuous, when $\phi$ is zero, $F_{1}=F_{2}$.

Hence, when $\phi$ is zero, the derivatives ${\frac {dF_{1}}{d\phi }}$ and ${\frac {dF_{2}}{d\phi }}$ may be different, but the derivatives with respect to any of the other variables, such as ${\frac {dF_{1}}{dy}}$ and ${\frac {dF_{2}}{dy}}$ must be the same. The discontinuity is therefore confined to the derivative with respect to $\phi$, 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 $x$, $x+a$, $x+na$, and all values of $x$ differing by $a$, $u$ is called a periodic function of $x$, and $a$ is called its period.

If $x$ is considered as a function of $u$, then, for a given value of $u$, 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.

The differential coefficient ${\frac {dx}{du}}$ has only a finite number of values 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 geometry 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.

But for many purposes in physical reasoning, 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 primitive 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 words 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 displacement 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.

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 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 expressed 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 composition of forces. In fact, the proof which Poisson gives of the 'parallelogram of forces' is applicable to the composition of any quantities such that a reversal of their sign is equivalent to turning them end for end.

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 ${\mathfrak {A}}$, ${\mathfrak {B}}$, &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 at that point of space 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 the point, the velocity of a fluid at that 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 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 displacement.

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 force and magnetic force belong to the first class, being defined with reference to lines. When we wish to indicate this fact, we may refer to them as Forces.

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 force produces electric currents in conductors, and tends to produce them in dielectrics. It produces electric induction in dielectrics, and probably in conductors also. In the same sense, magnetic force 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 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. 296. 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 between 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 magnetization 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, gives a result of scientific importance, and this is always a scalar quantity.

14.] There are two mathematical operations of frequent occurrence 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 position 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 represents 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.

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 longitudinal 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 rotation 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 treatment of the two classes, but there may be physical circumstances which indicate to which class we must refer a particular phenomenon. Thus, electrolysis consists of the transfer of certain substances along a line in one direction, and of certain other substances 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 magnetism is a longitudinal phenomenon, while the effect of magnetism in rotating the plane of polarized light distinctly shews that magnetism 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 value of the vector quantity at $P$, and let the tangent to the curve at $P$ make with the direction of $R$ the angle $\epsilon$, then $R\cos \epsilon$ is the resolved part of $R$ along the line, and the integral

$L=\int _{0}^{s}R\cos \epsilon ds$

is called the line-integral of $R$ along the line $s$.

We may write this expression

$L=\int _{0}^{s}{\Big (}X{\frac {dx}{ds}}+Y{\frac {dy}{ds}}+Z{\frac {dz}{ds}}{\Big )}ds$,

where $X$, $Y$, $Z$ are the components of $R$ parallel to $x$, $y$, $z$ respectively.

This quantity is, in general, different for different lines drawn between $A$ and $P$. When, however, within a certain region, the quantity

$X{dx}+Y{dy}+Z{dz}=-D\Psi \,\!$

that is, is an exact differential within that region, the value of $L$ becomes

$L=\Psi _{A}-\Psi _{P}$

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 $\Psi$ 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 components are $X$, $Y$, $Z$ is said to have a potential $\Psi$, if

$X=-({\frac {d\Psi }{dx}}),\quad Y=-({\frac {d\Psi }{dy}}),\quad Z=-({\frac {d\Psi }{dz}}).$

When a potential function exists, surfaces for which the potential is constant are called Equipotential surfaces. The direction of $R$ at any point of such a surface coincides with the normal to the surface, and if $n$ be a normal at the point $P$, then $R=-{\frac {d\Psi }{dn}}$.

The method of considering the components of a vector as the first derivatives of a certain function of the coordinates with respect to these coordinates was invented by Laplace^[4] in his treatment of the theory of attractions. The name of Potential was first given to this function by Green^[5] who made it the basis of his treatment of electricity. Green's essay was neglected by mathematicians till 1846, and before that time most of its important theorems had been rediscovered by Gauss, Chasles, Sturm, and Thomson^[6].

