The Alternate Current Transformer Vol. 1: The Induction of Electric Currents (1896) — part 21 of 35

there may be an equality in the quantities flowing past any sections in each of the conductors, as evidenced by equality in the magnetic fields produced by the first-named current and the moving electric charge ? This velocity is evidently a concrete velocity, which depends on the very nature of the qualities of the medium which determine magnetic and electrostatic attraction, and this velocity may be called the ratio of the magnitude of the electro-magnetic to the electro- static unit of quantity. This velocity is evidently one which is determined by the nature of the medium, and not by the particular units of length, time, and mass selected for use in the measurements. This comparison assumes that a moving electrostatic charge is in effect the equivalent of an electric current. This has been put to the test of experiment by Prof. Bowland.* A rigid gilt ebonite disc was fixed to an axis, and could be rotated between two gilt glass discs. One member of a very delicate astatic system of magnetic needles was placed near the disc and shielded from electrostatic disturbance. On charging the gilt ebonite disc and setting it in rapid rotation it was found to affect the magnetic needle whilst rotating just as a current of electricity would have done if flowing in a circular conductor coinciding in form with the periphery of the disc. Since 1876 Prof. Rowland has again in the United States repeated the experiment and confirmed the general result. There is, therefore, experimental founda- tion for the view that a static charge of electricity conveyed on a moving body creates a magnetic field whilst it is in movement. This kind of electric current, in which a static charge is bodily moved on a conductor, is called a convection current. The experiment of comparing the magnitudes of an electrostatic and an electro-magnetic unit of electric quantity as above defined was first made by Profs. Weber and Kohlrausch, and the value of that ratio for a medium such as air, in which approximately we have K and u. both equal to unity, gave as a result a velocity very nearly identical with the velocity of light. Since that time very many experimentalists have determined the value of this ratio, which is denoted by

• See Phil. Mag., 1876, Vol. II., Fifth Series, p. 233 : Dr. Helmholtz, «' On the Electro-Magnetic Action of Electric Convection." These experi- ments of Prof. Rowland were carried out a,t Berlin,

358 DYNAMICAL THEORY OF INDUCTION.

the symbol " v." The names and the results of the observa- tions made by some of the principal observers are set out in the Table on opposite page.

One of the best determinations of the velocity of light is that made by Prof. Newcomb, at Washington, in 1882. The method employed was the revolving mirror method of Foucault, the distance between the revolving and fixed mirror being in one portion of the experiments 2,550 metres, and in the other portion 3,720 metres. The resulting velocity of light in vacua is 2-99860 x 1010 centimetres per second.

The following results of other observations are abstracted from Prof. Everett's book, " Units and Physical Constants," 2nd edition : —

Observe, "*££"'

Michelson, at Naval Academy, 1879 .................. 2'99910 x 1010

Michelson, at Cleveland, 1882 ...................... 2'99853 x 10l°

Newcomb, at Washington, 1882 (best results) ... 2*99860 x 10W Newcomb (other results) .............................. 2'99810 x 10l°

Foucault, at Paris, 1862 ................................ 2'98COO x 10i°

Cornu, at Paris, 1874 .................................. 2 98500 x 1010

Cornu, at Paris, 1878 .................................. 3'004 x 1010

Last result discussed by Listing ..................... 2'9999 x 10i°

Young and Forbes, 1880-81 .................. : ........ 3'01382xl010

Earlier observations gave as follows : —

Koemer's method, by Jupiter's satellites ............ 3 '000 x 1010

Bradley's method, by stellar aberration ............ 2'977 x 1010

Fizeau ..................................................... 3'142 xlO*>

The general result of the best determinations is that the velocity of light is very close to 3-000 x 1010 centimetres per second, or nearly one thousand million feet per second.

We have, therefore, the following facts : — The velocity VTO of light of definite wave length in any medium is connected with the velocity Yp of the same ray in vacuo by an equation —

P-

where /* is the refractive index of that medium for the par- ticular wave length considered, and also that the velocity V is very nearly 3 x 1010 centimetres per second. Also we find that the ratio of the electro-magnetic to the electrostatic unit of electric quantity or current in any dielectric and magnetic

DYNAMICAL THEORY OF INDUCTION. 359

1!

SJ

x 8

s!

