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V= / — — through the field ; and since



djT dG^ dir=o

dx dy '^- '



dz



the resultant of these disturbances is in the plane of the wave.

(99) The remaining part of the total disturbances F, G, H being the part
depending on x. is subject to no condition except that expressed in the equation

dt ^ df



582 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

If we perform the operation V" on this equation, it becomes

ke = ^^-hV^4>(x, y, z) (79).

Since the medium is a perfect insulator, e, the free electricity, is immove-
able, and therefore - 7- is a function of x, y, z, and the value of J is either

constant or zero, or uniformly increasing or diminishing with the time ; so that
no disturbance depending on J can be propagated as a wave.

(100) The equations of the electromagnetic field, deduced from purely
experimental evidence, shew that transversal vibrations only can be propagated.
If we were to go beyond our experimental knowledge and to assign a definite
density to a substance which we should call the electric fluid, and select either
vitreous or resinous electricity as the representative of that fluid, then we might
have normal vibrations propagated with a velocity depending on this density.
We have, however, no evidence as to the density of electricity, as we do not
even know whether to consider vitreous electricity as a substance or as the
absence of a substance.

Hence electromagnetic science leads to exactly the same conclusions as
optical science with respect to the direction of the disturbances which can be
propagated through the field ; both affirm the propagation of transverse vibra-
tions, and both give the same velocity of propagation. On the other hand, both
sciences are at a loss when called on to affirm or deny the existence of normal
vibrations.

Relation between the Index of Refraction and the Electromagnetic Character

of the substance.

(101) The velocity of light in a medium, according to the Undulatory
Theory, is

If

where i is the index of refraction and V^ is the velocity in va<;uum. The
velocity, according to the Electromagnetic Theory, is

where, by equations (49) and (71), h = jjk^, and k„ = AnV^\



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 583



Hence i) = - (80),

or the Specific Inductive Capacity is equal to the square of the index of refrac-
tion divided by the coefficient of magnetic induction.



Propagation of Electromagnetic Disturbances in a Crystallized Medium.

(102) Let -us now calculate the conditions of propagation of a plane wave
in a medium for which the values of k and /x are different in different direc-
tions. As we do not propose to give a complete investigation of the question
in the present imperfect state of the theory as extended to disturbances of
short period, we shall assume that the axes of magnetic induction coincide in
direction with those of electric elasticity.

(103) Let the values of the magnetic coefficient for the three axes be
X, fi, V, then the equations of magnetic force (B) become

dH dG



dy dz

^ dF dH
f'^^'d^- d^

^dG_dF
^ dx dy



(81).



The equations of electric currents (C) remain as before.
The equations of electric elasticity (E) will be

^=47r6V (82),

R = A'nc'h\

where 47ra', 47r6', and Attc^ are the values of k for the axes of x, y, z.

Combining these equations with (A) and (D), we get equations of the form

\ [,d'F d'F d'F\ I d (JF dG dH\ I fd'F , d'^\ ,^,-
jr.[^d^-^f'df-^''^)-]ruMd^-^f'dJ^'' dz) = a^[-de^~d^tH^-^^-

(104) If /, m, n are the direction-cosines of the wave, and V its velocity,
and if

lx + my + 7iz- Vt=w (84),



584



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.



then F, G, H, and ^ will be functions of w\ and if we put F\ G\ H\ '^'
for the second differentials of these quantities with respect to w, the equations
will be






X V



{..-.gV?)}.-

If we now put



(85).



(86),



we shall find

F'V'U-WVU=0

with two similar equations for G' and H', Hence either

F =

U=Q



(87),

(88),

(89),

or

VF' = W, F(?' = m^ and VH' = n']!' (90).

The third supposition indicates that the resultant of F\ G', H' is in the
direction normal to the plane of the wave ; but the equations do not indicate
that such a disturbance, if possible, could be propagated, as we have no other
relation between ^' and F', G\ H'.

The solution F=0 refers to a case in which there is no propagation.

''• The solution Z7= gives two values for F^ corresponding to values of F\
G\ H', which are given by the equations



-2^' + ri^' + -2^' =

a' 0' c*

-p (6> - cV) + -^ (c-v - a'\) + -j^, (a'k - b'fi) =



(91).
(92).



