A Treatise on Electricity and Magnetism, Vol. 1 (1881) — part 18 of 28

small except near the boundaries of the plates, so that the new distribution may be approximately represented by what actually exists, namely a certain superficial distribution near the edges of the plates.

If therefore we integrate ijpdafd/ between the limits /= 0 and /=-£, and from #'=— oo to x = +oc, we shall find the whole

additional charge on one side of the plates due to the curvature.

0.    d(f>          d\lf 
   

Since -:—, = -- f, > we have dy dx

(35) Integrating with respect to yf , we find

1 12R + B. 2R + B

-p- ^g - r-jjr- (36)

This is half the total quantity of electricity which we must suppose distributed in space near the edge of one of the cylindric plates per unit of circumference. Since it is only close to the edge of the plate that the density is sensible, we may suppose it all condensed on the surface of the plate without altering sensibly its action on the opposed plane surface, and in calculating the attraction between that surface and the cylindric surface we may suppose this electricity to belong to the cylindric surface.

200.] CIRCULAR GROOVES. 283

If there had been no curvature the superficial charge on the positive surface of the plate per unit of length would have been

Hence, if we add to it the whole of the above distribution, this

TO

charge must be multiplied by the factor (l + 4 -^-) to get the total

charge on the positive side.

• In the case of a disk of radius R placed midway between two infinite parallel plates at a distance -5, we find for the capacity

of the disk R2 log1 2 ,.

-p-H-2 *e R+\B. (38)

Jj 7T

• [In Art. 200, in estimating the total space distribution we might perhaps more correctly take for it the integral ffp 2 IT (R + y') dx'dy', which gives, per unit circum-

1 7?

ference of the edge of radius R, •— — -= , thus leading to the same correction as in the

text.

The case of the disk may be treated in like manner as follows :

Let the figure of Art. 195 revolve round a line perpendicular to the plates and at a

distance + R from the edge of the middle one. That edge will therefore envelope a

circle, which will be the edge of the disk. As in Art. 200, we begin with Poisson'a

equation, which in this case will be

d-V d*V 1 dV

We now assume that F = ^, the potential function of Art. 195. We must therefore suppose electricity to exist in the region between the plates whose volume density p is

47r R—x dx The total amount is

p.2*(R-x')dx'dy'.

4/ v %/

Now if R is large in comparison with the distance between the plates this result will be seen, on an examination of the potential lines in Fig. XI, to be sensibly the same as B

/*2 fao dti

I I -— dx'dy'; that is, —^vB. JO J-co^ The total surface distribution if we include both sides of the disk is

If, therefore, the volume distribution between the plates be supposed to be concen- trated on the disk the expression for the capacity, the difference of the potentials of the plates and disk being f, becomes

T)

a result differing from that in the text by j nearly,]

284: CONJUGATE FUNCTIONS. [20 1.

Theory of Thomson's Guard-ring.

201.] In some of Sir W. Thomson's electrometers, a large plane surface is kept at one potential, and at a distance a from this surface is placed a plane disk of radius R surrounded by a large plane plate called a Guard-ring with a circular aperture of radius Hf concentric with the disk. This disk and plate are kept at potential zero.

The interval between the disk and the guard-plate may be regarded as a circular groove of infinite depth, and of breadth R' — R) which we denote by B.

The charge on the disk due to unit potential of the large disk,

R2 supposing the density uniform, would be — — - •

4 a.

The charge on one side of a straight groove of breadth B and length L —2irR, and of infinite depth, may be estimated by the number of lines of force emanating from the large disk and falling upon the side of the groove. Referring to Art. 197 and footnote we see that the charge will therefore be

. RB i.e. J—j 7t

A + a

since in this case $=1,0 = 0, and therefore b = A + a'.

But since the groove is not straight, but has a radius of curvature

TD

7?, this must be multiplied by the factor (l + J -=-) •

The whole charge on the disk is therefore R2 RB , B^

_R2 + R'2 Rf2-R2 a' , ,

U ~8A A + a''

The value of a' cannot be greater than

— ^— , = 0.22.Z? nearly.

