A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 12 of 28

Now we have shewn that dM/dr is negative for certain directions of r, hence when the electricity is free to move the instability in these directions will be increased.

M ^

CHAPTER VII.

FORMS OF THE EQUIPOTENTIAL SURFACES AND LINES OF INDUCTION IN SIMPLE CASES.

117.] WE have seen that the determination of the distribution of electricity on the surface of conductors may be made to depend on the solution of Laplace's equation

V being a function of x9 y, and #, which is always finite and con- tinuous, which vanishes at an infinite distance, and which has a given constant value at the surface of each conductor.

It is not in general possible by known mathematical methods to solve this equation so as to fulfil arbitrarily given conditions, but it is easy to write down any number of expressions for the function V which shall satisfy the equation, and to determine in each case the forms of the conducting surfaces, so that the function V shall be the true solution.

It appears, therefore, that what we should naturally call the inverse problem of determining the forms of the conductors when the expression for the potential is given is more manageable than the direct problem of determining the potential when the form of the conductors is given.

In fact, every electrical problem of which we know the solution has been constructed by this inverse process. It is therefore of great importance to the electrician that he should know what results have been obtained in this way, since the only method by which he can expect to solve a new problem is by reducing it to one of the cases in which a similar problem has been con- structed by the inverse process.

This historical knowledge of results can be turned to account in two ways. If we are required to devise an instrument for making electrical measurements with the greatest accuracy, we may select those forms for the electrified surfaces which correspond to cases of which we know the accurate solution. If, on the other hand, we are required to estimate what will be the electrification of bodies

1 1 8.] USE OF DIAGRAMS. 165

whose forms are given, we may begin with some case in which one of the equipotential surfaces takes a form somewhat resembling the given form, and then by a tentative method we may modify the pro- blem till it more nearly corresponds to the given case. This method is evidently very imperfect considered from a mathematical point of view, but it is the only one we have, and if we are not allowed to choose our conditions, we can make only an approximate cal- culation of the electrification. It appears, therefore, that what we want is a knowledge of the forms of equipotential surfaces and lines of induction in as many different cases as we can collect together and remember. In certain classes of cases, such as those relating to spheres, there are known mathematical methods by which we may proceed. In other cases we cannot afford to despise the humbler method of actually drawing tentative figures on paper, and selecting that which appears least unlike the figure we require.

This latter method I think may be of some use, even in cases in which the exact solution has been obtained, for I find that an eye- knowledge of the forms of the equipotential surfaces often leads to a right selection of a mathematical method of solution.

I have therefore drawn several diagrams of systems of equi- potential surfaces and lines of induction, so that the student may make himself familiar with the forms of the lines. The methods by which such diagrams may be drawn will be explained in Art. 123.

118.] In the first figure at the end of this volume we have the sections of the equipotential surfaces surrounding two points charged with quantities of electricity of the same kind and in the ratio of 20 to 5.

Here each point is surrounded by a system of equipotential surfaces which become more nearly spheres as they become smaller, though none of them are accurately spheres. If two of these sur- faces, one surrounding each point, be taken to represent the surfaces of two conducting bodies, nearly but not quite spherical, and if these bodies be charged with the • same kind of electricity, the charges being as 4 to 1, then the diagram will represent the equipotential surfaces, provided we expunge all those which are drawn inside the two bodies. It appears from the diagram that the action between the bodies will be the same as that between two points having the same charges, these points being not exactly in the middle of the axis of each body, but each somewhat more remote than the middle point from the other body.

The same diagram enables us to see what will be the distribution

166 EQUIPOTENTIAL SURFACES [119.

of electricity on one of the oval figures, larger at one end than ^De °ther, which surround both centres. Such a body, if charged with 25 units of electricity and free from external influence, will have the surface-density greatest at the small end, less at the large end, and least in a circle somewhat nearer the smaller than the ^ t larger end.