* [Although it is not expressly stated in the text it should be noticed that in finding equations
(91) and (92) the quantity ^ is put equal to zero. See § 98 and also the corresponding treat-
ment of this subject in the Electricity and Magnetism il § 796. It may be observed that the





X


y


2


X




a'

V


a-


V


V




X


2




X





A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 585

(105) The velocitiea along the axes are as follows: —
Direction of propagation



Direction of the electric displacements •



Now we know that in each principal plane of a crystal the ray polarized
in that plane obeys the ordinary law of refraction, and therefore its velocity
is the same in whatever direction in that plane it is propagated.

If polarized light consists of electromagnetic disturbances in which the
electric displacement is in the plane of polarization, then

a' = h' = c' (93).

If, on the contrary, the electric displacements are perpendicular to the plane

of polarization,

\ = fi = u (94).

We know, from the magnetic experiments of Faraday, Pliicker, &c., that in
many crystals X, /x, v are unequal.

equatious referred to and the table given in § 105 may perhaps be more readily understood from
a different mode of elimination. If we write

X^" + fim' + vn* = PX/Av and XIF' + fimG' + vnll ' = <2X/xv,
it is readily seen that



F' = l



F*' - a'XQ



V - a'KP '

with similar expressions for G', H'. From these we readily obtain by reasoning similar to that in
§ 104, the equation corresponding to (86), viz. :



V\



VI \i.



-0.



r-a»XP V'-b'tiP V'-c'vP
This form of the equation agrees with that given in the Electricity and Magnetism ii. § 797.

By means of this equation the equations (91) and (92) readily follow when ♦' = 0. The
ratios oi F' -. G' : H' for any direction of propagation may also be determined.]



586 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

The experiments of Knoblauch* on electric induction through crystals seem
to shew that a, b and c may be different.

The inequality, however, of X, /x, v is so small that great magnetic forces
are required to indicate their difference, and the differences do not seem of
sufficient magnitude to account for th§ double refraction of the crystals.

On the other hand, experiments on electric induction are liable to error
on account of minute flaws, or portions of conducting matter in the crystal.

Further experiments on the magnetic and dielectric properties of crystals
are required before we can decide whether the relation of these bodies to mag-
netic and electric forces is the same, when these forces are permanent as when
they are alternating with the rapidity of the vibrations of light.

Relation between Electric Reshtance and Trayisparency.

(106) If the medium, instead of being a perfect insulator, is a conductor
whose resistance per unit of volume is p, then there will be not only electric
displacements, but true currents of conduction in which electrical energy is
transformed into heat, and the undulation is thereby weakened. To determine
the coefficient of absorption, let us investigate the propagation along the axis
of X of the transverse disturbance G.

By the former equations



4^/^(1+?) by (A),
d'G , , fld'G ldG\ , ,^. , ,„. ,^,.



d'G . nd'G ldG\

P

If G is of the form

G = e-'^ cos (qx + nt) (96),

we find that

2^n^2^F

^ p q p I

where V is the velocity of light in air, ^nd i is the index of refraction. J' he

proportion of incident light transmitted through the thickness x is

e-*' (98).

* PhUpnophicaX Magazine, 1852.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 587

Let R be the resistance in electromagnetic measure of a plate of the
substance whose thickness is x, breadth h, and length I, then

OX,

2px = 4.^Zjl-^ (99).

(107) Most transparent solid bodies are good insulators, whereas all good
conductors are very opaque.

Electrolytes allow a current to pass easily and yet are often very trans-
parent. "We may suppose, howaver, that in the rapidly alternating vibrations
of light, the electromotive forces act for so short a time that they are unable to
effect a complete separation between the particles in combination, so that when
the force is reversed the particles oscillate into their former position without
loss of energy.

Gold, silver, and platinum are good conductors, and yet when reduced to
suflSciently thin plates they allow light to pass through them. If the resistance
of gold is the same for electromotive forces of short period as for those witli
which we make experiments, the amount of light which passes through a piece
of o-old-leaf, of which the resistance was determined by Mr C. Hockin, would
be only lO""*" of the incident light, a totally imperceptible quantity. I find that
between -=-Jjo and xoVo" ^^ green hght gets through such gold-leaf. Much of thiis
is transmitted through holes and cracks ; there is enough, however, transmitter!
through the gold itself to give a strong green hue to the transmitted light.
This result cannot be reconciled with the electromagnetic theory of light, unless
we suppose that there is less loss of energy when the electromotive forces are
reversed with the rapidity of the vibrations of light than when they act for
sensible times, as in our experiments.