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 potential 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 coordinate 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 ${\mathfrak {F}}$ resolved parallel to these vectors, then

${\mathfrak {F}}=iX+jY+kZ;$

(1)

and by what we have said above, if $\Psi$ is the potential,

${\mathfrak {F}}=-(i{\frac {d\Psi }{dx}}+j{\frac {d\Psi }{dy}}+k{\frac {d\Psi }{dz}})$

(2)

If we now write $\nabla$ for the operator,

$i{\frac {d}{dx}}+j{\frac {d}{dy}}+k{\frac {d}{dz}},$

(3)

${\mathfrak {F}}=-\nabla \Psi .$

(4)

The symbol of operation $\nabla$ may be interpreted as directing us to measure, in each of three rectangular directions, the rate of increase of $\Psi$, 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 $\Psi$ increases fastest, and then to lay off in that direction a vector representing this rate of increase.

M. Lamé, in his Traité 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 Lamé 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 ${\mathfrak {F}}$ the Slope of the scalar function $\Psi$, using the word Slope to indicate the direction, as well as the magnitude, of the most rapid decrease of $\Psi$.

18.] There are cases, however, in which the conditions

${\frac {dZ}{dy}}-{\frac {dY}{dz}}=0,\quad {\frac {dX}{dz}}-{\frac {dZ}{dx}}=0,\quad$ and $\quad {\frac {dY}{dx}}-{\frac {dX}{dy}}=0,$

which are those of $Xdx+Ydy+Zdz$ being a complete differential, are fulfilled 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^[7].

Let there be $p$ points in space, and let $l$ 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 composed 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 $k=l-p+1$.

Any closed path drawn along the lines of the diagram is composed 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 more closed lines, which may in the limiting cases become 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 number being unity.

The cyclomatic number of a closed surface is twice that of the region 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

$Xdx+Ydy+Zdz=-D\Psi$,

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 region, so that a closed line within the region, if it cuts 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 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 $AQ^{'}PQA$. But the line-integral of the closed path is zero, therefore 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

$Xdx+Ydy+Zdz=-D\Psi$

is everywhere fulfilled, 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 Diaphragms, 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 aloug these lines will be equal. Suppose each equal to $Z$, then the line-integral of $A^{'}P^{'}$ is equal to that of $A^{'}A+AP+PP^{'}=-L+K+L=K=$ that of $AP$.

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 corresponding 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 p times in the negative direction, and let $p-p^{'}=n_{1}$ . Then the line-integral of the closed curve will be $n_{1}K_{1}$.

Similarly the line-integral of any closed curve will be

$n_{1}K_{1}+n_{2}K_{2}+...+n_{k}K_{k}$;

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 ^[8]

The condition that $Xdx+Ydy+Zdz$ is a complete differential of some function $\Psi$ 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 components of the displacement of a point of a continuous body whose original coordinates are x, y, z, then the condition expresses that these displacements constitute a non-rotational strain ^[9].

If X, Y, Z represent the components of the velocity of a fluid at the point x, 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 x, 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 d,S be the element of a surface, and $\epsilon$ 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 $\iint R\ cos\ \epsilon \ dS$ is called the surface-integral of R over the surface S.

THEOREM III. The surface-integral of the flux through a closed surface may be expressed as the volume-integral of its convergence taken within the surface. (See Art. 25.)

Let X, Y, Z' be the components of R, and let l, m, n, be the direction-cosines of the normal to S measured outwards. Then the surface-integral or R over S is

$\iint R\ cos\ \epsilon \ dS=\iint XldS+\iint YmdS+\iint ZndS$

$=\iint Xdydz+\iint Ydzdx+\iint Zdxdy$

(1)

1. The theory of dimensions was first stated by Fourier, Théorie de Chaleur, § 160.
2. See Prof. J. Loschmidt, 'Zur Grösse der Luftmolecule', Academy of Vienna, Oct. 12, 1865; G. J. Stoney on 'The Internal Motions of Gases', Phil. Mag., Aug. 1868; and Sir W. Thomson on 'The Size of Atoms', Nature, March 31, 1870.
3. If a foot and a second are taken as units, the astronomical unit of mass would be about $932,000,000$ pounds.
4. Méc. Céleste, liv. iii.
5. Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism, Nottingham, 1828. Reprinted in Crelle's Journal, and in Mr. Ferrer's edition of Green's Works
6. Thomson and Tait, Natural Philosophy, $\S$ 483.
7. Der Census Raumlicher Complexe, Gött. Abh., Bd. x. S. 97 (1861).
8. See Sir W. Thomson 'On Vortex Motion,' Trans, R. S. Edin., 1869.
9. See Thomson and Tait's Natural Philosophy, § 190 (i).

