A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 3 of 28

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, J0 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 and -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 nt K: .

21.] SURFACE-INTEGRALS. 1 9

Similarly the line-integral of any closed curve will be

where nK 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 , 0 -, ,

Ida — —dydz.

Let a point travelling from a? = — oo to # = + oo first enter the space when so = sel9 then leave it when x = %2> an^ so on> and let the values of X at these points be X1)X2, &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 asl and xz) 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) = 0 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 xl 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 Slt S0, and S2. Let $! be a surface of any form bounded by a closed line L^. Let S0 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 S0 , 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 S2 be another surface of any form bounded by the closed curve I/2 in which it meets the surface S0.

Let Q1} Q0, Q2 be the surface-integrals of the surfaces Slt $0, S2) and let Q be the surface-integral of the closed surface S. Then

Q= Qi+Qo+ «2=0; (13)

and we know that Q0 = 0 ; (14)

therefore Q2= — 5i ; (15)

or, in other words, the surface-integral over the surface S2 is equal and opposite to that over Sl whatever be the form and position of $2, provided that the intermediate surface $0 is one for which R is always tangential.

If we suppose L± a closed curve of small area, S0 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 Lt 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 Slt the solenoidal condition is not satisfied, let

be the surface-integral for the surface enclosing this region, and let Q2, Q3, &c. be the corresponding quantities for the other en- closed regions S2, S3, &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 Slt S2i &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 Llt L2> &c* joking the internal surfaces Slt S2, &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 S1 and the surface-integral

IB«I.

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 r3 r3 rs

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 <?, and let f, 77, f be the components of another vector quantity S3, related to X, J, ^ by the equations

---

dy dz* r]~~fa fa' ~fa" dy' Then the surface-integral of S3 taken over the surface S is equal to the line-integral of 21 taken round the curve s. It is manifest that £ 77, f satisfy of themselves the solenoidal condition

^ + ^ + ^ = 0. dx dy dz

Let I, 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 S3 may be written

(2)

In order to form a definite idea of the meaning of the element dS, we shall suppose that the values of the coordinates #, y, z for every point of the surface are given as functions of two inde- pendent variables a and ft. If ft is constant and a varies, the point (#, y^ z) will describe a curve on the surface, and if a series of values is given to ft, 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 a, 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,

dz dy dz \ .

(3)

dft dft

The expressions for m dS and n dS are obtained from this by sub- stituting x, y, z in cyclic order.

The surface-integral which we have to find is

(4)

or, substituting the values of £, ry, £ in terms of X, Y, Zt

• dX dX dY 7dY 7dZ dZ^ 7C . N

?# — n-: — -n— i T= — h^-5 m-^-}ab. (5)

</0 ^fy ^ dk % »^y

The part of this which depends on X may be written