£•8

da a

I

I

IB

i ij

if.

ts-

I*

ig

*§

P

EH 02

360 . DYNAMICAL THEORY OF INDUCTION.

medium Em is connected with the same ratio measured in vacuo R, by an equation —

E -

""

where K is the dielectric constant and fi the magnetic per- meability.* Experiment has also indicated that within narrow limits, taking best results, EB and V0 have the same value, namely, 8 x 1010 centimetres per second, and that ^/K has the same value as p (refractive index) for media, for which p, (per- meability) has the value unity. We are led, therefore, to infer that this close relationship is not a matter of accident, but that it indicates a very intimate connection between electricity and light, and that the hypothesis that light is a disturbance propagated through an elastic medium may be supplemented with some considerable show of reason by the hypothesis that electro-magnetic phenomena are the result of actions taking place in identically the same medium or ether. There are no transparent media for which the magnetic permeability differs by more than a very small quantity from unity, and hence the approximate identity of the values of the ratio of the units compared in air with the value of the velocity of light waves of very long wave-length ; and the approximate identity for true dielectrics of the value of the refractive index and of the square root of the dielectric constant furnishes a test of the proba- bility of the truth of the electro-magnetic theory of light. Maxwell's mathematical method of arriving at this theory consisted hi forming certain equations expressing the velocity of propagation of vector potential, and noticing that these equations were mathematically of the same form as those which determine the velocity of propagation of a disturbance through an elastic medium. The physical meaning of this term, vector potential, may be arrived at as follows : —

Suppose a regiment of soldiers to set off marching down a street, the ranks being well spaced out. At any place in the street let two lines be drawn across the street parallel to each other and a few yards apart. Let two observers take

• It is unfortunate that usage has consecrated the same Greek letter /* for refractivity in optics and magnetic inductivity in electro-magnetics. In some respects it would be an advantage in electro-optics if these quantities were differently symbolised.

DYNAMICAL THEORY OF INDUCTION. 361

note of how many soldiers cross each line. At any instant the total number of soldiers which are contained between the two lines is equal to the difference between the numbers which have crossed each line respectively. However irregular the movement may be, the total number of soldiers at any instant in the area or the product of the area, and the number of soldiers per unit of area within the boundary, will be equal to the number obtained by reckoning the algebraic sum of the soldiers which have from the beginning of the time crossed the whole boundary line, calling those numbers jiositive when soldiers have stepped into the area and negative when they have stepped out of it. We have here a simple example of the way in which a line integral may be the equivalent of a surface inter/ml. If the area be irregular in shape and contain A square yards, and if the perimeter be I linear yards, then if % nz, &c., are the number of men which have stepped across each yard length of the boundary, and if Nx N2, &c., are the number of men in respective square yards within the area at any instant, then Ni + N2 + , &c., to A terms or 2N is called a surface integral and will be equal to n1 + n.i + , &c., to I terms, which is a line integral, provided that each n is reckoned positive when men step in, and negative when men step out of the area over each yard of the boundary. The algebraic sum of all the stepping over the boundary all the way round the area is equal to the sum of the men per square yard all over the area. We have here given an illustration of an important proposition in mathematical physics, viz., that a surface integral, or the sum- mation of a certain quantity over an area, can be replaced by a line integral, or the summation of another relative quantity all along the boundary line of that area. We proceed to illustrate it from an electrical point of view.

Let C (Fig. 182) be the circular cross-section of an infinite straight wire conveying a current C. Eound C describe a circle of radius r. The magnetic force at p is known to be

equal to — units, and is directed along the circumference of the circle ; the line integral of the magnetic force along the dotted line is equal to _ x27rr = 47rC, and the surface integral of the current through the area enclosed by the dotted

362

DYNAMICAL THEORY OF INDUCTION.

circle is C. Hence we have generally that the line integral of the magnetic force is equal to 4?r times the surface integral of the current. This proposition is generally true, and it is easy to show that if A be any area (see Fig. 133) traversed normally by a current, such that the current density is u over any element

FIG. 132.

of area d s, then the integral of u d s all over the area, or \uds,

is equal to the line integral of the magnetic force taken along the boundary line. The mathematical operation of taking a line integral has been called by Maxwell curling, and we express

FIG. 133.

the above proposition by saying that 4:r times the total current through the area is equal to the curl of the magnetic force round it. On the theory that lines of magnetic force do not spring suddenly into existence in a field, but are propagated onwards from point to point in ihe field, it is possible to show

DYNAMICAL THEORY OF INDUCTION. 363

that just as the current is the curl of the magnetic force so the magnetic force is the curl of another quantity called the vector potential.