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.

Let a point travelling from $x=-\infty$ to $x=+\infty$ first enter the space when $x=x_{1}$ then leave it when $x=x_{2}$, and so on; and let the values of X at these points be $X_{1},X_{2}$, &c., then

$\iint Xdydz=\iint \{(X_{2}-X_{1})+(X_{4}-X_{3})+\&c.+(X_{2n}-X_{2n-1})\}dy\,dz$

(2)

If $X$ is a quantity which is continuous, and has no infinite values between $x_{1}$ and $x_{2}$, then

$X_{2}-X_{1}=\int _{x_{1}}^{x_{2}}{\frac {dX}{dx}}dx$

(3)

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

$\iint Xdydz=\iiint {\frac {dX}{dx}}dx\,dy\,dz$

(4)

the double integration being confined to the closed surface, but the triple integration being extended to the whole enclosed space. Hence, if $X,Y,Z$ are continuous and finite within a closed surface $S$, the total surface-integral of $R$ over that surface will be

$\iint R\cos \epsilon dS=\iiint ({\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}})dx\,dy\,dz$

(5)

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 $F(x,y,z)=0$ the values of $X,Y,Z$ alter abruptly from $X,Y,Z$ on the negative side of the surface to $X',Y',Z'$ on the positive side.

If this discontinuity occurs, say, between $x_{1}$ and $x_{2}$ , the value of $X_{2}-X_{1}$ will be

$\int _{x_{1}}^{x_{2}}{\frac {dX}{dx}}dx+(X'-X)$

(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 $R$ over the closed surface will be expressed by $\iint R\cos \epsilon dS=\iint {\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}dx\,dy\,dz+\iint (X'-X)dy\,dz$

$+\iint (Y'-Y)dx\,dz+\iint (Z'-Z)dx\,dy;$

(7)

or, if $l',m',n'$ are the direction-cosines of the normal to the surface of discontinuity, and $dS'$ an element of that surface,

$\iint R\cos \epsilon dS=\iiint ({\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}})dx\,dy\,dz$

$+\iint \{l'(X'-X)+m'(Y'-Y)+n'(Z'-Z)\}dS'$,

(8)

where the integration of the last term is to be extended over the surface of discontinuity.

If at every point where $X,Y,Z$ are continuous

${\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}=0$

(9)

and at every surface where they are discontinuous

$l'X'+m'Y'+n'Z'=l'X+m'Y+n'Z$,

(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 condition, 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

${\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}=0$

(11)

is fulfilled. 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 $S_{1}$, $S_{0}$ and $S_{2}$. Let $S_{1}$ be a surface of any form bounded by a closed line $L_{1}$. Let $S_{0}$ be formed by drawing lines from every point of $L_{1}$ always coinciding with the direction of $R$. If $l,m,n$ are the direction cosines of the normal at any point of the surface $S_{0}$, we have

$R\cos \epsilon =Xl+Ym+Zn=0.$

(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 $L_{2}$ in which it meets the surface $S_{0}$. Let $Q_{1}Q_{0},Q_{2}$ be the surface-integrals of the surfaces $S_{1},S_{0},S_{2}$, and let $Q$ be the surface-integral of the closed surface $S$. Then

$Q=Q_{1}+Q_{0}+Q_{2}=0$;

(13)

and we know that

$Q=0$;

(14)

therefore

$Q_{2}=-Q_{1}$;

(15)

or, in other words, the surface-integral over the surface $S_{2}$ is equal and opposite to that over $S_{1}$ whatever be the form and position of $S_{2}$ , provided that the intermediate surface $S_{0}$ is one for which $R$ is always tangential.

