The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 16 of 39

it is found that the curve f{x, y) = 0 corresponds to the constant value V = C, then clearly the general value of V obtained from equation (234) will be a solution of Laplace's equation subject to the condition of having the constant value V = C over the boundary / (x, y) = 0. It will therefore be the potential in an electrostatic field in which the curve fix, y) = 0 may be taken to be a conductor raised to potential C.

266 Methods for the Solution of Special Problems [ch. viii

316. From a given transformation it is obviously always possible to deduce the corresponding electrostatic field, but on being given the con- ductors and potentials in the field, it is by no means always possible to deduce the required transformation. We shall begin by the examination of a few fields which are given by simple known transformations.

Special Transformations.

I. W = z11.

317. Considering  the  transformation  W  =  z11,  we  have 
     

U + iV = (x + iy)n = rn (cos nd + i sin nd),

so that V=rnsin nd. Thus any one of the surfaces rn sin nd — constant may be supposed to be an equipotential, including as a special case

rn sin nd = 0,

IT

in which the equipotential consists of two planes cutting at an angle - .

This transformation can be further discussed by assigning particular values to n.

n = 1. This gives simply V — x, a uniform field of force.

n = 2. This gives V = 2xy, so that the equipotentials are rectangular hyperbolic cylinders, including as a special case two planes intersecting at right angles (fig. 85).

Fig. 85.

Fig. 86.

316, 317]

Conjugate Functions

267

This transformation gives the field in the immediate neighbourhood of two conducting planes meeting at right angles in any field of force. It also gives the field between two coaxal rectangular hyperbolas.

Fig. 87.

n = . This gives x + iy = (U + iV)2, so that

x=U*-V\ y = 2UV, and on eliminating U we obtain

2/2 = 4F2 (x + V2).

Thus the equipotentials are confocal and coaxal parabolic cylinders, in- cluding as a special case ( V = 0) a semi-infinite plane bounded by the line of foci.

This transformation clearly gives the field in the immediate neighbour- hood of a conducting sharp straight edge in any field of force (fig. 86).

n = — 1. This gives

U+iV= - (cos 0 - 1 sin 0 V r

and the equipotentials are

rF=sin<9 or ^ + ^--^=0.

Thus the equipotentials are a series of circular cylinders, all touching the plane y = 0 along the axis x = 0, y = 0 (fig. 87).

268 Methods for the Solution of Special Problems [ch. vm

II. IF = log *

318. The  transformation  W  =  logz  gives 
     

U+iV=\ogr + i0,

so that the equipotentials are the planes 6 = constant, a system of planes all intersecting in the same line. As a special case, we may take 0 = 0 and 6 = ir to be the conductors, and obtain the field when the two halves of a plane are raised to different potentials. The lines of force, U = constant, are circles (fig. 88).

Fig. 88.

If we take U to be the potential, the equipotentials are concentric circular cylinders, and the field is seen to be simply that due to a uniform line-charge, or uniformly electrified cylinder.

It may be noticed that the transformation

W = log (z — a) gives the transformation appropriate to a line-charge at z=a.

Also we notice that

z — a

F=log

z + a

gives a field equivalent to the superposition of the fields given by

W = log (z - a) and W = — log (z + a).

This transformation is accordingly that appropriate to two equal and opposite •line-charges along the parallel lines z = a and z = — a.

This last transformation gives U = 0 when y = 0, so that it gives the transformation for a line -charge in front of a parallel infinite plane.

318-320] Conjugate Functions 269

General Methods. I. Unicursal Curves.

319. Suppose that the coordinates of a point on a conductor can he expressed as real functions of a real parameter, which varies as the point moves over the conductor, in such a way that the whole range of variation of the parameter just corresponds to motion over the whole conductor. In other words, suppose that the coordinates x, y can be expressed in the form

x=f(p), y = F(p)>

and that all real values of p give points on the conductor, while, conversely, all points on the conductor correspond to real values of p.

Then the transformation

z=f(W) + iF(W) (235)

will give V= 0 over the conductor. For on putting V= 0 in equation (235) we obtain

x + iy=f(U) + iF(U),

so that x=f{U\ y = F(U),

and by hypothesis the elimination of U will lead to the equation of the conductor.

