VA VB — — - ... —

Ri R2 R\ R%

The arrangement of conductors in parallel is therefore seen to offer the same resistance to the current as a single conductor of resistance

1

D + r> T •••

Mi JX^

The reciprocal, of the resistance of a conductor is called the " conductivity " of the conductor. The conductivity of the system of conductors arranged

in

parallel is -ft+ rr+-", and is therefore equal to the sum of the

314

Steady Currents in Linear Conductors [ch. ix

conductivities of the separate conductors. Also we have seen that the current divides itself between the different conductors in the ratio of their conductivities

Measurements.

The Measurement of Current.

349. The instrument used for measuring the current passing in a circuit at any given instant is called a galvanometer. The theory of this instrument will be given in a later chapter (Chap. xiu).

For measuring the total quantity of electricity passing within a given time an instrument called a voltameter is sometimes used. The current, in passing through the voltameter, encounters a number of discontinuities of potential in crossing which electrical energy becomes transformed into chemical energy. Thus a voltameter is practically a voltaic cell run back- wards. On measuring the amount of chemical energy which has been stored in the voltameter, we obtain a measure of the total quantity of electricity which has passed through the instrument.

The Measurement of Resistance.

350. The Resistance Box. A resistance box is a piece of apparatus which consists essentially of a collection of coils of wire of known resistances, arranged so that any combination of these coils can be arranged in series. The most usual arrangement is one in which the two extremities of each coil are brought to the upper surface of the box, and are there connected to a thick band of copper which runs over the surface of the box. This

Fig. 97,

band of copper is continuous, except between the two terminals of each coil, and in these places the copper is cut away in such a way that a copper plug can be made to fit exactly into the gap, and so put the two sides of the gap in electrical contact through the plug. The arrangement is shewn diagram- matically in fig. 97. When the plug is inserted in any gap DE, the plug and the coil beneath the gap DE form two conductors in parallel connecting

348-351] Measurements 315

the points D and E. Denoting the resistances of the coil and plug by RC) Rp, the resistance between D and E will be

Rc Rp

and since Rp is very small, this may be neglected. When the plug is removed, the resistance from D to E may be taken to be the resistance of the coil. Thus the resistance of the whole box will be the sum of the resistances of all the coils of which the plugs have been removed.

351. The Wheatstone Bridge. This is an arrangement by which it is possible to compare the resistances of conductors, and so determine an unknown resistance in terms of known resistances.

The " bridge " is represented diagrammatically in fig. 98. The current enters it at A and leaves it at D, these points being connected by the lines

ABD, ACD arranged in parallel. The line AD is composed of two con- ductors AB, BD of resistances Rly R2, and the line ACD is similarly composed of two conductors AG, CD of resistances R3, R4.

If current is allowed to flow through this arrangement of conductors, it will not in general happen that the points B and C will be at the same potential, so that if B and G are connected by a new conductor, there will usually be a current flowing through BG. The method of using the Wheatstone bridge consists in varying the resistances of one or more of the conductors R1} R2, R3, R4 until no current flows through the conductor BG.

When the bridge is adjusted in this way, the points B, G must be at the same potential, say v. Let VA, VD denote the potentials at A and D, and let the current through ABD be G. Then, by Ohm's Law,

VA-v=CR1, v-Vj>=GR2,

,, , Ri Xi-v

so that -^ = ■ Tr .

R2 v-Vj)

From a similar consideration of the flow in AGD, we obtain

R3 _ VA - v R* v-VD'

7? 7? so that we must have W = -d"3 (272),

316 Steady Currents in Linear Conductors [oh. ix

as the condition to be satisfied between the resistances when there is no current in BC.

Clearly by adjusting the bridge in this way we can determine an unknown resistance R^ in terms of known resistances R2, R3, R4. In the simplest form of Wheatstone's bridge, the line AGD is a single uniform wire, and the position of the point 0 can be varied by moving a "sliding contact" along the wire. The ratio of the resistances Rs : R4 is in this case simply the ratio of the two lengths A G, CD of the wire, so that the ratio R^ : R2 can be found by sliding the contact G along the wire AGD until there is observed to be no current in BG, and then reading the lengths AG and GD.