There is one equipotential surface, indicated by a dotted line, which consists of two lobes meeting at the conical point P. That point is a point of equilibrium, and the surface-density on a body of the form of this surface would be zero at this point.

The lines of force in this case form two distinct systems, divided from one another by a surface of the sixth degree, indicated by a dotted line, passing through the point of equilibrium, and some- what resembling one sheet of the hyperboloid of two sheets.

This diagram may also be taken to represent the lines of force and equipotential surfaces belonging to two spheres of gravitating matter whose masses are as 4 to 1.

119.] In the second figure we have again two points whose charges are as 20 to 5, but the one positive and the other negative. In this case one of the equipotential surfaces, that, namely, corre- sponding to potential zero, is a sphere. It is marked in the diagram by the dotted circle Q. The importance of this spherical surface will be seen when we come to the theory of Electrical Images.

We may see from this diagram that if two round bodies are charged with opposite kinds of electricity they will attract each other as much as two points having the same charges but placed somewhat nearer together than the middle points of the round bodies.

Here, again, one of the equipotential surfaces, indicated by a dotted line, has two lobes, an inner one surrounding the point whose charge is 5 and an outer one surrounding both bodies, the two lobes meeting in a conical point P which is a point of equilibrium.

If the surface of a conductor is of the form of the outer lobe, a roundish body having, like an apple, a conical dimple at one end of its axis, then, if this conductor be electrified, we shall be able to determine the surface-density at any point. That at the bottom of the dimple will be zero.

Surrounding this surface we have others having a rounded' dimple which flattens and finally disappears in the equipotential surface passing through the point marked M.

The lines of force in this diagram form two systems divided by a surface which passes through the point of equilibrium.

121.] AND LINES OF INDUCTION. 167

If we consider points on the axis on the further side of the point J2, we find that the resultant force diminishes to the double point P, where it vanishes. It then changes sign, and reaches a maximum at Jf, after which it continually diminishes.

This maximum, however, is only a maximum relatively to other points on the axis, for if we consider a surface through H per- pendicular to the axis, M is a point of minimum force relatively to neighbouring points on that surface.

120.] Figure III represents the equipotential surfaces and lines of induction due to a point whose charge is 10 placed at A, and surrounded by a field of force, which, before the introduction of the charged point, was uniform in direction and magnitude at every part.

The equipotential surfaces have each of them an asymptotic plane. One of them, indicated by a dotted line, has a conical point and a lobe surrounding the point A. Those below this surface have one sheet with a depression near the axis. Those above have a closed portion surrounding A and a separate sheet with a slight depression near the axis.

If we take one of the surfaces below A as the surface of a con- ductor, and another a long way below A as the surface of another conductor at a different potential, the system of lines and surfaces between the two conductors will indicate the distribution of electric force. If the lower conductor is very far from A its surface will be very nearly plane, so that we have here the solution of the distribution of electricity on two surfaces, both of them nearly plane and parallel to each other, except that the upper one has a protuberance near its middle point, which is more or less prominent according to the particular equipotential surface we choose.

121.] Figure IV represents the equipotential surfaces and lines of induction due to three points A, B and C, the charge of A being 15 units of positive electricity, that of B 12 units of negative electricity, and that of C 20 units of positive electricity. These points are placed in one straight line, so that

AB = 9, BC=16, AC =25.

In this case, the surface for which the potential is zero consists of two spheres whose centres are A and £and their radii 15 and 20. These spheres intersect in the circle which cuts the plane of the paper at right angles in D and J7, so that B is the centre of this circle and its radius is 12. This circle is an example of a line

168 EQUIPOTENTIAL SUEFACES [l22.

of equilibrium, for the resultant force vanishes at every point of this line.

If we suppose the sphere whose centre is A to be a conductor with a charge of 3 units of positive electricity, and placed under the influence of 20 units of positive electricity at C, the state of the case will be represented by the diagram if we leave out all the lines within the sphere A. The part of this spherical surface within the small circle DIf will be negatively charged by the influence of C. All the rest of the sphere will be positively charged, and the small circle Dl/ itself will be a line of no charge.