320. For example, consider the parabola (referred to its focus as origin),

t/2 = 4a (x + a). We can write the coordinates of any point on this parabola in the form

x + a = am2, y = 2am, and the transformation is seen to be

z = aW2 - a + 2aiW = a (W - if,

or W-i= (£)*

agreeing with that which has already been seen in § 317 to give a parabola as a possible equipotential.

270 Methods for the Solution of Special Problems [ch. vni

321. As  a  second  example  of  this  method,  let  us  consider  the  ellipse 
     

a? + 62 " i* The coordinates of a point on the ellipse may be expressed in the form

x = a cos <£, y = b sin (f>, and the transformation is seen to be

z = a cos W + ib sin W.

Fio. 89.

We can take a= c cosh a, b = c sinh a, where c^ = a7— 52, and the trans* formation becomes

z = c cos ( W + ia) = c cos { U + i ( F + a)}.

The same transformation may be expressed in the better known form

z = c cosh W.

The equipotentials are the confocal ellipses

a? y2

-1.

a2 + X 62 + X

while the lines of force are confocal hyperbolic cylinders. On taking 7 as the potential, we get a field in which the equipotentials are confocal hyperbolic cylinders.

321, 322] Conjugate Functions 271

II. Schwarz's Transformation.

322. Schwarz has shewn how to obtain a transformation in which one equipotential can be any linear polygon.

At any angle of a polygon it is clear that the property that small elements remain unchanged in shape can no longer hold. The reason is easily seen to be that the modulus of transformation is either infinite or zero (cf. figs. 24 and 25, p. 61). Thus, at the angles of any polygon,

dW

dz

= 0 or oo .

The same result is evident from electrostatic considerations. At an angle of a conductor, the surface-density <r is either infinite or zero (§ 70), while we have the relation (§ 313),

dW

47T 4ff

dz

Let us suppose that the polygon in the 2-plane is to correspond to the line V = 0 in the W-plane, and let the angular points correspond to

U=uJ} JJ=u2, etc.

Then, when W = ult W = w2> etc.,

dz

-r-jjr must either vanish or become infinite. We must accordingly have

dz ^^FiW-u^iW-u^ (236),

where Xi, X-,, ... are numbers which may be positive or negative, while F denotes a function, at present unknown, of W.

Suppose that, as we move along the polygon, the values of U at the angular points occur in the order ult u2, .... Then, on passing along the side of the polygon which joins the two angles U=u1} U = u2, we pass along a range for which V = 0, and v^kUku^. Thus, along this side of the polygon, W — ult W —u2, W — u3, etc. are real quantities; positive or negative, which retain the same sign along the whole of this edge. It follows that, as

we pass along this edge, the change in the value of arg (-ttjt), as given

KdWJ

by equation (236), is equal to the change in arg F, the arguments of the factors

(W-u^{W-u^... undergoing no change.

Now arg [-T^n] measures the inclination of the axis V = 0 to the edge of

the polygon at any point, so that if the potygon is to be rectilinear, this must remain constant as we pass along any edge. It follows that there must be no change in arg F as we pass along any side of the polygon.

272 Methods for the Solution of Special Problems [oh. vm

This condition can be satisfied by supposing F to be a pure numerical constant. Taking it to be real, we have, from equation (236),

arg \dw) = Xl &TS(W- Wi) + X2arg(TT- m2) + (237).

On passing through the angular point at which W=u2, the quantities W — Ui, W — u3, etc. remain of the same sign, while the single quantity W — u2 changes sign. Thus arg ( W — u2) increases by tt, whence, by equa- tion (237), arg (-t™-) increases by "Kir.

The axis V=0 does not turn in the TT- plane on passing through the

value W = u2, while arg (-Trs-) measures the inclination of the element of

the polygon in the ^-plane to the corresponding element of the axis V = 0 in the JF-plane.

Hence, on passing through the value W = u2, the perimeter of the polygon in the ^-plane must turn through an angle equal to the increase in

arg (-TTiv) , namely X27r, the direction of turning being from Ox to Oy. Thus

Xx7r, XjTt, ... must be the exterior angles of the polygon, these being positive when the polygon is convex to the axis Ox. Or, if a1} ct2, ... are the interior angles, reckoned positive when the polygon is concave to the axis of x, we must have

X, = — — 1, etc.

7T

Thus the transformation required for a polygon having internal angles

Of i y Gt'z j • • • IS

^ = G{W-uiy~\W-u2y-X (238),

where Wj, u2, ... are real quantities, which give the values of U at the angular points.