Examples of Currents in a Network.

I. Wheatstone's Bridge not in adjustment.

352. The condition that there shall be no current in the " bridge " BG in fig. 98 has been seen to be that given by equation (272).

B

Suppose that this condition is not satisfied, and let us examine the flow of currents which then takes place in the network of conductors. Let the conductors AB, BD, AG, GD as before be of resistances R1} R2, R3, Ri} and let the currents flowing through them be denoted by x1} x2, x3, x4. Let the bridge BG be of resistance Rb, and let the current flowing through it from B to G be xb.

From Kirchhoff 's Laws, we obtain the following equations :

(Law I, point B) x1 — x2 — xb = 0 (273),

(Law I, point G) x3 — x4 + xb = 0 (274),

(Law II, circuit ABG) x^ +xbRb — x3R3 = 0 (275),

(Law II, circuit BCD) xbRb + x4R4 - x2R, = 0 (276).

These four equations enable us to determine the ratios of the five currents xx, x2, x3, x4, xb. We may begin by eliminating #2 and xt from equations (273), (274) and (276), and obtain

xb (Rb + R2 + R4) + x3R4 — xxR2 = 0,

and from this and equation (275),

Xb X3 Xi

R%R3 — RxRt R1 (Rb + R2 + R4) + RbRa R3 (Rb -f R3 + R4) + RbR4

(277).

351-353]

Flow of Currents in a Network

317

The ratios of the other currents can be written down from symmetry.

If the total current entering at A is denoted by X, we have X = xx + x3. Thus if each of the fractions of equations (277) is denoted by 6,

X = 6 {(R, + R3) (R, + R4) + Rb(R1 + R2 + Rs + R4)} (278),

and this gives 6, and hence the actual values of the currents, in terras of the total current entering at A.

The fall of potential from A to D is given by

VA - Vj) = Rxxx + R2x2}

and from equations (277) this is found to reduce to

vA-Vj>=\e,

where

X = RXR3 (Ri + R4) + R2Ri (R3 + Ri) + Rb (R1R3 + R^Ri + RiRt + R2R3),

so that \ is the sum of the products of the five resistances taken three at a time, omitting the two products of the three resistances which meet at the points B and C.

There is now a current X flowing through the network, and having a fall of potential VA — Vj). Hence the equivalent resistance of the network

Va-Vj,

xd x

(R, + R3) (R2 + R4) + Rb (R, + R2 + RS + R4) ' by equation (278).

II. Telegraph wire with faults.

353. As a more complex example of the flow of electricity in a system of linear conductors, we may examine the case of a telegraph wire, in which there are a number of connexions through which the current can leak to earth. Such leaks are technically known as "faults."

Fi

Fo

To

R;

F«-j

R,

R.

Fio. 100

^

»+i

B

Let AB be the wire, and let Flt F2) ... Fn-lt Fn be the points on it at which faults occur, the resistances through these faults being Ru R2>...

318

Steady Currents in Linear Conductors [oh. ix

Rn-i, Rn> and the resistances of the sections AFlt F1F2, ... Fn_rFn and FnB being ru r2, ... rn, rn+1. Let the end B be supposed put to earth, and let the current be supposed to be generated by a battery of which one terminal is connected to A while the other end is to earth.

The equivalent resistance of the whole network of conductors from A to

earth can be found in a very simple way. Current arriving at Fn from the

section Fn-xFn passes to earth through two conductors arranged in parallel,

of which the resistances are Rn and rn+1. Hence the resistance from Fn to

earth is

1

-K-n rn+i and the resistance from Fn^ to earth, through Fn) is

1

.(279).

■f

■t*"n rn+1

Current reaching 2^_, can, however, pass to earth by two paths, either through the fault at Fn_lt or past Fn. These paths may be regarded as arranged in parallel, their resistances being Rn-r and expression (279) respectively. Thus the equivalent resistance from Fn_x is

1

• K-'

itn_!

rn +

or, written as a continued fraction,

1 1

We can continue in this way, until finally we find as the whole resistance from A to earth,

J_ 1 J_ 111

Tl + jRf1 + n + R2-1 +'" +rn + Ru'1 + rn+1 '

If the currents or potentials are required, it will be found best to attack the problem in a different manner.