Absolute Valves of the Electromotive and Magnetic Forces called into play in tin

Propagation of Light.

(108) If the equation of propagation of light is

i^=^cos^(2-F0.

74—2



588 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

the electromotive force will be

F=-A^Vsm^{z-Vt);
and the energy per unit of volume will be

Stt/xP'

where P represents the greatest value of the electromotive force. Half of this
consists of magnetic and half of electric energy.

The energy passing through a unit of area is

so that

P = s/S7riJiVW,

where V is the velocity of light, and W is the energy communicated to unit
of area by the Ught in a second.

According to Pouillet's data, as calculated by Professor W. Thomson*, the
mechanical value of direct sunlight at the Earth is

83'4 foot-pounds per second per square foot.

This gives the maximum value of P in direct sunlight at the Earth's distance
from the Sun,

P = 60,000,000,

or about 600 Darnell's cells per metre.

At the Sun's surface the value of P would be about
13,000 Daniell's cells per metre.

At the Earth the maximum magnetic force would be 193 f.

At the Sun it would be 4*13.

These electromotive and magnetic forces must be conceived to be reversed
twice in every vibration of Hght ; that is, more than a thousand miUion million
times in a second.

* Transactions of Uie Royal Society of Edirthurgh, 1854 ("Mechanical Energies of the Solar
System").

t The horizontal magnetic force at Kew is about 1'76 in metrical units.



A DYNAMICAL THEORY OF THE ELECmOMAONETIO FIELD. 589



PART VII.
CALCULATION OF THE COEFFICIENTS OF ELECTROMAGNETIC INDUCTION.

General Methods.

(109) The electromagnetic relations between two conducting circuits, A and
B, depend upon a function M of their form and relative position, as has been
already shewn.

M may be calculated in several different ways, which must of course all
lead to the same result.

First Method. M is the electromagnetic momentum of the circuit B when
A carries a unit current, or



^=/(^l-4-^S-)*''



where F, G, H are the components of electromagnetic momentum due to a unit
current in A, and ds is an element of length of B, and the integration is
performed round the circuit of B.

To find F, (t, H, we observe that by (B) and (C)

d?F d'F d'F

with corresponding equations for G and Hy p\ q, and / being the components
of the current in A.

Now if we consider only a single element ds of A, we shall have

p=s*' 'i=±'^' ^=s;*'

and the solution of the equation gives

F^t^ds, G=f^i^ds. H=t<i^ds,
p ds p as pels



590 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

where p is the distance of any point from ds. Hence
]] p \ds da' ds ds' ds ds'



= - cos Odsds',
Jj P



where 6 is the angle between the directions of the two elements ds, ds', and
p is the distance between them, and the integration is performed round both
circuits.

In this method we confine our attention during integration to the two linear
circuits alone,

(110) Second Method. M is the number of lines of magnetic force which
pass through the circuit B when A carries a unit current, or

M= t (fial + p.fim + ixyn) dS',

where fia, p.^, py are the components of magnetic induction due to unit current
in A, S' is a surface bounded by the current B, and I, m, n are the direction-
cosines of the normal to the surface, the integration being extended over the
surface.

We may express this in the form

M= 11% — sin 6 sin 6' sin (bdS'ds,
r p.

where dS' is an element of the surface bounded by B, ds is an element
of the circuit ^, ^ is the distance between them, and 9' are the angles
between p and ds and between p and the normal to dS respectively, and (f) is
the angle between the planes in which and 0' are measured. The integration
is performed round the circuit A and over the surface bounded by B.

This method is most convenient in the case of circuits lying in one plane,
in which case sin^ = l, and sin<^=l.

(111) Third Method. M is that part of the intrinsic magnetic energy of
the whole field which depends on the product of the currents in the two
circuits, each current being unity.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. j'.H

Let a, yS, y be the components of magnetic intensity at any point due to
the first circuit, a', yS', y the same for the second circuit ; then the intrinsic-
energy of the element of volume dV of the field is

''■ {(a + aT + (^ + yS-r + iy + /)'}</ r.

OTT

The -part which depends on the product of the currents is

^^-[o.o:^m'ryy')dV.

477

Hence if we know the magnetic intensities / and /' due to the unit current
in each circuit, we may obtain M by integrating

f-S/i/rcos^c^F

over all space, where 6 is the angle between the directions of / and /'.