323. As an illustration of the use of Schwarz's transformation, suppose the conducting system to consist of a semi-innnite plane placed parallel to an infinite plane.

In fig. 90, let the conductor be supposed to be a polygon ABODE, which is described by following the dotted line in the direction of the arrows. The points A, B, 0, E are all supposed to be at infinity, the points B and 0 coinciding. Let us take A to be W = — oo , B or C to be W = 0, jD to be W = 1 and ^tobe W = + oo . The angles of the polygon are zero at (BO) and 2ir at D. Thus the transformation is

dz _GW-1

dW W

322-325] Conjugate Functions 273

giving upon integration

z=C{W-\ogW+D} (239),

where C, D are constants of integration which may be obtained from the

E

VV=+co

■ <=. <-.„

w=+i c \

->- >~~*s

W=-oo

Fro. 90.

condition that the two planes are to be, say, y = 0 and y = h. From these

conditions we obtain G = — , D = iir, so that the transformation is

z = -{W-\ogW+iir} (240).

IT

On replacing z, W by — z, — W, the transformation assumes the simpler form

z=-(W+\ogW) (241).

7T

III. Successive Transformations.

324. If £= <f>(z), W=f(0 are any two transformations, then by elimi- nation of £, a relation

W=F(z) (242)

is obtained, which may be regarded as a new transformation.

We may regard the relation £ = </> (z) as expressing a transformation from the 2-plane into a £- plane, while the second relation W=f(%) expresses a further transformation from the £-plane into a T7-plane. Thus the final transformation (242) may be regarded as the result of two successive trans- formations.

Two uses of successive transformations are of particular importance.

325. Conductor influenced by line-charge. The transformation

gives, as we have seen (§ 318) the solution when a line-charge is placed at £ = a in front of the plane represented by the real axis of f, Let the further transformation £=f(z) transform the real axis of £ into a surface S, and the point f = a into the point z = z0, so that a =/(^0)« Then the transformation

j.

18

274 Methods for the Solution of Special Problems [ch. vm

gives the solution when a line-charge is placed at z = z0va. the presence of the surface S. In this transformation it must be remembered that U, and not V, is the potential (cf. § 318).

326. Conductors at different potentials. Let us suppose that the trans- formation %=<f)(z) transforms a conductor into the real axis of £. The further transformation W = G + D log £ (§ 318) will give the solution when the two parts of this plane on different sides of the origin are raised to different potentials G and C + ttD.

Thus the transformation obtained by elimination of £, namely

W=G + D\ogcf>(z),

will transform two parts of the same conductor into two parallel planes, and so will give the solution of a problem in which two parts of the same conductor are raised to different potentials.

Examples of the use of Conjugate Functions.

327. Two  examples  of  practical  importance  will  now  be  given  to  illus- 
     

trate the use of the methods of conjugate functions.

Example I. Parallel Plate Condenser. 328. The transformation

• = ^(t-log£-MV)

has been found to transform the two plates in fig. 90 into the positive and negative parts of the real axis of £. The further transformation W = log £ gives the solution when these two parts of the real axis of £ are at potentials 0 and it respectively (§ 326).

Thus the transformation obtained by the elimination of £, namely

z = -(ew-W + iir)

TT

•(243),

will transform the two planes of fig. 90 — one infinite and one semi-infinite — into two infinite parallel planes. Thus equation (243) gives the trans- formation suitable to the case of a semi-infinite plane at distance h from a parallel infinite plane, the potential difference being it.

By the principle of images it is obvious that the distribution on the iipper plate is the same as it would be if the lower plate were a semi- infinite plane at distance 2/i instead of an infinite plane at distance h. The equipotentials and lines of force for either problem are shewn in fig. 91.

325-328]

Conjugate Functions

275

Separating real and imaginary parts in equation (243),

x

IT h

(eucosV-U),

y = - (eusmV -V+tt).

IT

Thus the equipotential V — 0 is the line y = h, the equipotential V = it is the line y — 0.

Fig. 91. On the former equipotential, the relation between x and U is

h

TT

.(244).

When TJ — — co , a; = + co; as t/ increases, x decreases until it reaches a minimum value x — h/ir when U = 0 ; and as J7 further increases through positive values x again increases, reaching x=qc when Z7 = + oo . Thus as U varies while V=0, the path described is the path PQR in fig. 91.