Let VA, Vx, V2, ... be the potentials at the points A, F1} F2, ..., then, by Ohm's Law,

the current from Fs^ to Fg

rs

„ Fs to Fs

K-K

S 'S+l

S+l

r,

S+l

Fg through the fault =

-5. Rg'

353, 354] Flow of Currents in a Network 319

Hence, by Kirckhoff's first law,

Z-V... V

= 0,

V —V V —V

or T^+1 rs+f1 - Vs (Rs-> + rr1 + n+r1) + Vs., rg~l = 0,

and from this and the system of similar equations, the potentials may be found.

If all the R's are the same, and also all the r's are the same, the equation reduces to a difference equation with constant coefficients. These conditions might arise approximately if the line were supported by a series of similar imperfect insulators at equal distances apart. The difference equation is in this case seen to be

TSh-E^+IQ + IU-O.

and if we put 1 + — = cosh a,

the solution is known to be

l^ = ^coshsa + B sinhsa (280),

in which A and B are constants which must be determined from the conditions at the ends of the line. For instance to express that the end B is to earth, we have Vn+1 = 0, and therefore

A = - B tanh (n + l)a.

III. Submarine cable imperfectly insulated.

354. If we pass to the limiting case of an infinite number of faults, we have the analysis appropriate to a line from which there is leakage at every point. The conditions now contemplated may be supposed to be realised in a submarine cable in which, owing to the imperfection of the insulating sheath, the current leaks through to the sea at every point.

The problem in this form can also be attacked by the methods of the infinitesimal calculus. Let V be the potential at a distance x along the cable, V now being regarded as a continuous function of x. Let the resistance of the cable be supposed to be R per unit length, then the re- sistance from x to x + dx will be Rdx. The resistance of the insulation from

o x to x + dx, being inversely proportional to dx, may be supposed to be -r- .

Let G be the current in the cable at the point x, so that the leak from

dC dx

dC the cable between the points x and x + dx is — -j- dx. This leak is a current

320 Steady Currents in Linear Conductors [ch. ix

dx

which flows through a resistance -5- with a fall of potential V. Hence by

Ohm's Law,

V=-^dx(®-) dx \dxj '

35 — iff (281)-

dV Also, the fall of potential along the cable from x to x + dx is — -r— dx, the

(too

current is C, and the resistance is Rdx. Hence by Ohm's Law,

dV

-(^~ = RC (282).

dx

Eliminating G from equations (281) and (282), we find as the differential

equation satisfied by V,

d_(IdV\ V dx\R dx)~ S'

If R and S have the same values at all points of the cable, the solution of this equation is

V = A cosh ^/ -<= x + B sinh */ -5 x, which is easily seen to be the limiting form assumed by equation (280).

Generation of Heat in Conductors. The Joule Effect.

355. Let P, Q be any two points in a linear conductor, let Vp, Vq be the potentials at these points, R the resistance between them, and x the current flowing from P to Q. Then, by Ohm's Law,

Vp-Vq = Rx '. (283).

In moving a single unit of electricity from Q to P an amount of work is done against the electric field equal to Vp — Vq. Hence when a unit of electricity passes from P to Q, there is work done on it by the electric field of amount Vp — Vq. The energy represented by the work shews itself in a heating of the conductor.

The electron theory gives a simple explanation of the mechanism of this transforma- tion of energy. The electric forces do work on the electrons in driving them through the field. The total kinetic energy of the electrons can, as we have seen (§ 345 a), be regarded as made up of two parts, the energy of random motion and the energy of forward motion. The work done by the electric field goes directly towards increasing this second part of the kinetic energy of the electrons. But after a number of collisions the direction of the velocity of forward motion is completely changed, and the energy of this motion has become indistinguishable from the energy of the random motion of the electrons. Thus the collisions are continually transforming forward motion into random motion, or what is the same thing, into heat.