7T

If ^5 is small compared with either A or R this expression will give a sufficiently good approximation to the charge on the disk due to unity of difference of potential. The ratio of A to R may have any value, but the radii of the large disk and of the guard-ring must exceed R by several multiples of A.

202-] A CASE OF TWO PLANES. 285

EXAMPLE VII.— Fie-. XII

202.] Helmholtz, in his memoir on discontinuous fluid motion *, has pointed out the application of several formulae in which the coordinates are expressed as functions of the potential and its conjugate function.

One of these may be applied to the case of an electrified plate of finite size placed parallel to an infinite plane surface connected with the earth.

Since x^—A<^ and yl=A\r,

and also x2=Ae* cos ^ and y% = A e* sin //-,

are conjugate functions of $ and x/r, the functions formed by adding XL to x2 and yx to y2 will be also conjugate. Hence, if

x = A^ + Ae^cos\l/, y = A \fr + A efi sin \j/.

then x and y will be conjugate with respect to $ and ^-, and <£ and \f/ will be conjugate with respect to x and y.

Now let x and y be rectangular coordinates, and let kty be the potential, then kfy will be conjugate to /fcx/f, k being any constant.

Let us put \l/ = TT, then y = ATI, x = A (0 — ^).

If 0 varies from — oo to 0, and then from 0 to +00, x varies from -co to —A and from — A to — oo. Hence the equipotential surface, for which \j/ = TT, is a plane parallel to a? at a distance b = i7 A from the origin, and extending from — oo to x = — A.

Let us consider a portion of this plane, extending from

x = — (A + a) to x = —A and from z = 0 to z = c,

let us suppose its distance from the plane of xz to be y = b = A ir, and its potential to be F= k^r = kit.

The charge of electricity on the portion of the plane considered is found by ascertaining the values of </> at its extremities.

We have therefore to determine $ from the equation

<£ will have a negative value <^>1 and a positive value $2 ; at the edge of the plane, where x = — A, 0 = 0.

Hence the charge on the one side is — ckfa-*- 4?:, and that on the other side is c/£<-H 477.

Konigl. Akad. der WissenscJiaften, zu Berlin, April 23, 1868.

286 CONJUGATE FUNCTIONS. [203.

Both these charges are positive and their sum is

47T

If we suppose that a is large compared with A,

-- -T-1 + &C. A

^2 = log{z+ 1 +log Oj + x +&cO j-

If we neglect the exponential terms in fa we shall find that the charge on the negative surface exceeds that which it would have if the superficial density had been uniform and equal to that at a distance from the boundary, by a quantity equal to the charge on a

strip of breadth A = - with the uniform superficial density. The total capacity of the part of the plane considered is

The total charge is CV, and the attraction towards the infinite plane, whose equation is y = 0 and potential x/r = 0, is

dC ac,

2^

db i + — 10 —

P

The equipotential lines and lines of force are given in Fig. XII.

EXAMPLE VIII. Theory of a Grating of Parallel Wires. Fig. XIII.

203.] In many electrical instruments a wire grating is used to prevent certain parts of the apparatus from being electrified by induction. We know that if a conductor be entirely surrounded by a metallic vessel at the same potential with itself, no electricity can be induced on the surface of the conductor by any electrified body outside the vessel. The conductor, however, when completely surrounded by metal, cannot be seen, and therefore, in certain cases, an aperture is left which is covered with a grating of fine wire. Let us investigate the effect of this grating in diminishing the effect of electrical induction. We shall suppose the grating to consist of a series of parallel wires in one plane and at equal intervals, the diameter of the wires being small compared with the

204.] INDUCTION THHOUGH A GRATING. 287

distance between them, while the nearest portions of the electrified bodies on the one side and of the protected conductor on the other are at distances from the plane of the screen, which are considerable compared with the distance between consecutive wires.