We may also consider the diagram to represent the sphere whose centre is (?, charged with 8 units of positive electricity, and in- fluenced by 15 units of positive electricity placed at A.

The diagram may also be taken to represent a conductor whose surface consists of the larger segments of the two spheres meeting in DD', charged with 23 units of positive electricity.

We shall return to the consideration of this diagram as an

illustration of Thomson's Theory of Electrical Images. See Art. 168.

122.] These diagrams should be studied as illustrations of the

language of Faraday in speaking of ' lines of force,' the ' forces of an

electrified body,' &c.

The word Force denotes a restricted aspect of that action between two material bodies by which their motions are rendered different from what they would have been in the absence of that action. The whole phenomenon, when both bodies are contemplated at once, is called Stress, and may be described as a transference of momentum from one body to the other. When we restrict our attention to the first of the two bodies, we call the stress acting on it the Moving Force, or simply the Force on that body, and it is measured by the momentum which that body is receiving per unit of time.

The mechanical action between two charged bodies is a stress^ and that on one of them is a force. The force on a small charged body is proportional to its own charge, and the force per unit of charge is called the Intensity of the force.

The word Induction was employed by Faraday to denote the mode in which the charges of electrified bodies are related to each other, every unit of positive charge being connected with a unit of negative charge by a line, the direction of which, in fluid dielectrics, coincides at every part of its course with that of the electric intensity. Such a line is often called a

I23-] AND LINES OF INDUCTION. 169

line of Force, but it is more correct to call it a line of In- duction.

Now the quantity of electricity in a body is measured, according to Faraday's ideas, by the number of lines of force, or rather of induction, which proceed from it. These lines of force must all terminate somewhere, either on bodies in the neighbourhood, or on the walls and roof of the room, or on the earth, or on the heavenly bodies, and wherever they terminate there is a quantity of elec- tricity exactly equal and opposite to that on the part of the body from which they proceeded. By examining the diagrams this will be seen to be the case. There is therefore no contradiction between Faraday's views and the mathematical results of the old theory, but, on the contrary, the idea of lines of force throws great light on these results, and seems to afford the means of rising by a con- tinuous process from the somewhat rigid conceptions of the old theory to notions which may be capable of greater expansion, so as to provide room for the increase of our knowledge by further researches.

123.] These diagrams are constructed in the following manner : —

First, take the case of a single centre of force, a small electrified body with a charge e. The potential at a distance r is V—e/r\ hence, if we make r = e/P~t we shall find r, the radius of the sphere for which the potential is V. If we now give to V the values 1, 2, 3, &c., and draw the corresponding spheres, we shall obtain a series of equipotential surfaces, the potentials corresponding to which are measured by the natural numbers. The sections of these spheres by a plane passing through their common centre will be circles, each of which we may mark with the number denoting its potential. These are indicated by the dotted semi-circles on the right hand of Fig. 6.

If there be another centre of force, we may in the same way draw the equipotential surfaces belonging to it, and if we now wish to find the form of the equipotential surfaces due to both centres together, we must remember that if T[ be the potential due to one centre, and 7£ that due to the other, the potential due to both will be 7^-f- J^=F. Hence, since at every intersection of the equipotential surfaces belonging to the two series we know both ^ and 7£, we also know the value of V. If therefore we draw a surface which passes through all those intersections for which the value of Pis the same, this surface will coincide with a true equipotential surface at all these intersections, and if the original systems of surfaces

170 EQUIPOTENTIAL SURFACES. [123.

are drawn sufficiently close, the new surface may be drawn with any required degree of accuracy. The equipotential surfaces due to two points whose charges are equal and opposite are represented by the continuous lines on the right hand side of Fig. 6.

This method may be applied to the drawing of any system of equipotential surfaces when the potential is the sum of two potentials, for which we have already drawn the equipotential surfaces.