354-356] Generation of Heat 321

We are supposing that x units of electricity pass per unit time from P to Q. Hence the work done by the electric field per unit time within the region PQ isx(Vp- VQ), and this again, by equation (283), is equal to Rx2.

Thus in unit time, the heat generated in the section PQ of the con- ductor represents Rx2 units of mechanical energy. Each unit of energy is

equal to -j units of heat, where J is the " mechanical equivalent of heat."

Thus the number of heat-units developed in unit time in the conductor PQ will be

Rx2

-J- (284).

It is important to notice that in this formula x and R are measured in electrostatic units. If the values of the resistance and current are given in practical units, we must transform to electrostatic units before using formula (284).

Let the resistance of a conductor be R' ohms, and let the current flowing through it

be xf amperes. Then, in electrostatic units, the values of the resistance R and the current

x are given by

/?' R = x-^r-r. and x=3x I09a/. Ox 1011

Thus the number of heat-units produced per unit time is

R* (3 x 103)2 J ~ 9x10". J '

and on substituting for J its value 4-2 x 107 in c.G.s.- centigrade units, this becomes

0-24 fi'A

Generation of Heat a minimum.

356. In general the solution of any physical problem is arrived at by the solution of a system of equations, the number of these equations being equal to the number of unknown quantities in the problem. The condition that any function in which these unknown quantities enter as variables shall be a maximum or a minimum, is also arrived at by the solution of an equal number of equations. If it is possible to discover a function of the unknown quantities such that the two systems of equations become identical, — i.e. if the equations which express that the function is a maximum or a minimum are the same as those which contain the solution of the physical problem — then we may say that the solution of the problem is contained in the single statement that the function in question is a maximum or a minimum.

Examples of functions which serve this purpose are not hard to find. In

§ 189, we proved that when an electrostatic system is in equilibrium, its

potential energy is a minimum. Thus the solution of any electrostatic

problem is contained in the single statement that the function which

j. 21

322 Steady Currents in Linear Conductors [ch. ix

expresses the potential energy is a minimum. Again, the solution of any dynamical problem is contained in the statement that the "action" is a minimum, while in thermodynamics the equilibrium state of any system can be expressed by the condition that the " entropy " shall be a maximum. It will now be shewn that the function which expresses the total rate of generation of heat plays a similar r61e in the theory of steady electric currents.

357. Theorem. When a steady current flows through a network of conductors in which no discontinuities of potential occur {and which, therefore, contains no batteries), the currents are distributed in such a way that the rate of generation of heat in the network is a minimum, subject only to the conditions imposed by Kirchhoff's first law ; and conversely.

To prove this, let us select any closed circuit PQR ... P in the network, and let the currents and resistances in the sections PQ, QR, ... be xl, x%, ... and R1} R2, Let the currents and resistances in those sections of the net- work Avhich are not included in this closed circuit be denoted by xa, %i, ... and Ra, Rf,, .... Then the total rate of production of heat is

XRa^ + tR^ (285).

A different arrangement of currents, and one moreover which does not violate Kirchhoff's first law, can be obtained in imagination by supposing all the currents in the circuit PQR ... P increased by the same amount e. The total rate of production of heat is now

XRaXa* + ^R1 Oz + *)V

and this exceeds the actual rate of production of heat, as given by expression

(285), by

2R1(2x1e + e*) (286).

Now if the original distribution of currents is that which actually occurs in nature, then

XR.x, = 0,

by Kirchhoff's second law. Thus the rate of production of heat, under the new imaginary distribution of currents, exceeds that in the actual distribu- tion by e'XRi, an essentially positive quantity.

The most general alteration which can be supposed made to the original system of currents, consistently with Kirchhoff's first law remaining satisfied, will consist in superposing upon this system a number of currents flowing in closed circuits in the network. One such current is typified by the current e, already discussed. If we have any number of such currents, the resulting increase in the rate of heat-production

= XR, (^ + e + e' + e" + . . .)2 - ^RlXl\

356-358] Generation of Heat 323

where e, e', e", ... are the additional currents flowing through the resistance i2j. As before this expression

= 2%R1x1 (e + e'+ e" + ...) + XR, (e + / + e" + ...)2 = 2^1(e + e/ + e,/ + ...)3,

by Kirchhoff's second law. This is an essentially positive quantity, so that any alteration in the distribution of the currents increases the rate of heat- production. In other words, the original distribution was that in which the rate was a minimum.