204.] The potential at a distance / from the axis of a straight wire of infinite length charged with a quantity of electricity A per unit of length is F = - 2 A log / + (7. (l)

We may express this in terms of polar coordinates referred to an axis whose distance from the wire is unity, in which case we must make /2 = 1 - 2 r cos 0 + r2, (2)

and if we suppose that the axis of reference is also charged with the linear density A', we find

V— — Alog(l— 2/COS0 + /2)— 2A'logr + C. (3)

If we now make

H r = «"!, 0 = ^, (4)

then, by the theory of conjugate functions,

*)_

F= — Alog (l — 2e * cos- - + e a ) — 2A'log* « + 0, (5)

where x and y are rectangular coordinates, will be the value of the potential due to an infinite series of fine wires parallel to z in the plane of xz, and passing through points in the axis of x for which so is a multiple of a.

Each of these wires is charged with a linear density A.

The term involving A' indicates an electrification, producing a

constant force in the direction ofy.

a

The forms of the equipotential surfaces and lines of force when A'= 0 are given in Fig. XIII. The equipotential surfaces near the wires are nearly cylinders, so that we may consider the solution approximately true, even when the wires are cylinders of a diameter which is finite but small compared with the distance between them.

The equipotential surfaces at a distance from the wires become more and more nearly planes parallel to that of the grating.

If in the equation we make y = 6lt a quantity large compared with «, we find approximately,

^ = - ~p (A + A') + C nearly. (6)

If we next make y — — b2, where #2 is a positive quantity large compared with a, we find approximately,

288 CONJUGATE FUNCTIONS. [205.

/+ C nearly. (7)

If c is the radius of the wires of the grating, c being small compared with a, we may find the potential of the grating itself by supposing that the surface of the wire coincides with the equi- potential surface which cuts the plane of xz at a distance c from the axis of z. To find the potential of the grating we therefore put x = c, and y — 0, whence

V=— 2 A log 2 sin— + C. (8)

205.] We have now obtained expressions representing the elec- trical state of a system consisting of a grating of wires whose diameter is small compared with the distance between them, and two plane conducting surfaces, one on each side of the grating, and at distances which are great compared with the distance between the wires.

The surface-density o^ on the first plane is got from the equa-

tion(6)

That on the second plane cr2 from the equation (?)

4^=^ = 1^'. (10)

db2 a

If we now write a . vc

« = -2-loSe(2sm-),

and eliminate X and A/ from the equations (6), (7), (8), (9), (10), we find

^4^ +^).F1(l + 4)-F'i-rJ, (12)

v^ + --r1+r1(i + -ri. (is)

When the wires are infinitely thin, a becomes infinite, and the terms in which it is the denominator disappear, so that the case is reduced to that of two parallel planes without a grating in- terposed.

If trie grating is in metallic communication with one of the planes, say the first, V— FI} and the right-hand side of the equation for oi becomes Fl — F2. Hence the density o-j induced on the first plane when the grating is interposed is to that which would have been induced on it if the grating were removed, the second plane

being maintained at the same potential, as 1 to 1 H

206.] METHOD OF APPROXIMATION. 289

We should have found the same value for the effect of the grating in diminishing the electrical influence of the first surface on the second, if we had supposed the grating connected with the second surface. This is evident since dl and b2 enter into the expression in the same way. It is also a direct result of the theorem of Art. 88.

The induction of the one electrified plane on the other through the grating is the same as if the grating were removed, and the distance between the planes increased from bl + £2 to

If the two planes are kept at potential zero, and the grating electrified to a given potential, the quantity of electricity on the grating will be to that which would be induced on a plane of equal area placed in the same position as

Ma : M2 + a(£i + ^)-

This investigation is approximate only when bl and b2 are large compared with a, and when a is large compared with c. The quantity a is a line which may be of any magnitude. It becomes infinite when c is indefinitely diminished.

If we suppose c = \ a there will be no apertures between the wires of the grating, and therefore there will be no induction through it. We ought therefore to have for this case a = 0. The formula (ll), however, gives in this case

a = — — log 2, =-0-110,

2 7T

which is evidently erroneous, as the induction can never be altered in sign by means of the grating. It is easy, however, to proceed to a higher degree of approximation in the case of a grating of cylindrical wires. I shall merely indicate the steps of this process.