Let A B (Fig. 134) be a portion of a straight conductor in •which a current can be started. Let x x, y y' be two lines drawn a unit of distance apart, parallel to each other and at right angles to the conductor. These lines bound a strip of plane space taken in the plane of the current. Draw any two transverse lines a b, c d, parallel to the conductor and separated by a small distance. We know that when a current is started in the con- ductor the lines of magnetic force F will be circles formed round A B as axis, and having their planes perpendicular to the plane x x', y y'. Let us now assume that if a current is suddenly started in the conductor A B the magnetic force is

B

propagated outwards from the conductor with a finite velocity v. In other words, each circular line of force must be con- sidered to expand outwards like a circular ripple on the surface of water. When once the field has arrived everywhere at its normal value the magnetic force at a distance r from the wire

20

is — , where C is the value of the current, and we shall sup- pose, as usual, that the magnetic field is indicated as to value by the density of the lines of force, or that the number per square centimetre traversing normally the plane x x', y y' is at any point proportional or numerically equal to the magnetic force at that point. If, then, we neglect for the moment all effect of self-induction, and suppose the current in the wire to rise up instantaneously to its full value, we may yet regard the

364 DYNAMICAL THEORY OF INDUCTION.

circular lines of force as expanding outwards with a certain velocity of enlargement, and attaining or taking up their final positions after a short interval of time. If we represent the intersections of these rings of force on the plane of x x y y by dots, these dots will march forward like the soldiers in the pre- vious illustration. The total number of lines of force which at any instant are found traversing the area a b d c is equal numeri- cally to the difference in the number between those which from the beginning of the epoch have intersected or cut through the Jine a b and those which have cut through c d. In other words, the surface integral of the magnetic force over abed may be represented by, or is equal to, the line integral round abed of a certain quantity called the vector potential, which, physi- cally interpreted, is the total number of lines of force which have cut through a unit element of the boundary in the process of expansion or propagation outwards. This term vector poten- tial is justified as follows : — If F be the total number of lines of force per unit of length of a ft which have cut through a b from the instant of beginning the current, and if the small distance b d is called Sa*, the length x b being called x, then by Taylor's theorem (Diff. Calc.), the number which have cut

dF through unit of length of c d is F — -3— 8 x, and hence the

difference between F and this last quantity is -j-, 8x, and this last when multiplied by 8 y, which we may take for the length

T TTt

of a ft or c d — that is -j-^ 8x8 y — is the total number of lines of

force included in the area abed. If we call the induction through this area B — that is to say, the number of lines of force per square centimetre is B — it follows that the number through ab c d is B 8x8y. Hence, equating the two values,

we have -^ 8 x 8 y = B 8 x 8 y,

*I-R

dx

Hence, the mean magnetic force over the small area is numeri- cally equal to the space variation of a certain quantity F. In electrostatics the electric force X at any point in the electric

DYNAMICAL THEORY OF INDUCTION. 365

field is the space variation of a certain quantity V, called the electrostatic or scalar potential — that is to say,

~~^T = X;

and accordingly by anology that quantity F whose space varia- tion gives the magnetic force under the circumstances considered above is called the vector potential of the current. From Ampere's investigations it is known that the magnetic force due to an element of a current C of length 8 s at a distance r from this

fl £ «

element, has the value — — , and is along a line at right angles

to the plane containing & s and r. The space variation of -

/i o r

is — — ; hence the vector potential of an element of current at

any point is proportional to the length of that element divided by its distance from that point.

In electrostatic phenomena we obtain the static potential at any point due to any charge Q by taking each element q of the charge, and dividing the magnitude of this element of charge by its distance from the point at which the potential is required,

and taking the sum 2 3. of all such quotients. In electrostatics

the potential at a point is a scalar or directionless quantity, and the summation is merely an algebraic sum ; but in dealing c* /?

with currents the quotients are vectors, or directed quan-

r

tities, and have to be added together according to the laws for the addition of vector quantities just as forces and velocities are added. Hence the potential of a current at any point is a vector or directed quantity. The lines of vector potential of a straight current are lines described in space parallel to the current, and the lines of vector potential of a circular current are circles described on planes parallel to the plane of the cur- rent. Returning to the simple case of a straight current, let us suppose that a unit of length is described somewhere parallel to the current, and that on starting the current suddenly cir- cular lines of magnetic force are propagated outwards with a velocity V ; these lines will, as they expand, cut perpendicularly through the element of length just as the expanding ripples on water due to a stone dropped into it would "cut through" a