Application to a Coil.

(112) To find the coefiicient (M) of mutual induction between two circular
linear conductors in parallel planes, the distance between the cur\ es being every-
where the same, and small compared with the radius of either.

If r be the distance between the curves, and a the radius of either, then
when r is very small compared with a, we find by the second method, as a
first approximation,

M=A7ra(\og~-2

To approximate more closely to the value of M, let a and a, be the radii of
the circles, and h the distance between their planes ; then

T^ = {a-a,y + b\
We obtain M by considering the following conditions: —
1st. M must fulfil the difierential equation

d'M d'M IdM^^
da^ dJ/ a da

This equation being true for any magnetic field symmetrical with respect to the
common axis of the circles, cannot of itself lead to the determination of 3/ ;vy
a function of a, a^, and h. We therefore make use of other conditions.



592 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

2ndly. The value of M must remain the same when a and a, are exchanged.
3rdlj. The first two terms of M must be the same as those given above.
M may thus be expanded in the following series : —

(113) We may apply this result to find the coeflScient of self-induction
(//) of a circular coil of wire whose section is small compared with the radius
of the circle.

Let the section of the coil be a rectangle, the breadth in the plane of
the circle being c, and the depth perpendicular to the plane of the circle being h.

Let the mean radius of the coil be a, and the number of windings n;
then we find, by integrating,



^-6



|-2 j J J J M{xy xy) dx dy dx' dy\



where M(xy x'y') means the value of M for the two windings whose coordinates
are xy and xy respectively; and the integration is performed first with respect
to X and y over the rectangular section, and then with respect to x and y'
over the same space.

L = 47rn^a|log.^ + 1^ - | (^-f)cot2^-|cos2^-icot^^logcos^-itan'^logsin^|

Here a= mean radius of the coil.
„ r= diagonal of the rectangular section = Jb" + c".
,, 6= angle between r and the plane of the circle.
„ n= numbei of windings.
The logarithms are Napierian, and the angles are in circular measure.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. o9:{

In the experiments made by the Committee of the British Association for
determining a standard of Electrical Resistance, a double coil was used, con-
sisting of two nearly equal coils of rectangular section, placed parallel to each
other, with a small interval between them.

The value of L for this coil was found in the following way.

The value of L was calculated by the preceding formula for six different
cases, in which the rectangular section considered has always the same breadth,
while the depth was

A, B, C, A + B, B+C, A+B+C,
and n = 1 in each case.

Calling the results L{A), L(B), L{C), &c.,

we calculate the coefficient of mutual induction M(AC) of the two coils thus,

2ACM{AC) = {A+B+CYL{A+B + C)-(A+BYL(A-\-B)

-{B+CyL{B-^C) + RL(B).

Then if n^ is the number of windings in the coil A and w, in the coil C, the
coefficient of self-induction of the two coils together is

L = n,'L{A) + 2n,n,M{ACr)-\-n.^L{C).

(114) These values of L are calculated on the supposition that the windings
of the wire are evenly distributed so as to fill up exactly the whole section.
This, however, is not the case, as the wire is generally circular and covered with
insulating material. Hence the current in the wire is more concentrated than it
would have been if it had been distributed uniformly over the section, and the
currents in the neighbouring wires do not act on it exactly as such a uniform
current would do.

The corrections arising from these considerations may be expressed as nu-
merical quantities, by which we must multiply the length of the wire, and they
are the same whatever be the form of the coil.

Let the distance between each w^ire and the next, on the supposition that
they are arranged in square order, be D, and let the diameter of the wire
be d, then the correction for diameter of wire is

VOL. I. 75



594 A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD.

The correction for the eight nearest wires is

+ 0-0236.
For the sixteen in the next row +0-00083.

These corrections being multiplied by the length of wire and added to the
former result, give the true value of L, considered as the measure of the
potential of the coil on itself for unit current in the wire when that current
has been established for some time, and is uniformly distributed through the
section of the wire.

(115) But at the commencement of a current and during its variation the
current is not uniform throughout the section of the wire, because the induc-
tive action between different portions of the current tends to make the current
stronger at one part of the section than at another. When a uniform electro-
motive force P arising from any cause acts on a cylindrical wire of specific
resistance p, we have

where F is got from the equation

d'F 1 dF

r being the distance from the axis of the cylinder.