To prove the converse it is sufficient to notice that if the rate of heat- production is given to be a minimum, then expression (286) must vanish as far as the first power of e, so that we have

tR.x, = 0,

and of course similar equations for all other possible closed circuits. These, however, are known to be the equations which determine the actual dis- tribution.

358. Theorem. When a system of steady currents flows through a net- work of conductors of resistances RltR2, ..., containing batteries of electromotive forces E1} E2) ..., the currents x1} x2, ... are distributed in such a way that the function

ZRx*-2'$Ex (287)

is a minimum, subject to the conditions imposed by Kirchhoff's first law ; and, conversely.

As before, we can imagine the most general variation possible to consist of the superposition of small currents e, e, e", ... flowing in closed circuits. The increase in the function (287) produced by this variation is

2i2 [(x + e + e' + ...)2 - a2] - 2XE [(x + e+ e' +...)- x]

= 2e . (%Rx - IE) + 2e' (...) + ...

• XR(e + e' + ...y (288).

If the system of currents x, x, ... is the natural system, then the first line of this expression vanishes by Kirchhoff's second law (cf. equations (270)), and the increase in heat-production is the essentially positive quantity

2E(e+e'+...)V shewing that the original value of function (287) must have been a minimum.

Conversely, if the original value of function (287) was given to be a

minimum, then expression (288) must vanish as far as first powers of e, e, ...,

so that we must have

2ifo = E, etc.,

shewing that the currents x, x, ... must be the natural system of currents.

21—2

324 Steady Currents in Linear Conductors [ch. ix

359. Theorem. If two points A , B are connected by a network of con- ductors, a decrease in the resistance of any one of these conductors will decrease (or, in special cases, leave unaltered) the equivalent resistance from A to B.

Let x be the current flowing from A to B, R the equivalent resistance of the network, and VA — VB the fall of potential. The generation of heat per unit time represents the energy set free by x units moving through a potential-difference VA - V£. Thus the rate of generation of heat is

*(VA-VS),

or, since Vi — VB = Rx, the rate of generation of heat will be Rx2.

Let the resistance of any single conductor in the network be supposed decreased from Rx to RJ, and let x1 be the current originally flowing through the network. If we imagine the currents to remain unaltered in spite of the change in the resistance of this conductor, then there will be a decrease in the rate of heat-production equal to (R1 — R/) xf. The currents now flowing are not the natural currents, but if we allow the current entering the network to distribute itself in the natural way, there is, by § 357, a further decrease in the rate of heat-production. Thus a decrease in the resistance of the single conductor has resulted in a decrease in the natural rate of heat- production.

If R, R' are the equivalent resistances before and after the change, the two rates of heat-production are Rx2 and R'x2. We have proved that R'x2 < Rx2, so that R' < R, proving the theorem.

General Theory of a Network.

360. In addition to depending on the resistances of the conductors, the flow of currents through a network depends on the order in which the con- ductors are connected together, but not on the geometrical shapes, positions or distances of the conductors. Thus we can obtain the most general case of flow through any network by considering a number of points 1, 2, ... n, con- nected in pairs by conductors of general resistances which may be denoted by

R12) R23, If, in any special problem, any two points P, Q are not joined

by a conductor, we must simply suppose RPQ to be infinite. Discontinuities of potential must not be excluded, so we shall suppose that in passing through the conductor FQ, we pass over discontinuities of algebraic sum Epq. This is the same as supposing that there are batteries in the arm PQ of total electromotive force EPQ. We shall suppose that the current flowing in PQ from P to Q is xPQ) and shall denote the potentials at the points 1, 2, ... by

The total fall of potential from P to Q is VP— VQ, but of this an amount

359, 360] General Theory of a Network 325

— EPQ is contributed by discontinuities, so that the aggregate fall from P to Q which arises from the steady potential gradient in conductors will be

VP-VQ + EPQ. Hence, by Ohm's Law,

Vp— Vq + HjpQ = KpqXpQ.