366 DYNAMICAL THEORY OF INDUCTION.

stick held perpendicularly in the water a little way from the place where the "splash " was made. Suppose that after N lines of force have cut through the element of length this little line is made to move forward parallel to itself, so that there is no further increase in the number of lines of force which after- wards cut through it, it is evident that it must move with the velocity of propagation of the expanding rings of force. But the number expressing the number of lines of force which have cut through the element of length already is the value of the vector potential at that point where the element is at that instant ; hence the velocity of propagation of the vector potential is the velocity of propagation of an electro-magnetic disturbance. Maxwell's general mathematical method of investigating the propagation of an electro-magnetic disturbance consisted in forming equations expressing the change of the value of the vector potential of a current or system of currents at any point in the field, and deducing equations which mathematically are of the same type as those which express the propagation of a disturbance through an elastic solid or fluid, and his result was that the velocity of propagation of the vector potential through a medium of electrostatic and magnetic inductivities

K and p. was equal to — — , or to (K/*)-i.

N/KM The complete proof of the above proposition as given by

Maxwell in all its generality requires some elaborate analysis, but is is not difficult to give a simple illustration by treating a reduced case, and which will exhibit the principles of the more complete problem.

Let an infinite straight conductor be supposed situated in a dielectric medium of specific inductive capacity (electrostatic inductivity) K and of permeability (magnetic inductivity) p.. We proceed to investigate the velocity of lateral propagation of electro-magnetic induction on the supposition that if a current is instantaneously started at its full value in the conductor, supposing this possible, the magnetic force travels outwards laterally from the conductor in all directions with a velocity v. This amounts to the supposition that the circular lines of magnetic force surrounding the conductor swell out or expand outwards from the surface of the conductor, so that the radius of any determinate circular line of force increases or grows

DYNAMICAL THEORY OF INDUCTION. 367

with a velocity v. It must be borne in mind that the magnetic force at any point in the field at any instant is

defined by the density or concentration of the lines of force

that is, by the number passing normally through a unit of area. If we complicate the problem by supposing the strength of the current in the conductor to gradually increase, then the concentration of the lines at any point must be supposed to increase gradually, but the rate of increase of concentration — that is, of the force — is a different thing from the rate of outward movement of the lines of force.

We might in imagination suppose each line of force to be labelled so as to recognise it. All the lines travel outward from the conductor at the same rate, but some go out farther than others. The first ones shed off expand out to reach

FIG. 135.

positions in the most distant portions of the field, and the succeeding ones reach intermediate positions, and as the current strength grows up fresh arrivals or deliveries of lines of force happen which pack the space fuller, and increase the concentration at all points of the field, at a rate depending on the rate of growth of the current.

Let 0 C (Fig. 135) be a portion of the straight conductor. In the plane of 0 C take any little rectangular area abed, with side a c equal to unit of length, and side a b equal to 8 x, 8 x being a very small quantity compared with the distance between OC and ac, that is, let the distance Oc = x and Od = x + 8x, and let the distance Sx be the distance by which the radius of any circular line of force of the conductor OC increases

368 DYNAMICAL THEORY OF INDUCTION.

in a small time 8 1. At any instant the number of lines of force which pass normally through the small area abed is equal to the difference between the number which have "cut" across ac and those which have cut across Id in consequence of our supposition as to the outward growth or expansion of the circular lines of force. Let F be the total number of lines of force due to the current in 0 C which have from the beginning of the current flow " cut across " a c, then, by the principles of the Differential Calculus, the number which have cut across bd is represented by the quantity

7 -[7\

F - — 8 x, and the number existing in, or perforating through,

dx ,p

the area abed is the difference between F and F- — 8x, or

,p equal to 8 x. Let B stand for the induction through

unit of area of the rectangle abed, or to the number of lines of force per unit of area, then the total number of lines of force through abed is represented also by B 8 x, since the area of a b c d is 8x square units, a c = b d being unity.

Hence, H=B (109)

or the induction is represented by the space rate of change of the vector potential of the current at that point in the direc- tion of x. In this case let it be borne in mind that the vector potential signifies the number of lines of force which have from the beginning of the epoch cut through unit length taken parallel to the current. Again, since by supposition each line of force moves outwards parallel to itself through a distance

8x in a time 8t, — is the velocity of propagation v of the o t

electro-magnetic disturbance or of the vector potential. The rate of " cutting across " a c at any instant is represented by

7 T71

— ; hence the number of lines of force added to the area in d t

a time 8t must be — 8t, and this must be equal to the accu- dt

mulation of the lines in a b c d in the same time in the area a b c d.