Method of Approximation.

206.] Since the wires are cylindrical, and since the distribution of electricity on each is symmetrical with respect to the diameter parallel to ^, the proper expansion of the potential is of the form

7= C'0logr-f2<7<r<cosi0, (14)

where r is the distance from the axis of one of the wires, and Q the angle between r and ^, and, since the wire is a conductor, when ;• is made equal to the radius V must be constant, and therefore the coefficient of each of the multiple cosines of 0 must vanish.

VOL. I. U

290 CONJUGATE FUNCTIONS. [206.

For the sake of conciseness let us assume new coordinates £, 77, &c. such that

a£ = 27T#, at] = 27ry, ap = 2Trr, aft = 27r#, &c., (15) and let Fft = log (^+^-fe-(TJ+^-2 cos £). (16)

Then if we make

7 X7 ..72 7^

r=4,^+4^+4j-£+&c. (17)

by giving proper values to the coefficients A we may express any potential which is a function of 77 and cos f, and does not become infinite except when ?? + /3 = 0 and cos f = 1.

When (3 = 0 the expansion of .F in terms of p and 0 is

FQ = 2logp + TVp2 cos20— TTVoP4cos40 + &c. (18)

For finite values of (3 the expansion of .F is

. (19)

In the case of the grating with two conducting planes whose equations are q = /^ and 77 = — /32, that of the plane of the grating being 77 = 0, there will be two infinite series of images of the grating. The first series will consist of the grating itself together with an infinite series of images on both sides, equal and similarly electrified. The axes of these imaginary cylinders lie in planes whose equations are of the form

1?= ±2»(/31 + /32), (20)

n being an integer.

The second series will consist of an infinite series of images for which the coefficients A0, A.2, A^ &c. are equal and opposite to the same quantities in the grating itself, while Aly A^ &c. are equal and of the same sign. The axes of these images are in planes whose equations are of the form

rj= 2{32± 2a»(/31 + j8a), (21)

m being an integer.

The potential due to any finite series of such images will depend on whether the number of images is odd or even. Hence the potential due to an infinite series is indeterminate, but if we add to it the function Brj--C, the conditions of the problem will be suffi- cient to determine the electrical distribution.

We may first determine V^ and T2, the potentials of the two conducting planes, in terms of the coefficients A0, Alt &c., and of j5 and C. We must then determine o^ and <r2, the surface-density at any point of these planes. The mean values of o^ and <r2 are given by the equations

206.] METHOD OF APPROXIMATION. 291

(4, + A) (22)

We must then expand the potentials due to the grating itself and to all the images in terms of p and cosines of multiples of 0, adding to the result fip cos Q + c.

The terms independent of 0 then give V the potential of the grating, and the coefficient of the cosine of each multiple of 0 equated to zero gives an equation between the indeterminate co- efficients.

In this way as many equations may be found as are sufficient to eliminate all these coefficients and to leave two equations to determine o^ and o-2 in terms of 7^, V^ and T.

These equations will be of the form

62 + a--y). (23)

The quantity of electricity induced on one of the planes protected by the grating, the other plane being at a given difference of potential, will be the same as if the plates had been at a distance

instead of 6 b

The values of a and y are approximately as follows, a 5

a f = ^(

U 2

CHAPTER XIII.

ELECTROSTATIC INSTRUMENTS.

On Electrostatic Instruments.

THE instruments which we have to consider at present may be divided into the following" classes :

(1) Electrical machines for the production and augmentation of electrification.

(2) Multipliers, for increasing electrification in a known ratio.

(3) Electrometers, for the measurement of electric potentials and charges.

(4) Accumulators, for holding large electrical charges.

Electrical Machines.

207.] In the common electrical machine a plate or cylinder of glass is made to revolve so as to rub against a surface of leather, on which is spread an amalgam of zinc and mercury. The surface of the glass becomes electrified positively and that of the rubber negatively. As the electrified surface of the glass moves away from the negative electrification of the rubber it acquires a high positive potential. It then comes opposite to a set of sharp metal points in connexion with the conductor of the machine. The posi- tive electrification of the glass induces a negative electrification of the points, which is the more intense the sharper the points and the nearer they are to the glass.