If we introduce a symbol KPQ to denote the conductivity -p — , we have

the current given by

xPQ = Kpq(Vp-Vq + EpQ) (289).

Suppose that currents Xx, X2, ... enter the system from outside at the points 1, 2, ..., then we must have

4i = #12 T #13 T #14 T • • • t

since there is to be no accumulation of electricity at the point 1, and so on for the points 2, 3, .... Substituting from equations (289) into the right hand of this equation,

Zi - Kn (K- K+ El2) + K1S (K- V3 + E13) + ... = V1(K12 + K13 + ...)

-(K12V2 + K13V3 + ...) + K12E12 + K13El3 + (290).

The symbol KPP has so far had no meaning assigned to it. Let us use it to denote — (KP1 + KP2 + KP3 + ...); then equation (290) may be written in the more concise form

X1 = -(K11V1 + K12V3 + ...) + K12EW + K13E19+ (291).

There are n equations of this type, but it is easily seen that they are not all independent. For if we add corresponding members we obtain

Zi + X2+ ... + Xn = -ZV1(KU + K12 + ... + Km) + 22 (KPQEPQ + KQPEQP).

i

The first term on the right vanishes on account of the meaning which has been assigned to Ku, etc.; while the second term vanishes because EPQ = — EQP, while KPQ = Kqp. Thus the equation reduces to

X1 + X2 + ...+Xn = 0,

which simply expresses that the total flow into the network is equal to the total flow out of it, a condition which must be satisfied by Xu X2, ... Xn at the outset. Thus we arrive at the conclusion that the equations of system (291) are not independent.

This is as it should be, for if the equations were independent, we should have n equations from which it would be possible to determine the values of 1^, V2, ... in terms of X1} X2, ...; whereas clearly from a knowledge of the currents entering the network, we must be able to determine differences of potential only, and not absolute values.

326 Steady Currents in Linear Conductors [ch. ix

To the right-hand side of equation (291), let us add the expression

(Kn + Kls + ... + Km)Vn, of which the value is zero by the definition of Kn. The equation becomes

Kn(V1-Vn) + Kia(Vi-Vn) + ... + K1>n-1(Vn-1-Vn)

= — XY + KnE12 + Kl3E13 + ... + KmEm.

There are n equations of this type in all. Of these the first (n — 1) may be regarded as a system of equations determining

V-V V— V V —V

That these equations are independent will be seen a posteriori from the fact

that they enable us to determine the values of the n — 1 independent

quantities

V— V V- V V ,-V

'i 'n> '2 'n> ••'> 'n—l 'n-

Solving these equations, we have

— X\ + K\i E\i + • • • + Km Em ,

— X2 + K2l E21 + ... + Km E2n ,

it 22 > 1 1 23 )

-"-i,n-i

■*» 2,71—1

— -^n— l r" -^ti— 1,1-^n— 1,1 + • •• + J^-n~z,n ^Jn—,ni -**-n— i,a> -l-n— i,3j •••> -"-n— i,ti— l

Km Ki3>

K

••> ■'M.n-i

it2l> -^22) -^23) •'•> ^ 2, 71—1

-**-n— 1,1 j J-^-n— 1,2> i* 7i— 1,3> •••> 1*- n—l, n—l

The current flowing in conductor In follows at once from equation (289), and the currents in the other conductors can be written down from symmetry.

If we denote the determinant in the denominator of the foregoing equation by A, and the minor of the term Kpq by AP<2, we find that the value of Vy — V,x can be expressed in the form

K-K= (-X1 + K12E12+... + KmEm)^

• (-X2 + K21E21+ ... + K2nE2n)^ + (292).

361. Suppose first that the whole system of currents in the network is produced by a current X entering at P and leaving at Q, there being no batteries in the network. Then all the E's vanish, and all the X's vanish except XP and XQ, these being given by

AP = — 2L n = .A .