DYNAMICAL THEORY OF INDUCTION. 300 If in a small time interval the rate of cutting across a c is

J TJ1

_—, then the rate at which " cutting" is taking place across a length b d, removed by a distance 8 x, is

dt dx \ dt) X'

and the rate at which accumulation of lines of induction is going on in the area is

dx \dt

Hence, since B is the induction per unit of area and the area of abed is 8x square units, the rate of increase of induction through a b c d is

Accordingly we have

£<B«.)---*£

rf«V ; dx\dt

or since 8 # is constant,

rfB= __^ dt d»\dt' _ d fdF\dt dt\dt) dae dE dx= _d2F dt . or' ~dx di dt? dx '

but — =v = velocity of propagation of the impulse. Hence, dt

(no)

dx

T T71

or, generally, sinco B = — — ,

wehave *| + ^|I.O .... (HI)

d t* d x2

as the equation of motion of the vector potential. This equation, which is a reduced case of the general one, is of the same type as that obtained in the theory of sound for the

370 DYNAMICAL THEORY OF INDUCTION.

propagation of an impulse along a tube or canal. In the case of sound the symbol F would be the velocity potential.* In the electro-magnelic problem the F is the rector potential. It might perhaps be more expressively called the induction potential.

The rate of cutting, or the value of — , also expresses the

dt

electromotive force acting along the unit of length a c in the dielectric. On Maxwell's hypothesis this electromotive force in the dielectric acting parallel to the current in the conductor produces a displacement in the dielectric, such that if E is the electromotive force we have as above

dt K

where D is the displacement through unit of area ; hence,

£-£# <112>

and — - is the rate of displacement or the displacement current dt

flowing through unit of area taken perpendicularly to the cur- rent in 0 C at the point considered. Let this displacement

current be denoted by u. We have then that — r = -^> K

rfr Iv

being the dielectric constant of the medium.

Consider now a small parallelopipedon (Fig. 136) or solid rectangle described in the dielectric, of which the sides are respectively ac = l, cd = Sx, c e = 8y.

The effect of the cutting across of this solid rectangle by expanding lines of induction will be to generate in it a displace- ment current such that the total displacement current parallel to a c and through c d fe will be u d x d y. By a previous theorem the line integral of magnetic force round any line is equal to 4?r times the surface integral of the current through the area bounded by that line, and this is true whether the magnetic force be produced by that current, or whether it is a current produced by a certain changing magnetic force. Apply the theorem to the small rectangle bounded by the lines c efd. The surface in- tegral of the current through c efd is u d x d y. The magnetic

• See Besant's " Hydromechanics," p. 251 (Third Edition).

DYNAMICAL THEORY OF INDUCTION. B

371

force along c e is -, where B is the induction at c and //, is the

magnetic permeability of the medium, since by a fundamental theorem the magnetic induction B at any place is equal to p. times the magnetic force at that point. The magnetic force

along df removed by a distance 8 x from ce is -(B- — 8x]

p\ dx /'

and there is no magnetic force along c d and ef, for these sides are perpendicular to the direction of the magnetic force of the

FIG. 136.

current in 0 C. Hence, the line integral of magnetic force round cefdis

hence,

4;r u 8 x 8 y = _ — 8 x 8 y, d x

(113)

Accordingly, in the equations (112) and (113) above, we have

^F dE . obtained values for the quantities -^ and -^ in terms of

the permanent constants of the medium ; and by substitution

372 DYNAMICAL THEORY OF INDUCTION.

of these values in equation (111) above, we see that the square of the velocity of propagation of the vector potential is d-F 4:r M

*»-£*JL. *

d B 4?r p. u K fj.'

V IT'

that is, the velocity of propagation of the magnetic force is the square root of the reciprocal of the product of the magnetic and electrostatic inductive constants of the medium. We have above proved that the ratio of the electro-magnetic to the electrostatic units of electric current is expressed by the same quantity, and indicated that accurate experiment shows this ratio to be numerically the same as the velocity of light.

Hence, the velocity of an electro-magnetic disturbance or magnetic force is the same as the velocity of light, and the conclusion is urged upon us with great force that the medium concerned in both phenomena is the same.