When the machine works properly there is a discharge through the air between the glass and the points, the glass loses part of its positive charge, which is transferred to the points and so to the insulated prime conductor of the machine, and to any other body with which it is in electric communication.

The portion of the glass which is advancing towards the rubber has thus a smaller positive charge than that which is leaving it at the same time, so that the rubber, and the conductors in com- munication with it, become negatively electrified.

208.] ELECTROPHORUS. 293

The highly positive surface of the glass where it leaves the rubber is more attracted by the negative charge of the rubber than the partially discharged surface which is advancing towards the rubber. The electrical forces therefore act as a resistance to the force employed in turning the machine. The work done in turning the machine is therefore greater than that spent in overcoming ordinary friction and other resistances, and the excess is employed in pro- ducing a state of electrification whose energy is equivalent to this excess.

The work done in overcoming friction is at once converted into heat in the bodies rubbed together. The electrical energy may be also converted either into mechanical energy or into heat.

If the machine does not store up mechanical energy, all the energy will be converted into heat, and the only difference between the heat due to friction and that due to electrical action is that the former is generated at the rubbing surfaces while the latter may be generated in conductors at a distance *.

We have seen that the electrical charge on the surface of the glass is attracted by the rubber. If this attraction were sufficiently intense there would be a discharge between the glass and the rubber, instead of between the glass and the collecting points. To prevent this, flaps of silk are attached to the rubber. These become negatively electrified and adhere to the glass, and so diminish the potential near the rubber.

The potential therefore increases more gradually as the glass moves away from the rubber, and therefore at any one point there is less attraction of the charge on the glass towards the rubber, and consequently less danger of direct discharge to the rubber.

In some electrical machines the moving part is of ebonite instead of glass, and the rubbers of wool or fur. The rubber is then elec- trified positively and the prime conductor negatively.

The Electrophorus of Tolta.

208.] The electrophorus consists of a plate of resin or of ebonite backed with metal, and a plate of metal of the same size. An insulating handle can be screwed to the back of either of these plates. The ebonite plate has a metal pin which connects the metal

• It is probable that in many cases where dynamical energy is converted into heat by friction, part of the energy may be first transformed into electrical energy and then converted into heat as the electrical energy is spent in maintaining currents of short circuit close to the rubbing surfaces. See Sir W. Thomson, 'On tho Electro- dynamic Qualities of Metals.' Phil. Trans., 1856, p. 650.

294 ELECTROSTATIC INSTRUMENTS. [209.

plate with the metal back of the ebonite plate when the two plates are in contact.

The ebonite plate is electrified negatively by rubbing it with wool or cat's skin. The metal plate is then brought near the ebonite by means of the insulating handle. No direct discharge passes between the ebonite and the metal plate, but the potential of the metal plate is rendered negative by induction, so that when it comes within a certain distance of the metal pin a spark passes, and if the metal plate be now carried to a distance it is found to have a positive charge which may be communicated to a con- ductor. The metal at the back of the ebonite plate is found to have a negative charge equal and opposite to the charge of the metal plate.

In using the instrument to charge a condense? or accumulator one of the plates is laid on a conductor in communication with the earth, and the other is first laid on it, then removed and applied to the electrode of the condenser, then laid on the fixed plate and the process repeated. If the ebonite plate is fixed the condenser will be charged positively. If the metal plate is fixed the condenser will be charged negatively.

The work done by the hand in separating the plates is always greater than the work done by the electrical attraction during the approach of the plates, so that the operation of charging the con- denser involves the expenditure of work. Part of this work is accounted for by the energy of the charged condenser, part is spent in producing the noise and heat of the sparks, and the rest in overcoming other resistances to the motion.

On Machines producing Electrification by Mechanical Work.