The intensity at any point is

\dW

R =

dz

h\ e

IT W

II"

At a point on the 'equipotential V = 0, the surface-density is

R _ 1

a47r4A(e^-iy

IS— 2

276 Methods for the Solution of Special Problems [ch. vni

At P, U = — oo , so that <7 = -tj ; as we approach Q, a increases and finally

becomes infinite at Q, while after passing Q and moving along QR, the upper side of the plate, a decreases, and ultimately vanishes to the order of e~ u.

The total charge within any range U1} U2 is, by equation (233),

It therefore appears that the total charge on the upper part of the plate QR is infinite.

Let us, however, consider the charges on the two sides of a strip of the plate of width I from Q, i.e. the strip between x — hjir and x = l + K\tt. The two values of U corresponding to the points in the upper and lower faces at which this strip terminates, are from equation (244) the two real roots of

l+hJ±{ev_U) (245).

Of these roots we know that one, say Ult is negative and the other (U^) is positive. If I is large, we find that the negative root U^ is, to a firsfr approximation, equal to

ir ,' h

and this is its actual value when I is very large. Thus the charge on the lower plate within a large distance I of the edge is

h /. h

and therefore the disturbance in the distribution of electricity as we approach Q results in an increase on the charge of the lower plate equal to what would be the charge on a strip of width k/ir in the undisturbed state.

If I is large the positive root of equation (245) is

«t.-i*(i+t).

so that the total charge on a strip of width I of the upper plate approximates, when I is large, to

loo- [ 1 + _ 1

4tt °V hj'

Thus although the charge on the upper plate is infinite, it vanishes in comparison with that on the lower plate.

328, 329]

Conjugate Functions

277

Example IT. Bend of a Ley den Jar.

329. The method of conjugate functions enables us to approximate to the correction required in the formula for the capacity of a Leyden Jar, on account of the presence of the sharp bend in the plates.

A

^=-a

F

f=&

_D

B

Fig. 92.

As a preliminary, let us find the capacity of a two-dimensional condenser formed of two conductors, each of which consists of an infinite plate, bent into an L-shape, the two L's being fitted into one another as in fig. 92.

Let us assume the five points A, B, (CD)/E, F to be £ = — oo , — a, 0,

• b, + oo respectively, and let us for convenience suppose the potential difference which occurs on passing through the value £*=0 to be ir. Then the transformation is

where W = log £ (cf. § 326).

To integrate, we put u = (%+a)~- (£ — b)^, and obtain

-/M^W^.

.(246),

where C is a constant of integration.

To make C vanish, we must have z = Q when u = 0, i.e. at the point E. We shall accordingly take E as origin, so that G = 0.

278 Methods for the Solution of Special Problems [oh. viii

At B, we now have £ = — a, u = oo , and therefore

z = ± IT A a/- + 17tJ..

V a

Thus the distances between the pairs of arms are it a / - A and irA respectively.

Let P be any point in EF which is at a distance from E great compared with EB. Let the value of £ at P be £p, so that t,P is positive and greater than b.

We have Tf = U + iV = log £ so that along the conductor FED, V = 0 and U = log £

The total charge per unit width on the strip EP is, by formula (233),

jydS = ±(UP-UE) = ±(}og!;P-\ogb) (247).

If P is far removed from E, the value of £P is very great, and since

? = ~ (248),

the value of it? will be nearly equal to unity at P. From equation (246),

z = - 2A J - tan-1 lS/j u + 2 A log (1 + u) - A log (1 - u%

so that log (1 - t<2) = 2 log (1 +«) - 2 a/- tan"1 ^/| m - -| (249),

in which the terms log (1 — u2), — z/A, are large at P in comparison with the others. Again, from equation (248), we have

log £= log (em2 + &)- log(l-w2) (250),

in which log £, log (1 — v?) are large at P, in comparison with the term log (ait2 + b). Combining equations (249) and (250),

log £ = log (av? + b) - 2 log (1 + «) + 2 ^ tan"1 y^ it + -J

(251),

in which the terms log £ and -j are large at P in comparison with the other

terms. At P we may put u = 1 in all terms except log £ and z/A, and obtain as an approximation

log & = log (o + 6)- 2 log 2 + 2 Jj/| tan"1 /y/| + §.

329, 330] Multiple-valued Potentials 279

The value of zP -is of course xP + iyP, or EP. Thus, from the equation just obtained, equation (247) may be thrown into the form

/

p 1

<rds = — (log £, - log &)

= I S i1 + ?)-21og2 + 2 ^ tan- J\ + f }...(252).