3G0-362] General Theory of a Network 327

Equation (292) now becomes

V V — Y Api Y ^Q1

"l— Yn= ~ Ap-^- — Ac -£-

= ^ (* ~ APl)'

so that K- K = (X- K) - 0£- K)

= ^ (AQ1 - Ag2 - AP1 + Ap,) (293).

Replacing 1, 2 by P, Q and P, Q by 1, 2, we find that if a current X enters the network at 1 and leaves it at 2, the fall of potential from P to Q is

VP-VQ = ^(A2P- A2Q- A1P + A1Q) (294),

and since Arg = Ag,., it is clear that the right-hand members of equations (293) and (294) are identical.

From this we have the theorem :

The potential-fall from A to B when unit current traverses the network from G to D is the same as the potential-fall from G to D when unit current traverses the network from A to B.

362. Let it now be supposed that the whole flow of current in the network is produced by a battery of electromotive force E placed in the conductor PQ. We now take all the Z's equal to zero in equation (292) and all the E's equal to zero except EPQ which we put equal to E, and EQP which we put equal to — E. We then have

Ap, . rr „ AC

A

Vl-Vn = KPQEPQ ~p + EQPEQP -

= ^(AP1-Ayi).

Hence K- K=^%^(AP1- Ap,- AQ1 + A^) (295),

and, by equation (289), the current flowing in the arm 12 is

a12 = K»K*E (Api - Ap, - AQ1 + AQ2) (296).

This expression remains unaltered if we replace 1, 2 by P, Q and P, Q by 1, 2. From this we deduce the theorem :

The current which flows from A to B when an electromotive force E is introduced into the arm GD of the network, is equal to the current which flows from G to D when the same electromotive force is introduced into the arm AB.

328

Steady Currents in Linear Conductors [ch. ix

Conjugate Conductors.

363. The same expression occurs as a factor in the right-hand members of each of the equations (293), (294), (295), and (296), namely,

Api+A^-A^-Apa (297).

If this expression vanishes, the two conductors 12 and PQ are said to be " conjugate."

By examining the form assumed by equations (293) to (296), when expression (297) vanishes, we obtain the following theorems.

Theorem I. If the conductors AB and CD are conjugate, a current entering at A and leaving at B will produce no current in CD. Similarly, a current entering at C and leaving at D will produce no current in AB.

Theorem II. If the conductors AB and CD are conjugate, a battery introduced into the arm AB produces no current in CD. Similarly, a battery introduced into the arm CD produces no current in AB.

As an illustration of two conductors which are conjugate, it may be noticed that when the Wheatstone's Bridge (§ 352) is in adjustment, the conductors AD and BC are conjugate.

Equations expressed in Symmetrical Form.

364. The determinant A is not in form a symmetric function of the n points 1, 2, ..., n, so that equations and conditions which must necessarily involve these n points symmetrically have not yet been expressed in symmetrical form.

We have, for instance,

A„ =

M3

li. 21 , ■"■&) -*£ 24 ) -"- 25 > • • • > ■"■ 2, n— 1

•"-31 > "-32 J -"-34 J A35, •••> -^3,n— 1

-"-n-i,i> AM_ 1,2> An-i,4) -K-n—i.S) •••> &-n—,n—\

in which the points which enter unsymmetrically are not only 1 and 3, but also n. Similarly, we have

Au = -

jfi2l) -^22> A23, A25, •••; -"2,71— 1

A31, A32, A 33, A35, •••> **-3,n—l

-tt-n-i,i> An_i)2, J^n-i,3> An_i>6, •••> -K-n- l,n-l

so that, on subtraction,

A13-A14 =

A21, A22, ^23+^24,

A31, A32, A 33 + A34,

#25,

^35,

. . . , IV 2> n— 1 • • • > **- 3, n—i

ii»_i,ij An-l,2, An-1,3 + An_li4, iln_ii5, ..., An_lin_i

363, 364] General Theory of a Network 329

From the relation

KP1 + KP2 + ... + KPin_l + KPin = 0 (298),

it follows that the sum of all the terms in the first row of the above deter- minant is equal to — K2>n, the sum of all the terms in the second row is equal to — K3)n, and so on. Thus the equation may be replaced by

AU-AM «(-!)»