209.] In the ordinary frictional electrical machine the work done in overcoming friction is far greater than that done in increasing the electrification. Hence any arrangement by which the elec- trification may be produced entirely by mechanical work against the electrical forces is of scientific importance if not of practical value. The first machine of this kind seems to have been Nicholson's Revolving Doubler, described in the Philosophical Transactions for 1788 as 'an instrument which by the turning of a Winch produces the two states of Electricity without friction or communication with the Earth/

210.] It was by means of the revolving doubler that Volta succeeded in developing from the electrification of the pile an

210.] THE REVOLVING DOUBLEK. 295

electrification capable of affecting his electrometer. Instruments on the same principle have been invented independently by Mr. C. F. Varley * and Sir W. Thomson.

These instruments consist essentially of insulated conductors of various forms, some fixed and others moveable. The moveable conductors are called Carriers, and the fixed ones may be called Inductors, Receivers, and Regenerators. The inductors and receivers are so formed that when the carriers arrive at certain points in their revolution they are almost completely surrounded by a con- ducting body. As the inductors and receivers cannot completely surround the carrier and at the same time allow it to move freely in and out without a complicated arrangement of moveable pieces, the instrument is not theoretically perfect without a pair of re- generators, which store up the small amount of electricity which the carriers retain when they emerge from the receivers.

For the present, however, we may suppose the inductors and receivers to surround the carrier completely when it is within them, in which case the theory is much simplified.

We shall suppose the machine to consist of two inductors A and C, and of two receivers B and D, with two carriers F and G.

Suppose the inductor A to be positively electrified so that its potential is A, and that the carrier J^is within it and is at potential F. Then, if Q is the coefficient of induction (taken positive) between A and F, the quantity of electricity on the carrier will be Q (F—A).

If the carrier, while within the inductor, is put in connexion with the earth, then F = 0, and the charge on the carrier will be — QA, a negative quantity. Let the carrier be carried round till it is within the receiver B, and let it then come in contact with a spring so as to be in electrical connexion with B. It will then, as was shewn in Art. 32, become completely discharged, and will com- municate its whole negative charge to the receiver B.

The carrier will next enter the inductor C, which we shall suppose charged negatively. While within C it is put in connexion with the earth and thus acquires a positive charge, which it carries off and communicates to the receiver D, and so on.

In this way, if the potentials of the inductors remain always constant, the receivers B and D receive successive charges, which are the same for every revolution of the carrier, and thus every revolution produces an equal increment of electricity in the re- ceivers.

• Specification of Patent, Jan. 27, I860, No. 206.

296 ELECTROSTATIC INSTRUMENTS. [2IO.

But by putting the inductor A in communication with the re- ceiver D, and the inductor C with the receiver B, the potentials of the inducto.-s will be continually increased, and the quantity of electricity communicated to the receivers in each revolution will continually increase.

For instance, let the potential of A and D be U, and that of B and C, 7, then, since the potential of the carrier is zero when it is within A, being in contact with earth, its charge is z = — QU. The carrier enters B with this charge and communicates it to B. If the capacity of B and C is B, their potential will be changed

from

If the other carrier has at the same time carried a charge — Q V from C to D, it will change the potential of A and D from U to

Q'

U— -j- F9 if Q' is the coefficient of induction between the carrier A

and C, and A the capacity of A and D. If, therefore, Un and Vn be the potentials of the two inductors after n half revolutions, and Un+1 and Fn+l after ^+1 half revolutions,

V V TT

' — > ~ ~ U-

If we write p2 = ~ and <f =. -— > we find

+pq) = Hence

4 pgY) + r. ((1

It appears from these equations that the quantity pU+qV con- tinually diminishes, so that whatever be the initial state of elec- trification the receivers are ultimately oppositely electrified, so that the potentials of A and B are in the ratio of p to — q.

On the other hand, the quantity pU—q~P continually increases, so that, however little joC/may exceed or fall short of ^Fat first, the difference will be increased in a geometrical ratio in each

211.] THE RECIPROCAL ELECTROPHORUS. 297

revolution till the electromotive forces become so great that the insulation of the apparatus is overcome.

Instruments of this kind may be used for various purposes.

For producing a copious supply of electricity at a high potential, as is done by means of Mr. Varley's large machine.