The lines of force due to a single centre of force are straight lines radiating from that centre. If we wish to indicate by these lines the intensity as well as the direction of the force at any point, we must draw them so that they mark out on the equipotential surfaces portions over which the surface-integral of induction has definite values. The best way of doing this is to suppose our plane figure to be the section of a figure in space formed by the revolution of the plane figure about an axis passing through the centre of force. Any straight line radiating from the centre and making an angle 6 with the axis will then trace out a cone, and the surface-integral of the induction through that part of any surface which is cut off by this cone on the side next the positive direction of the axis is 2ne (l — cos0).

If we further suppose this surface to be bounded by its inter- section with two planes passing through the axis, and inclined at the angle whose arc is equal to half the radius, then the induction through the surface so bounded is

e (l — cos0) = 23>, say;

and 0 = cos"1 ( 1 — 2 — V v e '

If we now give to 4> a series of values 1, 2, 3 ...<?, we shall find a corresponding series of values of 0, and if e be an integer, the number of corresponding lines of force, including the axis, will be equal to e.

We have thus a method of drawing lines of force so that the charge of any centre is indicated by the number of lines which diverge from it, and the induction through any surface cut off in the way described is measured by the number of lines of force which pass through it. The dotted straight lines on the left hand side of Fig. 6 represent the lines of force due to each of two electrified points whose charges are 10 and —10 respectively.

If there are two centres of force on the axis of the figure we may draw the lines of force for each axis corresponding to values

To face P. 170.

Fig. (5.

Lines offeree* and EquipolenMi&l Surfaces.

1 2 3.] AND LINES OF INDUCTION. 171

of cf^ and <l>2, and then, by drawing lines through the consecutive intersections of these lines for which the value of 4>j -f 4>2 is the same, we may find the lines of force due to both centres, and in the same way we may combine any two systems of lines of force which are symmetrically situated about the same axis. The con- tinuous curves on the left hand side of Fig. 6 represent the lines of force due to the two charged points acting at once.

After the equipotential surfaces and lines of force have been constructed by this method the accuracy of the drawing may be tested by observing whether the two systems of lines are every- where orthogonal, and whether the distance between consecutive equipotential surfaces is to the distance between consecutive lines of force as half the mean distance from the axis is to the assumed unit of length.

In the case of any such system of finite dimensions the line of force whose index number is 4> has an asymptote which passes through the electric centre (Art. 89 d) of the system, and is inclined to the axis at an angle whose cosine is 1 — 2 <£/<?, where e is the total elec- trification of the system, provided <J> is less than e. Lines of force whose index is greater than e are finite lines. If e is zero, they are all finite.

The lines of force corresponding to a field of uniform force parallel to the axis are lines parallel to the axis, the distances from the axis being the square roots of an arithmetical series.

The theory of equipotential surfaces and lines of force in two dimensions will be given- when we come to the theory of conjugate functions*.

• See a paper 'On the Flow of Electricity in Conducting Surfaces,' by Prof. W. R. Smith, Proc. M.S. Edin., 1869-70, p. 79.

CHAPTER VIII.

SIMPLE CASES OF ELECTRIFICATION.

Two Parallel Planes.

124.] WE shall consider, in the first place, two parallel plane conducting surfaces of infinite extent, at a distance e from each other, maintained respectively at potentials A and B.

It is manifest that in this case the potential V will be a function of the distance z from the plane A, and will be the same for all points of any parallel plane between A and .Z?, except near the boundaries of the electrified surfaces, which by the supposition are at an infinitely great distance from the point considered.

Hence, Laplace's equation becomes reduced to

the integral of which is

7= Q+<72*; and since when z = 0, 7= A, and when z = c, 7= B,

For all points between the planes, the resultant intensity is normal to the planes, and its magnitude is

-n A — B

R = -- j- fi^ **v

In the substance of the conductors themselves, R = 0. Hence the distribution of "electricity on the first plane has a surface- density o-, where /y-

477(7=72 = - — •

On the other surface, where the potential is B, the surface- density </ will be equal and opposite to o-, and

1 24.] SIMPLE CASES. PAKALLEL PLANES. 173

Let us next consider a portion of the first surface whose area is S, taken so that no part of S is near the boundary of the surface.