If we suppose $L_{1}$ a closed curve of small area, $S_{0}$ 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

${\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}=0$,

(16)

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 $S$ 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 $L$, 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 $S_{1}$ the solenoidal condition

${\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}=0$

is fulfilled, 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 fulfilled 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 any of these enclosed regions, $S_{1}$ the solenoidal condition is not fulfilled, let

$Q_{1}=\iint R\cos \epsilon \,dS_{1}$

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.

Then, if a closed surface $S^{'}$ is drawn within the region $S_{1}$ the value of its surface-integral will be zero only when this surface $S^{'}$ does not include any of the enclosed regions $S_{1},S_{2}$, &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 $S$.

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 joining the different bounding surfaces. 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. When these lines have been drawn we may assert that if the solenoidal condition is fulfilled in the region $S$, any closed surface drawn entirely within $S$, and not cutting any of the lines, has its surface-integral zero.

In drawing these lines we must remember that any line joining surfaces which are already connected does not diminish the periphraxy, but introduces cyclosis.

The most familiar example of a periphractic region within which the solenoidal condition is fulfilled is the region surrounding a mass attracting or repelling inversely as the square of the distance.

In this case we have

$X=m{\frac {x}{r^{3}}},\qquad X=m{\frac {y}{r^{3}}},\qquad X=m{\frac {z}{r^{3}}}$;

where $m$ is the mass supposed to be at the origin of coordinates.

At any point where $r$ is finite

${\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}=0$,

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 $4\pi m$.

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 $4\pi m$ 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 ^[1].

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.

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. The opposite, or left-handed system, is adopted in Hamilton's and Tait's Quaternions. 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 $x,y,z$, 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 $x,y$ may be written either

$\int x\ dy\qquad or\qquad -\int y\ dx$;

the order of integration being $x,y$ in the first expression, and $y,x$ in the second.

This relation between the two products $dxdy$ and $dydx$ may be compared with that between the products of two perpendicular vectors in the doctrine 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 exchanged.

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 establishing 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 taken over a surface bounded by the curve.

Let $X,Y,Z$ be the components of a vector quantity ${\mathfrak {A}}$ whose line-integral is to be taken round a closed curve s.

Let $S$ be any continuous finite surface bounded entirely by the closed curve $s$, and let $\xi ,\eta ,\zeta$ be the components of another vector quantity ${\mathfrak {B}}$, related to $X,Y,Z$ by the equations

$\xi ={\frac {dZ}{dy}}-{\frac {dY}{dz}}\qquad \eta ={\frac {dX}{dz}}-{\frac {dZ}{dx}}\qquad \zeta ={\frac {dY}{dx}}-{\frac {dX}{dy}}$

Then the surface-integral of ${\mathfrak {B}}$ taken over the surface $S$ is equal to the line-integral of ${\mathfrak {A}}$ taken round the curve $s$. It is manifest that $\xi ,\eta ,\zeta$ fulfil of themselves the solenoidal condition

${\frac {d\xi }{dx}}+{\frac {d\eta }{dy}}+{\frac {d\zeta }{dz}}=0$

Let $l,m,n$ be the direction-cosines of the normal to an element of the surface dS reckoned in the positive direction. Then the value of the surface-integral of ${\mathfrak {B}}$ may be written

$\iint (l\xi +m\eta +n\zeta )dS$

(2)

In order to form a definite idea of the meaning of the element $dS$, we shall suppose that the values of the coordinates $x,y,z$ for every point of the surface are given as functions of two inde pendent variables $\alpha$ and $\beta$. If $\beta$ is constant and $\alpha$ varies, the point $(x,y,z)$ will describe a curve on the surface, and if a series of values is given to $\beta$, a series of such curves will be traced, all lying on the surface $S$. In the same way, by giving a series of constant values to $\alpha$, a second series of curves may be traced, cutting the first series, and dividing the whole surface into elementary portions, any one of which may be taken as the element $dS$.

The projection of this element on the plane of $y,z$ is, by the ordinary formula,

$ldS=({\frac {dy}{d\alpha }}{\frac {dz}{d\beta }}-{\frac {dy}{d\beta }}{\frac {dz}{d\alpha }})d\beta \,d\alpha$.