If the lines of force were not disturbed by the bend, we should have

ads

!

1 fEP\ 4>tt\A )'

Equation (252) shews that I ads is greater than this, by an amount

J E

I K i1 + 1) - - ios 2 + 2 /l tan"' v7?} (253>-

Let us denote the distances between the plates, namely ttA a / - and irA, by h and A; respectively, so that a/ - = - . Expression (253) now becomes

so that the charge on the plate EP is the same as it would be in a parallel plate condenser in which the breadth of the strip was greater than EP by

When h = k, this becomes

£ (| - log, 2) or -279A.

Multiple-valued Potentials.

330. There are many problems to which mathematical analysis yields more than one solution, although it may be found that only one of these solutions will ultimately satisfy the actual data of the problem. In such a case it will often be of interest to examine what interpretation has to be given to the rejected solutions.

The problem of determining the potential when the boundary conditions are given is not of this class, for it has already been shewn (§§ 186 — 188) that, subject to specified boundary conditions, the termination of the poten- tial is absolutely unique. But it may happen that, in searching for the required solution, we come upon a multiple-valued solution of Laplace's equation. Only one value can satisfy the boundary conditions, but the interpretation of the other values is of interest, and in this way we arrive at the study of multiple-valued potentials.

280 Methods for the Solution of Special Problems [ch. viii

Conjugate Functions on a Riemann's Surface.

331. An obvious case of a multiple-valued potential arises from the conjugate function transformation

W = <f>(z) (254),

when (f) is not a single-valued function of z. Such cases have already occurred in §§ 317, 320, 323, etc.

The meaning of the multiple-valued potential becomes clear as soon as we construct a Riemann's surface on which $ (z) can be represented as a single-valued function of position. One point on this Riemann's surface must now correspond to each value of W, and therefore to each point in the IT-plane. Thus we see that the transformation (254) transforms the complete TT-plane into a complete Riemann's surface. Corresponding to a given value of z there may be many values of the potential, but these values will refer to the different sheets of the Riemann's surface. If any region on this surface is selected, which does not contain any branch points or lines, we can regard this region as a real two-dimensional region, and the corresponding value of the potential, as given by equation (254), will give the solution of an electrostatic problem.

332. To illustrate this by a concrete case, consider the transformation

F = .

.(255),

_ -a

TF-plane.

Fig. 93.

z-surface-

7?'

which has already been considered in § 317. The Riemann's surface appro- priate for the representation of the two-valued function z* may be supposed to be a surface of two infinite sheets connected along a branch line which extends over the positive half of the real axis of z.

To regard this surface as a deformation of the TF-plane, we must suppose that a slit is cut along the line OB (fig. 93) in the TF-plane, and that the

331-333] Multiple-valued Potentials 281

two edges of the slit are taken and turned so that the angle lir, which they originally enclosed in the W-plane, is increased to 4>ir, after which the edges are again joined together.

The upper sheet of the Riemann's surface so formed will now represent the upper half of the W-plane, while the lower sheet will represent the lower half. Two points i?, B„ which represent equal and opposite values of W, say ± Wn, will (by equation (255)) be represented by points at which z has the same value; they are accordingly the two points on the upper and lower sheet respectively for which z has the value W02.

A circular path pqrs surrounding 0 in the W-plane becomes a double circle on the ^-surface, one circle being on the upper sheet and one on the lower, and the path being continuous since it crosses from one sheet to the other each time it meets the branch-line.

A line a/3 in the upper half of the W-plane becomes, as we have seen, a parabola a/3 on the upper sheet of the ^-surface. Similarly a line a/3' in the lower half of the W-plane will become a parabola a'/3' on the lower sheet of the ^-surface. The space outside the parabola a/3 on the upper sheet of the ^-surface transforms into a space in the W-plane bounded by the line a/3 and the line at infinity. Consequently the transformation under consideration gives the solution of the electrostatic problem, in which the field is bounded only by a conducting parabola and the region at infinity. The same is not true of the space inside the parabola a/3, for this transforms into a space in the W-plane bounded by both the line a/3 and the axis AOB. It is now clear that the transformation has no application to problems in which the electrostatic field is the space inside a parabola.