The quantity of electricity on this surface is e1 = So; and, by Art. 79, the force acting on every unit of electricity is \R, so that the whole force acting on the area S, and attracting it towards the other plane, is

•

Here the attraction is expressed in terms of the area S, the difference of potentials of the two surfaces (A—JB), and the distance between them c. The attraction, expressed in terms of the charge e^ , on the area S, is _ __ 2 TT 2

The electrical energy due to the distribution of electricity on the area S, and that on the corresponding area S' on the surface B defined by projecting S on the surface B by a system of lines of force, which in this case are normals to the planes, is

8 (A-B)*

~

27T

The first of these expressions is the general expression of elec- trical energy (Art. 84).

The second gives the energy in terms of the area, the distance, and the difference of potentials.

The third gives it in terms of the resultant force R, and the volume So included between the areas S and /S", and shews that the energy in unit of volume is p where 8 up = R2.

The attraction between the planes ispS, or in other words, there is an electrical tension (or negative pressure) equal to p on every unit of area.

The fourth expression gives the energy in terms of the charge.

The fifth shews that the electrical energy is equal to the work which would be done by the electric force if the two surfaces were to be brought together, moving parallel to themselves, with their electric charges constant.

174 SIMPLE CASES. [125.

To express the charge in terms of the difference of potentials, we have 1 S , . „. , _„

The coefficient q represents the charge due to a difference of potentials equal to unity. This coefficient is called the Capacity of the surface S, due to its position relatively to the opposite surface.

Let us now suppose that the medium between the two surfaces is no longer air but some other dielectric substance whose specific inductive capacity is X, then the charge due to a given difference of potentials will be K times as great as when the dielectric is air, or KS .

6l

The total energy will be

27T

^ /»

The force between the surfaces will be

Hence the force between two surfaces kept at given potentials varies directly as K, the specific capacity of the dielectric, but the force between two surfaces charged with given quantities of elec- tricity varies inversely as K.

Two Concentric Spherical Surfaces.

125.] Let two concentric spherical surfaces of radii a and 3, of which b is the greater, be maintained at potentials A and B respectively, then it is manifest that the potential V is a function of r the distance from the centre. In this case, Laplace's equation becomes d27 2 d7

dr2 +r dr =

The solution of this is

and the condition that V—A when r = a, and V—E when r = ot gives for the space between the spherical surfaces,

1 25.] CONCENTRIC SPHERICAL SURFACES. 175

Aa—Bb A-B

dr

If o-j , (r2 are the surface-densities on the opposed surfaces of a solid sphere of radius a, and a spherical hollow of radius b, then 1 A-B 1 B-A

2~ 4-nb2 a^-b'1

If el and e2 are the whole charges of electricity on these surfaces,

A-B

The capacity of the enclosed sphere is therefore -^— •

b — a j g

If the outer surface of the shell be also spherical and of radius c, then, if there are no other conductors in the neighbourhood, the charge on the outer surface is

e3 = Be.

Hence the whole charge on the inner sphere is

and that of the outer shell

If we put b = oo, we have the case of a sphere in an infinite space. The electric capacity of such a sphere is a, or it is numeri- cally equal to its radius.

The electric tension on the inner sphere per unit of area is

La*

•T

The resultant of this tension over a hemisphere is Tra2p = F normal to the base of the hemisphere, and if this is balanced by a surface tension exerted across the circular boundary of the hemi- sphere, the tension on unit of length being T, we have

F=

Hence

„ b* (A- BY e* Jp==L_=_,

(A- B

:

176 SIMPLE CASES. [126.

If a spherical soap bubble is electrified to a potential A, then, if its radius is #, the charge will be Aa, and the surface- density will be l A

The resultant intensity just outside the surface will be 4770-, and inside the bubble it is zero, so that by Art. 79 the electrical force on unit of area of the surface will be 27rcr2, acting outwards. Hence the electrification will diminish the pressure of the air within the bubble by 2 wo-2, or

1 A2

87T 02 '

But it may be shewn that if T0 is the tension which the liquid film exerts across a line of unit length, then the pressure from within required to keep the bubble from collapsing is 2 TQ/a. If the electrical force is just sufficient to keep the bubble in equilibrium when the air within and without is at the same pressure,

Two Infinite Coaxal Cylindric Surfaces.