For adjusting the charge of a condenser, as in the case of {Thomson's electrometer, the charge of which can be increased or diminished by a few turns of a very small machine of this kind, which is then called a Replenishes

For multiplying small differences of potential. The inductors may be charged at first to an exceedingly small potential, as, for instance, that due to a thermo-electric pair, then, by turning the machine, the difference of potentials may be continually multiplied till it becomes capable of measurement by an ordinary electrometer. By determining by experiment the ratio of increase of this difference due to each turn of the machine, the original electromotive force with which the inductors were charged may be deduced from the number of turns and the final electrification.

In most of these instruments the carriers are made to revolve about an axis and to come into the proper positions with respect to the inductors by turning an axle. The connexions are made by means of springs so placed that the carriers come in contact with them at the proper instants.

211.] Sir W. Thomson *, however, has constructed a machine for multiplying electrical charges in which the carriers are drops of water falling out of the inside of an inductor into an insulated receiver. The receiver is thus continually supplied with electricity of opposite sign to that of the inductor. If the inductor is electrified positively, the receiver will receive a continually increasing charge of negative electricity.

The water is made to escape from the receiver by means of a funnel, the nozzle of which is almost surrounded by the metal of the receiver. The drops falling from this nozzle are therefore nearly free from electrification. Another inductor and receiver of the same construction are arranged so that the inductor of the one system is in connexion with the receiver of the other. The rate of increase of charge of the receivers is thus no longer constant, but increases in a geometrical progression with the time, the charges of the two receivers being of opposite signs. This increase goes on till the falling drops are so diverted from their course by

• Proc. E. S., June 20, 1867.

298

ELECTEOSTATIC INSTRUMENTS.

[212.

the electrical action that they fall outside of the receiver or even strike the inductor.

In this instrument the energy of the electrification is drawn from that of the falling drops.

212.] Several other electrical machines have been constructed in which the principle of electric induction is employed. Of these the most remarkable is that of Holtz, in which the carrier is a glass plate varnished with gum-lac and the inductors are pieces of pasteboard. Sparks are prevented from passing between the parts of the apparatus by means of two glass plates, one on each side of the revolving carrier plate. This machine is found to be very effective, and not to be much affected by the state of the atmo- sphere. The principle is the same as in the revolving doubler and the instruments developed out of the same idea, but as the carrier is an insulating plate and the inductors are imperfect conductors, the complete explanation of the action is more difficult than in the case where the carriers are good conductors of known form and are charged and discharged at definite points.

213.] In the electrical machines already described sparks occur

whenever the carrier comes in contact with a conductor at a different potential from its own.

Now we have shewn that whenever this occurs there is a loss of energy, and therefore the whole work employed in turning the machine is not con- verted into electrification in an available form, but part is spent in producing the heat and noise of electric sparks. I have therefore thought it desirable to shew how an electrical machine may be constructed which is not subject to this loss of efficiency. I do not propose it as a useful form of machine, but as an example of the method by which the contrivance called in heat-engines a regenerator may be applied to an electrical machine to prevent loss of work.

In the figure let A, B, <?, A', B', C' represent hollow fixed conductors, so arranged that the carrier P passes in succession within each of them. Of these A, A' and B< B' nearly surround the

Fig. 18.

2 I 3-] MACHINE WITHOUT SPAKKS. 299

carrier when it is at the middle point of its passage, but C, C' do not cover it so much.

We shall suppose A, B, C to be connected with a Leyden jar of great capacity at potential V, and A', I? ', Cf to be connected with another jar at potential — V .

P is one of the carriers moving in a circle from A to C', &c., and touching in its course certain springs, of which a and a' are connected with A and A' respectively, and e> e are connected with the earth.

Let us suppose that when the carrier P is in the middle of A the coefficient of induction between P and A is — A. The capacity of P in this position is greater than J, since it is not completely surrounded by the receiver A. Let it be A +a.

Then if the potential of P is U, and that of A, V, the charge on P will be (A + a) U—A7.

Now let P be in contact with the spring a when in the middle of the receiver A, then the potential of P is F, the same as that of A, and its charge is therefore aV.