In general it will be seen that two points, which are close to one another on one sheet of the ^-surface, but are on opposite sides, of a branch-line, will transform into two points which are not adjacent to one another in the W-plane, and which therefore correspond to different potentials. Conse- quently we cannot solve a problem by a transformation which requires a branch-line to be introduced into that part of the Riemann's surface which represents the electrostatic field.

Images on a Riemann's Surface.

333. In the theory of electrical images, a system of imaginary charges is placed in a region which does not form part of the actual electrostatic field. When a two-dimensional problem is solved by a conjugate function trans- formation, the electrostatic field must, as we have seen, be represented by a region on a single sheet of the corresponding Riemann's surface, and this region must not be broken by branch-lines. The same, however, is not true of the part of the field in which the imaginary images are placed, for this

282 Methods for the Solution of Special Problems [ch. viii

may be represented by a region on one of the other sheets of the Riemann's surface.

To take the simplest possible illustration, suppose that in the £-plane we have a line-charge e along the line represented by the point P, in front of

f- plane z- surface

P»+e P» (upper sheet)

A O B O A

P'*-e P • (lower sliect)

Fig. 94.

the uninsulated conducting plane represented by the real axis AB. The solution, as we know, is obtained by placing a charge — e at the point P', which is the image of P in AOB. The value of the potential (U) is given, as in § 818, by

U+iV=A log

£-&»

r- w

Let us now transform this solution by means of the transformation

£=z$ (256).

The conducting plane A OB transforms into a semi-infinite plane OB, which may be taken to coincide with the branch-line of the Riemann's surface. The charge e at P becomes a charge at a point P on the upper sheet of the surface, while the image at P' becomes a charge at a point P' on the lower sheet. Thus we can replace the semi-infinite conductor OB in the 2-plane by an image at a point P' on the lower sheet of a Riemann's surface, and we obtain the field due to a line-charge and a semi-infinite con cl actor in an ordinary two-dimensional space.

From the transformation used, the potential is found to be given by

/z — Va.

U + iV=A\oz

o

\Jz—"J-

a

in which U is the potential, z — a is the point (a, a) on the upper sheet, and z = — a is the image on the lower sheet.

In calculating a potential on a Riemann's surface, we must not assume the potential of a line-charge e at the point (a, a) to be

0-2e\ogR (257),

where R is the distance from the point (a, a). In fact, this potential would obviously have an infinity both at the point (a, a) on the upper sheet, and also at the point {a, a) on the lower sheet, and 0 would be the potential of two line-charges, one at the point (a, a) on each sheet.

333-335] MuUiple-valued Potentials 283

The appropriate potential-function for a single charge can easily be found.

As in the problem just discussed, it is clear that the potential due to the single line-charge at (a, a) on the upper sheet is the value of U given by

U+ i V= G + A log (VJ - Va)

= 6' + ;! log (rM - a* e¥)

== G + A log \ ( Vr cos -x — V« cos - J + i ( Vr sin - — Va sin - J I , so that

U = G + %A log -If Vr cos^ — Vacos ~] + ( Vr sin - — Va sin -

= C + \ A log {r - 2 Var cos ^ (0 - a) + a},

and if this is to be the potential due to a line-charge e, it is clear, on examining the value of U near the point (a, a), that the value of A must be — 2e. Thus the potential function must be

C-elog {r-2 Var cos £(0- a) + a} (258),

instead of that given by expression (257), namely,

C- e log {r2-2ar cos (#-«) + a2} (259).

It will be noticed that both expressions are single- valued for given values of (r, 6), but that for a given value of z, expression (258) has two values, corresponding to two values of 6 differing by 2tt, while expression (259) has only one value. Or, to state the same thing in other words, the expression (259) is periodic in 0 with a period 2tt, while expression (258) is periodic with a period 4)ir.

Potential in a Riemann's Space.

334. Sommerfeld* has extended these ideas so as to provide the solution of problems in three-dimensional space.

His method rests on the determination of a multiple-valued potential function, the function being capable of representation as a single-valued function of position in a " Riemann's space," this space being an imaginary space which bears the same relation to real three-dimensional space as a Riemann's surface bears to a plane.

335. The best introduction to this method will be found in a study of the simplest possible example, and this will be obtained by considering the three-dimensional problem analogous to the two-dimensional problem already discussed in § 333.

• "Ueber verzweigte Potentiate im Raum," Froc. Lund. Math. Soc. 28, p. 395, and 30, p. 161.