126.] Let the radius of the outer surface of a conducting cylinder be a, and let the radius of the inner surface of a hollow cylinder, having the same axis with the first, be b. Let their potentials be A and B respectively. Then, since the potential Fis in this case a function of r, the distance from the axis, Laplace's equation becomes

dr* + r dr = '

whence .V=

Since V— A when r — a, and V— B when r = b,

If o-l5 o-2 are the surface-densities on the inner and outer surfaces,

A-B B-A

12;.] COAXAL CYLINDERS. 177

If el and e2 are the charges on the portions of the two cylinders between two sections transverse to the axis at a distance I from each other, A — B

el = 2ital(Tl=: ^ —I = —e2.

lo&«

The capacity of a length I of the interior cylinder is therefore

I

If the space between the cylinders is occupied by a dielectric of

specific capacity K instead of air, then the capacity of the inner cylinder is ^ IK

~' ' ' ^ t»iL0»

The energy of the electrical distribution on the part of the infinite cylinder which we have considered is

&fa&)

/ A £

Fig. 5.

127.] Let there be two hollow cylindric conductors A and .5, Fig. 5, of indefinite length, having the axis of x for their common axis, one on the positive and the other on the negative side of the origin, and separated by a short interval near the origin of co- ordinates.

Let a hollow cylinder C of length 2 1 be placed with its middle point at a distance x on the positive side of the -origin, so as to extend into both the hollow cylinders.

Let the potential of the positive hollow cylinder be A, that of the negative one B, and that of the internal one <?, and let us put a for the capacity per unit of length of C with respect to A, and )3 for the same quantity with respect to B.

The surface densities of the parts of the cylinders at fixed points near the origin and at points at given small distances from the ends of the inner cylinder will not be affected by the

VOL. i. N

178 SIMPLE CASES. [127.

value of x provided a considerable length of the inner cylinder enters each of the hollow cylinders. Near the ends of the hollow cylinders, and near the ends of the inner cylinder, there will be distributions of electricity which we are not yet able to calculate, but the distribution near the origin will not be altered by the motion of the inner cylinder provided neither of its ends comes near the origin, and the distributions at the ends of the inner cylinder will move with it, so that the only effect of the motion will be to increase or diminish the length of those parts of the inner cylinder where the distribution is similar to that on an infinite cylinder.

Hence the whole energy of the system will be, so far as it depends on #,

Q = \a(l+x) (C-Af + \fi(l-x) (C—B)* + quantities

independent of x ; and the resultant force parallel to the axis of the cylinders will be

If the cylinders A and B are of equal section, a = /3, and X= a (B-A) (C-\ (A + B)).

It appears, therefore, that there is a constant force acting on the inner cylinder tending to draw it into that one of the outer cylinders from which its potential differs most.

If C be numerically large and A + B comparatively small, then

the force is approximately X = a (B A) C\

so that the difference of the potentials of the two cylinders can be measured if we can measure X, and the delicacy of the measurement will be increased by raising C, the potential of the inner cylinder.

This principle in a modified form is adopted in Thomson's Quadrant Electrometer, Art. 219.

The same arrangement of three cylinders may be used as a measure of capacity by connecting B and C. If the potential of A is zero, and that of B and C is F, then the quantity of electricity on A will be fi — (q 3_j_a (£+#)) F;

so that by moving £7 to the right till x becomes x + f the capacity of the cylinder C becomes increased by the definite quantity af , where

1

a a and b being the radii of the opposed cylindric surfaces.