(3)

The expressions for $mdS$ and $ndS$ are obtained from this by substituting $x,y,z$ in cyclic order.

The surface-integral which we have to find is

$\iint (l\xi +m\eta +n\zeta )dS$;

(4)

or, substituting the values of $\xi ,\eta ,\zeta$ in terms of $X,Y,Z$,

$\iint (m{\frac {dX}{dz}}-n{\frac {dX}{dy}}+n{\frac {dY}{dx}}-l{\frac {dY}{dz}}+l{\frac {dZ}{dy}}-m{\frac {dZ}{dx}})dS$;

(5)

The part of this which depends on $X$ may be written

$\iint \left\{{\frac {dX}{dz}}({\frac {dz}{d\alpha }}{\frac {dx}{d\beta }}-{\frac {dz}{d\beta }}{\frac {dx}{d\alpha }})-{\frac {dX}{dy}}({\frac {dx}{d\alpha }}{\frac {dy}{d\beta }}-{\frac {dx}{d\beta }}{\frac {dy}{d\alpha }}\right\rbrace d\beta \,d\alpha$;

(6)

adding and subtracting ${\frac {dX}{dx}}{\frac {dx}{d\alpha }}{\frac {dx}{d\beta }}$ this becomes

$\iint \left\{{\frac {dx}{d\beta }}({\frac {dX}{dx}}{\frac {dx}{d\alpha }}+{\frac {dX}{dy}}{\frac {dy}{d\alpha }}+{\frac {dX}{dz}}{\frac {dz}{d\alpha }})\right.$

$\left.-{\frac {dx}{d\alpha }}({\frac {dX}{dx}}{\frac {dx}{d\beta }}+{\frac {dX}{dy}}{\frac {dx}{d\beta }}+{\frac {dX}{dz}}{\frac {dz}{d\beta }})\right\rbrace d\beta \,d\alpha$;

(7)

$=\iint ({\frac {dX}{d\alpha }}{\frac {dx}{d\beta }}-{\frac {dX}{d\beta }}{\frac {dx}{d\alpha }})d\beta \,d\alpha$.

(8)

As we have made no assumption as to the form of the functions $\alpha$ and $\beta$, we may assume that $\alpha$ is a function of $X$, or, in other words, that the curves for which $\alpha$ is constant are those for which X is constant. In this case ${\frac {dX}{d\beta }}=0$, and the expression becomes by integration with respect to $\alpha$,

$\iint {\frac {dX}{d\alpha }}{\frac {dx}{d\beta }}d\beta \,d\alpha =\int X{\frac {dx}{d\beta }}d\beta$;

(9)

where the integration is now to be performed round the closed curve. Since all the quantities are now expressed in terms of one variable $\beta$ we may make $s$, the length of the bounding curve, the independent variable, and the expression may then be written

$\int X{\frac {dx}{ds}}ds$;

(10)

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

$\iint (l\xi +m\eta +n\zeta )dS=\int (X{\frac {dx}{ds}}+Y{\frac {dy}{ds}}+Z{\frac {dz}{ds}})ds$;

(11)

where the first integral is extended over the surface $S$, and the second round the bounding curve $s$^[2].

On the effect of the operator $\nabla$ on a vector function.

25.] We have seen that the operation denoted by $\nabla$ is that by which a vector quantity is deduced from its potential. The same operation, however, when applied to a vector function, produces results which enter into the two theorems we have just proved (III and IV). The extension of this operator to vector displacements, and most of its further development, is due to Professor Tait^[3].

Let $\sigma$ be a vector function of $\rho$, the vector of a variable point. Let us suppose, as usual, that

$\rho =ix+jy+kz,$

and

$\sigma =iX+jY+kZ;$

where $X,Y,Z$ are the components of $\sigma$ in the directions of the axes.

We have to perform on $\sigma$ the operation

$\nabla =i{\frac {d}{dx}}+j{\frac {d}{dy}}+k{\frac {d}{dz}}$

Performing this operation, and remembering the rules for the multiplication of $i,j,k,$ we find that $\nabla \sigma$ consists of two parts, one scalar and the other vector.