Let one term of the value of F be of the form T/'", where T is a
function of the time, then the term of p which produced it is of the form

1






Hence if we write



dT\ fiTT d-T fin



pp=\'^-dtrv^






\ d'T ^ ,
r* — &c.



V . 2* df
The total counter current of self-induction at any point is

from < = to « = Qo .



k:



A DYNAMICAL THEORY OF THE ELECTKOMAGNETIC FIELD. 595

When . = 0.^ = 0, ■••(§) =i',O.= 0,&c.







When t:


= 00 , p


_P

P'


■^■m."-


(f ).


= 0,


&c.






J Jo


(?■


-,)


rdrdt =


= 1 T.r
P


■*t?S-


-?'f


1
.2\


d'T
3 dt'


/ + &C.


from


t = to


= 00 .



















When ^ = 0, ^ = throughout the section, •'• (-7-) =P, (7777) =0» <^c.

When« = <.,^ = „ „ „ .•.(f)^ = 0, ('g)_ = 0.&c.

Also if Z be the length of the wire, and R its resistance,

PI

and if C be the current when established in the wire, C=~^.

K

The total counter current may be written

7 7 T C

Now if the current instead of being variable from the centre to the cir-
cumference of the section of the wire had been the same throughout, the value
of F would have been



where y is the current in the wire at any instant, and the total counter
current would have been

Hence L=^ L — ^/x/.



0:



or the value of L which must be used in calculating the self-induction of a
wire for variable currents is less than that which is deduced from the suppo-
sition of the current being constant throughout the section of the wire by ^/x/,

75—2



596 A DYNAMICAL THEORY OF THE ELECTTROMAGNETIC FIELD.

where I is the length of the wire, and /n is the coefficient of magnetic induc-
tion for the substance of the wire.

(116) The dimensions of the coil used by the Committee of the British
Association in their experiments at King's College in 1864 were as follows:—

metre.

Mean radius =a= -158194

Depth of each coil =6 = -01608

Breadth of each coil =c = '01841

Distance between the coils ='02010

Number of windings n= 313

Diameter of wire ="00126

The value of L derived from the first term of the expression is 437440
metres.

The correction depending on the radius not being infinitely great compared
with the section of the coil as found from the second term is -7345 metres.

The correction depending on the diameter of the wire is 1 , ...qq^

per unit of length J

Correction of eight neighbouring wires + '0236

For sixteen wires next to these + '0008

Correction for variation of current in difierent parts of section - "2500

Total correction per unit of length '22437

Length 311-236 metres.

Sum of corrections of this kind 70

Final value of i by calculation 430165

This value of L was employed in reducing the observations, according to
the method explained in the Report of the Committee*. The correction de-
pending on L varies as the square of the velocity. The results of sixteen
experiments to which this correction had been applied, and in which the velocity
varied from 100 revolutions in seventeen seconds to 100 in seventy-seven seconds,

* British Association Reports, 1863, p. 169.



A DYNAMICAL THEORY OF THE ELECTROMAGNETIC FIELD. 597

were compared by the method of least squares to determine what further cor-
rection depending on the square of the velocity should be applied to make the
outstanding errors a minimum.

The result of this examination shewed that the calculated value of L should
be multiplied by 1*0618 to obtain the value of L, which would give the most
consistent results.

We have therefore L by calculation 430165 metres.

Probable value of L by method of least squares 456748 „

Result of rough experiment with the Electric Balance (see § 46) 41 0000 „

The value of L calculated from the dimensions of the coil is probably much
more accurate than either of the other determinations.



[From the Philosophical Magazine, Vol. xxvii]



■"' XXVI. On the Calculation of the Equilibrium and Stiffness of Frames.

The theory of the equilibrium and deflections of frameworks subjected to
the action of forces is sometimes considered as more complicated than it really
is, especially in cases in which the framework is not simply stiff, but is
strengthened (or weakened as it may be) by additional connecting pieces.

I have therefore stated a general method of solving all such questions in
the least complicated manner. The method is derived from the principle of
Conservation of Energy, and is referred to in Lame's Legons sur VElasticite,
Lefon 7"'^ as Clapeyron's Theorem ; but I have not yet seen any detailed
application of it.

K such questions were attempted, especially in cases of three dimensions,
by the regular method of equations of forces, every point would have three



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