-^21> -»*22> A%5>

A-31) -^32) -*^3S>

. . . , A 2, n— 1 , J^-2,Jl

... t A 3> jj,—! , A 3> n

Am— i,D -ft-n— i,2> -"^n— 1,5) •••> -&n— i,n— 1> -K-n— i,n

and similarly,

^28 A24=(— ■•■) -*Mli -&12) -^15) »••> A1>n

-^31) -"-32) ^35> •••> -^3,n

Aw— 1,1» An— 1,2> An— 1,5) •••> -^n-i,n

These two determinants differ only in their first row, so that on sub- traction,

(Au-A^-CA^-AaO

= (-!)»

Kn + K^, Kl2 + K22, K16 + K2i, ..., Khn + K2>n

K3i, -»^32> ^35) •••) -"-3,n

-n-n-1,1)

A3I) ^32> -A

-n— 1,2>

An— 1,5) •••> -^-n— l,n

... , A;

3,71

-&n-i,i) A«— 1,2j -"-n— 1,5) •••> -K n— 1,1 -^n,i> ^n,2, -&n,5> •••> &~n,n

•(299),

the last transformation being effected by the use of relation (298).

The relation which has now been obtained is in a symmetrical shape. If D is a symmetrical determinant given by

D =

-^lD -*M2> -"-13) •••) Ai.n

A2i, -&22> -^-23) •••) -"-2,n

■Z*-n,> -^n,2> An,3) »••) A«,n

then the determinant on the right-hand of equation (299) is obtained from D by striking out the lines and columns which contain the terms Kn and K24. Thus equation (299) may be written in the form

A13 + A24 - A^ - A14 =

dKlsdK

13UJJ.24

330 Steady Currents in Linear Conductors

Again the determinant A given by

[CH. IX

A =

•*^n> -^12) -^13> •••> -"-i,n— 1

■^21) -^22> -^-23) •••> -^-2,n— 2

-^n-i,i> -"-n— 1,2> -**n-i,3> •••> -^-n— i,n— l

.(300)

may be written in the form

A =

dD

dKntn'

This is not of symmetrical form, for the point n enters unsymmetrically. We can, however, easily shew that the value of A is symmetrical, although its form is unsymmetrical.

By application of relation (298), we can transform equation (300) into

A =

-■ n, l > -"■ n, 2 > -- n, 3 > • • • > -"- n, n— l

■■ 21 > ■"■ 22 > ■■ 23 J "• > ■"■ 2, n— 1

■"■n— i,i> ■**-«— 1,2> -ti-n— 1,3» •••> -"-n— i,n— l = ( — -I) -^21) -^22) -&23) •••> l*-2,n— 1

J-*-n— 1,1> ■'in— 1,2) -'i-n— 1,3) •••) -K-n— i,n— 1 -&n,i) -^-n, 2) -^-n.S) •••» -^n,ra— 1

-ft-22) -"-23) •••) -"^2,n— 1) -**-2,n

--n— 1,2) --n— 1,3) •••» -'i-n— i,n— l) ^n-i,fl

7f

n,2)

n,3)

• . . , ii 7i, n— l )

«,n

8Z) 3^ii *

Thus A is the differential coefficient of Z) with respect to either ifu or Kn>n, or of course with respect to any other one of the terms in the leading diagonal of P. Thus, if K denote any term in the leading diagonal of P, we have

and this virtually expresses A in a symmetrical form.

We can now express in symmetrical form the relations which have been obtained in §§ 360 to 362, as follows :

I. (§ 362.) The conductors 1, 2 and P, Q will be conjugate if

d-P

dKhPdK.2tQ

= 0.

364-366] Slowly-varying Currents 331

II. (Equation 293.) If the conductors 1, 2 and P, Q are not conjugate, a current X entering at P and leaving at Q produces in 1, 2 a fall of potential given by

-—-* — a^— •

dK