CHAPTEE IX.

SPHEKICAL HARMONICS.

128.] The mathematical theory of spherical harmonics has been made the subject of several special treatises. The Handbuch der Kugelfunctionen of Dr. E. Heine, which is the most elaborate work on the subject, has now (1878) reached a second edition in two volumes, and Dr. F. Neumann has published his Beitrage zur Theorie der Kugelfunctionen (Leipzig, Teubner, 1878). The treat- ment of the subject in Thomson and Tait's Natural Philosophy is considerably improved in the second edition (1879), and Mr. Tod- hunter's Elementary Treatise on Laplace's Functions, Lame's Func- tions, and Bessel's Functions, together with Mr. Ferrers' Elementary Treatise on Spherical Harmonics and subjects connected with them, have rendered it unnecessary to devote much space in a book on electricity to the purely mathematical development of the subject.

I have retained however the specification of a spherical harmonic in terms of its poles.

On Singular Points at which the Potential becomes Infinite. 129 a] If a charge, AQ, of electricity is uniformly spread over the surface of a sphere the coordinates of whose centre are (#, 5, <?) the potential at any point (#, y, z) outside the sphere is, by Art. 125,

where r2 = (x-af + (y-3)2 + (z-c)\ (2)

As the expression for V is independent of the radius of the sphere, the form of the expression will be the same if we suppose the radius infinitely small. The physical interpretation of the expression would be that the charge A0 is placed on the surface of an infinitely small sphere, which is sensibly the same as a

N 2,

180 SPHERICAL HARMONICS. [129 I.

mathematical point. We have already (Arts. 55, 81) shewn that there is a limit to the surface-density of electricity, so that it is physically impossible to place a finite charge of electricity on a sphere of less than a certain radius.

Nevertheless as the equation (l) represents a possible distri- bution of potential in the space surrounding a sphere, we may for mathematical purposes treat it as if it arose from a charge A0 condensed at the mathematical point (at d, c) and we may call the point an infinite point of order zero.

There are other kinds of singular points, the properties of which we shall now investigate, but before doing so we must define certain expressions which we shall find useful in dealing with directions in space, and with the points on a sphere which cor- respond to them.

129$.] An axis is any definite direction in space. We may suppose it defined by a mark made on the surface of a sphere at the point where the radius drawn from the centre in the direction of the axis meets the surface. This point is called the Pole of the axis. An axis has therefore one pole only, not two.

If p is the cosine of the angle between the axis Ji and any vector r> and if p _ ^ (3)

p is the resolved part of r in the direction of the axis k.

Different axes are distinguished by different suffixes, and the cosine of the angle between two axes is denoted by Amn, where m and n are the suffixes specifying the axes.

Differentiation with respect to an axis, Ji> whose direction cosines are Z, M, N, is denoted by

d d ,_ d ^Td

-- = 1— +M-r+N-r. (4)

dh dx dy dz

From these definitions it is evident that

,fiv

dk

If we now suppose that the potential at the point (#, y, z) due to a singular point of any order placed at the origin is

I29C.] INFINITE POINTS. 181

then if such a point be placed at the extremity of the axis /&, the potential at (#, y, z) will be

Af \fr-Lk), (y-Mk), (z-Nh)l

and if a point in all respects the same, except that the sign of A is reversed, be placed at the origin, the potential due to the pair of points will be

V = Af[(x-Lk), (y-MA), (z-Nh)]-Af(x,y,z),

= — Ah — f(x, y, z) + terms containing k2.

If we now diminish h and increase A without limit, their pro- duct continuing finite and equal to A', the ultimate value of the potential of the pair of points will be

P=-^JS/(>«>- (")

If f (#, y> z) satisfies Laplace's equation, then, since this equation is linear, V , which is the difference of two functions, each of which separately satisfies the equation, must itself satisfy it.

1296*.] Now the potential due to an infinite point of order zero

V^A,, (9)