284 Methods for the Solution of Special Problems [oh. viii

We suppose that we have a single point-charge in the presence of an uninsulated conducting semi-infinite plane bounded by a straight edge. Let us take cylindrical coordinates r, 6, z, taking the edge of the plane to be r = 0, the plane itself to be 8 = 0, and the plane through the charge at right angles to the edge of the conductor to be z = 0. Let the coordinates of the point-charge be a, a, 0.

The Riemann's space is to be the exact analogue of the Riemann's surface described in § 332. That is to say, it is to be such that one revolu- tion round the line r — 0 takes us from one " sheet " to the other of the space, while two revolutions bring us back to the starting-point. Thus, for a function to be a single-valued function of position in this space, it must be a periodic function of 0 of period 4nr.

Let us denote by f(r, 6, z, a, a, 0) a function of r, 6, and z which is to satisfy the following conditions :

(i) it must be a solution of Laplace's equation ;

(ii) it must be a continuous and single-valued function of position in the Riemann's space ;

(iii) it must have one and only one infinity, this being at the point a, cl, 0 on the first " sheet " of the space, and the function

approximating near the point to the function -p, where B, is

the distance from this point ; (iv) it must vanish when r = co .

It can be shewn, by a method exactly similar to that used in § 186, that there can be only one function satisfying these conditions. Hence the func- tion f(r, 0, z, a, a, 0) can be uniquely determined, and when found it will be the potential in the Riemann's space of a point-charge of unit strength at the point a, a, 0.

Consider now the function

f(r, 0, z, a, a, 0) -/(r, 6, z, a, -a, 0) (260),

which is of course the potential of equal and opposite point-charges at the point a, a, 0, and at its image in the plane 6 = 0, namely, the point a, — a, 0.

This function, by conditions (i) and (iv), satisfies Laplace's equation and vanishes at infinity. On the first sheet of the surface, on which a varies from 0 to 2ir (or from 4>tt to 67r, etc.), it has only one infinity, namely, at

a, a, 0, at which it assumes the value -5.

Jlv

From the conditions which it satisfies, the function /(r, 0, z, a, a, 0) must clearly involve 6 and a only through 6 — a, and must moreover be an even function of 6 - cr. It follows that, when 6 = 0, expression (260) vanishes.

335, 336] Multiple-valued Potentials 285

Again, since the function f is periodic in 6 with a period 2tt, it follows that, when 6 = — 2ir, expression (2G0) may be written in the form

f(r, 2tt, z, a, a, 0) -/(r, - 2tt, z, a, - a, 0), and this clearly vanishes. Thus expression (260) vanishes when 0 = 0 and when 6 = 2tt. That is to say, it vanishes on both sides of the semi-infinite conducting plane.

It is now clear that expression (260) satisfies all the conditions which have to be satisfied by the potential. The problem is accordingly reduced to that of the determination of the function /(?*, 6, z, a, a, 0).

336.     Let  us  write 
     

r = ep, a = ep',

then the distance R from r, 0, z to a, a, 0 is given by R* _ r2 _ 2ar cos (6 - a) + a2 + z"

= 2ar {cos i (p — p) — cos (0 — a)] + z\ Take new functions R' and/(w) given by

Kl = 2ar (cos i (p - p) - cos (0 - it)} + z\

J \ ' gilt gta *

The function f(u) has infinities when u = a, a ± 2ir, a ± 4ur, ..., its residue being unity at each infinity. Also, when u = a, the value of R' becomes R. Hence the integral

£,f(u)du (261),

where the integral is taken round any closed contour in the it-plane which surrounds the value u = a, but no other of the infinities off(ii)} will have as

its value 2iir x ^ . We accordingly have

1 1 fl eiu

S = ^JB'i=37> (262).

The integral just found gives a form for the potential function in ordinary space which, as we shall now see, can easily be modified so as to give the potential function in the Riern ami's space which we are now considering.

We notice first that p> , regarded as a function of r, 6, and z, is a solution

of Laplace's equation, whatever value u may have. Hence the integral (261) will be a solution of Laplace's equation for all values of f(u), for each term of the integrand will satisfy the equation separately.

If we take

/:

e2 — e2

286 Methods for the Solution of Special Problems [ch. vin

we see that the infinities of /(w) occur when u — a, a± 4<7r, a ± S-n-, etc., and the residue at each is unity. Hence, if we take the integral round one infinity only, say u = a, the value of

is/iff/M*1 (263)

will become identical with -~ at the point at which R' = 0. Moreover,