§ 6. Electrical Oscillations.— A survey of the phenomena of electric current induction would be very incomplete if it did not contain some reference to the subject of electrical oscilla- tions. Recent researches have endowed this department of electrical investigation with fresh interest. We proceed to consider the manner in which electrical oscillations may arise. If a material body is subjected to elastic constraint, and is dis- turbed from a position of equilibrium, it returns when set free to its original position. If that body is endowed with mass, and hence possesses the quality of inertia, its motion of return to its position of equilibrium will, under certain circumstances, carry it beyond that point and set up oscillations, which decay gradually away. Two illustrations of this readily present themselves, one a mechanical and the other a pneumatical example. The first case is that of a pendulum or straight spring. Let this pendulum or spring be deflected from its position or condition of equilibrium and held in constraint. Next let it be set free — the elastic or restoring forces urge it back again to its first position. In virtue of its mass it will acquire

DYNAMICAL THEORY OF INDUCTION. 373

a certain momentum, and on reaching the position of equili- brium this momentum may carry it past this point, and the acquired kinetic energy will then be expended in making a displacement against the elastic forces. If there is nothing of the nature of friction present to fritter away the work expended on the body in making the first displacement, then the energy would remain associated with it for ever, being alternately potential and kinetic, and the oscillations continue with undi- minished amplitude. If the spring or pendulum vibrates in a viscous fluid, then a frictional retardation will be experienced, and in so far as this is present the energy is gradually dissi- pated, and the oscillations decay away, becoming gradually less and less in amplitude. It may so happen that the work done against frictional resistance during the first quarter of a com- plete oscillation in starting to return from the position of greatest displacement is just equal to the work done in origin- ally making the displacement. When this is the case the whole energy is dissipated by the time the deflected or dis- placed body reaches its original position of rest, and there are then no oscillations. Accordingly a pendulum or spring may be set in a viscous fluid of such a kind that the frictional resistance is just sufficient to secure that when the body is disturbed and then set free it returns to its original position without ever passing it ; in other words, there are no oscillations. Another illustration of oscillatory and non- oscillatory establishment of equilibrium is as follows : Sup- pose there be two large vessels, or reservoirs, connected by a pipe, closed or able to be closed in the middle by a stop- cock. Let one of these vessels, A, be exhausted of its air, and let the other, B, have air in it at the atmospheric, or a greater than the atmospheric pressure. First, let the connecting pipe be supposed to be long and narrow ; on opening the stopcock air will rush over from B into A, and the flow of air will con- tinue uniformly in the pipe in one direction until the pressure in A and B is equalised. Second, let the connecting pipe be very short and large, so that little tubular friction is offered to the flow of air. Under these circumstances the result of open- ing the tap would be that a rush of air would take place, which would be succeeded by a series of oscillations of the air in the tube. The air, in fact, rebounds from side to side, and the

374 DYNAMICAL THEORY OF INDUCTION.

equilibrium is only finally established after a series of gradually diminishing oscillations or backward or forward currents of air in the tube. This establishment of equilibrium or pressure by oscillatory movement takes place when the resistance to the flow is small. That this is no fanciful description is proved by the experience of MM. Clement and Desormes in their experiments to determine the ratio of the specific heats of gases. In these experiments a large glass vessel had a partial vacuum made in it. A stopcock was then quickly opened and closed, and the pressure of the air determined after a short time. These experiments were repeated by MM. Gay Lussac and ^Yelter. See Journal de Physique, LXXXIX., 1819, 428, and Ann. de Ch. et de Phys. [1], XIX., 1821, 436.

M. Cazin (Ann. de Ch. et de Phys. [3] LXVL, 1862, 206) first pointed out a source of error which resulted from these air oscillations, and showed that the final pressure depended upon the phase of the oscillation at which the stopcock is closed.

These examples are sufficient to indicate that when a material system of bodies having inertia is displaced against elastic forces which compel it to return, if free, to a definite position, whilst at the same time its motion is resisted by actions of the nature of frictional resistance which dissipate its energy, we have a resulting motion which may be oscillatory or non-oscillatory, according to the relation of the constants of the system. Under certain conditions as to mass, or inertia and friction, we have oscillations dying gradually away. Under other conditions we have a gradual return to the original position without ever passing it. The motion is then said to be perfectly dead-beat. "We shall investigate presently the conditions which must hold good, and the relation between the inertia factor, in virtue of which the moving system possesses kinetic energy, and the resistance factor, in virtue of which the energy bestowed upon the system at its first displacement is frittered away into heat, in order that the motion may be vibratory or dead-beat.

When a condenser or Ley den jar is discharged through a conductor, the potential energy runs down in the form of an electric current. In this case we have a similar state of things to that existing when a bent spring is released. This trans- formation of the potential energy may take place either by a vibratory current, that is, by a series of electrical oscillations — or

DYNAMICAL THEORY OF INDUCTION. 375