The scalar part is

$S\nabla \sigma =-({\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}})$, see Theorem III,

and the vector part is

$V\nabla \sigma =i({\frac {dZ}{dy}}-{\frac {dY}{dz}})+j({\frac {dx}{dz}}-{\frac {dX}{dx}})+k({\frac {dY}{dx}}-{\frac {dX}{dy}})$.

If the relation between $X,Y,Z$ and $\xi ,\eta ,\zeta$ is that given by equation (1) of the last theorem, we may write

$V\nabla \sigma =i\xi +j\eta +k\zeta$. See Theorem IV.

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

$\iiint S\nabla \sigma d\varsigma \ =\iint S.\sigma U\nu ds$, (III)

and

$\int S\sigma d\rho =\iint S.\nabla \sigma U\nu ds$; (IV)

where $d\varsigma$ is an element of a volume, $ds$ of a surface, $d\rho$ of a curve, and $U\nu$ a unit-vector in the direction of the normal. To understand the meaning of these functions of a vector, let us suppose that $\sigma _{0}$ is the value of $\sigma$ at a point $P$, and let us examine

Fig. 1

the value of $\sigma -\sigma _{0}$ in the neighbourhood of $P$.

If we draw a closed surface round $P$ then, if the surface-integral of $\sigma$ over this surface is directed inwards, $S\nabla \sigma$ will be positive, and the vector $\sigma -\sigma _{0}$ near the point $P$ will be on the whole directed towards $P$, as in the figure (1).

I propose therefore to call the scalar part of $\nabla \sigma$ the convergence of $\sigma$ at the point $P$.

To interpret the vector part of $\nabla \sigma$, let us suppose ourselves to be looking in the direction of the vector whose

Fig. 2

components are $\xi ,\eta ,\zeta ,$ and let us examine the vector $\sigma -\sigma _{0}$ near the point $P$. It will appear as in the figure (2), this vector being arranged on the whole tangentially in the direction opposite to the hands of a watch.

I propose (with great diffidence) to call the vector part of $\nabla \sigma$ the curl, or the version of $\sigma$ at the point $P$.

At Fig. 3 we have an illustration of curl combined with convergence.

Fig. 3

Let us now consider the meaning of the equation

$V\nabla \sigma =0$

This implies that $\nabla \sigma$ is a scalar, or that the vector $\sigma$ is the slope of some scalar function $\Psi$. These applications of the operator $\nabla$ are due to Professor Tait^[4]. A more complete development of the theory is given in his paper 'On Green's and other allied Theorems'^[5] to which I refer the reader for the purely Quaternion investigation of the properties of the operator $\nabla$.

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

$\nabla ^{2}=-({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}})$

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

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

If, with any point $P$ as centre, we draw a small sphere whose radius is $r$, then if $q_{0}$ is the value of $q$ at the centre, and ${\bar {q}}$ the mean value of $q$ for all points within the sphere,

$q_{0}-{\overline {q}}={\tfrac {1}{10}}r^{2}\nabla ^{2}q$;

so that the value at the centre exceeds or falls short of the mean value according as $\nabla ^{2}q$ is positive or negative.

I propose therefore to call $\nabla ^{2}q$ the concentration of $q$ at the point $P$, because it indicates the excess of the value of $q$ at that point over its mean value in the neighbourhood of the point.

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

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1. 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 Linneus, and of screw-makers in all civilized countries except Japan. De Candolle was the first who called the hop-tendril right-handed, and in this he is followed by Listing, and by most writers on the rotatory 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 ordinary carriages, but they are always called left-handed screws by those who use them.
2. This theorem was given by Professor Stokes. Smith's Prize Examination, 1854, question 8. It is proved in Thomson and Tait's Natural Philosophy, § 190 (j).
3. See Proc. R. S. Edin., April 28, 1862. On Green s and other allied Theorems, Trans. R. S. Edin., 1869-70, a very valuable paper ; and On some Quaternion Integrals, Proc. R. S. Edin., 1870-71.
4. Proceedings R. S. Edin., 1862
5. Trans. R. S. Edin., 1869-70.