. Let JT
and Y be the two solutions, making
w = X+qY, (153)
where q is a constant. The normal F"-function, say u, is got from w by
the first of (145), giving
if X' = dX/dz, Y' = dY/dz.
In X and F, which together make up the w in (153), p has the same
value. Therefore, in (147), supposing (7 X to be X and C 2 to be F, we
have disappearance of the right member, making
j g J) = 0, or FjOj - F 2 tfi = constant,
or XY' -YX' = S"* constant = hS", say, ........ (155)
leading to the well-known equation
connecting the two solutions of the class of equations (152); which we
see expresses the reciprocity of the mutual activities of the two parts
into which we may divide the electromagnetic state represented by a
single normal solution.
ON THE SELF-INDUCTION OF WIRES. PART IV. 225
Also, by (147), integrating with respect to z from to /,
( l
Jo
SUl u. 2 dz - lA- ...(156)
o Jo Pi-P* Pi -Pt
either member of which represents the complete U 12 - T 12 of the line.
The negative of this quantity, as in Part III., is the corresponding
7 12 - T 12 in the terminal arrangements; so that the value of 2(U T)
in a complete normal system, including the apparatus, is
2(U-T) = Stfdz- u?dz-w? + w*, ...(157)
Jo JoqP dp dp
if P r /C = Z l at z = l and Z at 2 = 0, (these being functions of p and
constants), and w v W Q are the values of w at z = l and 0. Or, which is
the same,
(158)
as before used.
The Solution for V and C due to an Arbitrary Distribution of e, subject
to any Terminal Conditions.
There is naturally some difficulty in expressing the state at time t
in this form :
due to an arbitrary initial state, on account of the difficulty connected
with
(JZf-BfHCft-ft),
and the unstated form of R". But when the initial state is such as can
be set up by any steadily-acting distribution of longitudinal impressed
force (e an arbitrary function of 2), so that whilst r is arbitrary, C is
only in a very limited sense arbitrary, and C, C, etc., are initially zero,
and certain definite distributions of electric and magnetic energy in the
terminal apparatus are also necessarily involved ; in this case we may
readily find the full solutions, and therefore also determine the effect of
any distribution of e varying anyhow with the time. In fact, by the
condenser-method of Part III., we shall arrive at the solution (135);
we have merely to employ the present u and w, and let M be the value
of the right member of (158). The following establishment, however,
is quite direct, and less mixed up with physical considerations.
To determine how V and C rise from zero everywhere to the final
state due to a steadily-acting arbitrary distribution of e put on at
the time 2 = 0. Start with e 2 at 2 = 2 2 and none elsewhere, and let
(X + q Q Y)A Q and (X+q l Y)A l be the currents on the left (nearest 2=0)
and right sides of the seat of impressed force. We have to find q Q , q l}
A Qt and A r The condition F=Z Q C at 2 = gives us, by (153), 154),
therefore q = - (X' + S{Z Q X ) + (Yl + S!Z Q YJ. .. .......... (159)
H.E.P. VOL. II. P
226 ELECTRICAL PAPERS.
Similarly, V Zfi at z = /, gives us
l )^(Y[ + S'{Z l Y l } ............. (160)
Here the numbers and l mean that the values of X, etc., and S" at
z = and atz = l are to be taken.
Now, at the place z = 2 the current is continuous, whilst the V rises
by the amount e 2 suddenly in passing through it. These two con-
ditions give us
where the 2 means that the values at z = z z are to be taken. These
determine A and A^ to be
or A -* i or a oa / 161 \
A *----
Now use (155), making the denominator in (161) be ^( -!?i)- We
have then, if G' and G l are the currents on the left and right sides of
the seat of impressed force,
These are, when the p is throughout treated as d/dt, the ordinary
differential equations of GO and C\ arising out of the partial differential
equation of C by subjecting it to the terminal conditions and to the
impressed-force discontinuity.
Now make use of the algebraical expansion *
/(?> = v_JW .. (163)
* [The limitations to which this expansion is subject render its use in the above
manner undesirable even when it gives correct results, and, of course, when it
gives incorrect results, as when the initial G is not zero, the manner of application
should necessarily be changed. We should rather proceed thus : Let
. ,...(1)
be the differential equation connecting G with e, where p stands for d/dt, and
0(2>) = is the determinantal equation of the system, that is, f means d^/dp, and the summation includes all the roots of (p) =
Therefore, by (1), using (2) and integrating,
e being zero before, and constant after t = 0. But also, by (2),
1 = 2 1 > (4)
ON THE SELF-INDUCTION OF WIRES. PART IV. 227
the summation being with respect to the p's which are the roots of
(.p) = 0, without inquiring too curiously into its strict applicability, or
troubling about equal roots. Here p Q has to be d/dt, and the p'a the
roots of
< = %o-?i) = ;
so that (162) expands to
e 2 (164)
where the single q takes the place of the previous q Q or q v which have
now equal values, and C has the same expression on both sides of the
seat of impressed force. But e 2 is constant with respect to /, whilst C
is initially zero ; hence
d/dt p
where means with p - 0, so that (3) becomes
o P
(5)
Now perform the operations indicated by J\p^ and we get
= 8^+6^-^1.6^, ...(6)
0o P'
where f means / with p = 0. (See also the investigation at the end of the (later)
paper on "Resistance and Conductance Operators.") Here e/" /0o i s the
steady current, when there is such a thing.
Thus, if we take = h(q - qj, and use (6), (162) lead to
e*><, .............. , ............... (7)
0o
instead of (165). In the first terms the p = values must be taken, with
w 2 = X,> + q 1 F 2 in (7), and w 2 = X 2 + q F 2 in (8).
Here q and q l are not the same, but they are the same in the summation ; because
then = 0. We may write (165) thus :
^=^0 + 2^- ", .................................... 0)
where <7 is the final steady current, to be got direct from the first or second of
(162) as the case may be. Therefore (166) should be
where C is the final steady current at z due to the whole impressed force.
In accordance with the above (167) is not always applicable, and in accordance
with the text (168) is incorrect. But the substituted method of finding (7 , viz.
(169), will do when (164) is applicable, and fail otherwise. The result (170),
however, is independent of this restriction, as it is immediately obtainable from
the differential equations (162).
So up to (162) inclusive the text is correct. Then pass on to (170), (172), as the
next clear results. Between these places modify the method as in the present
note.]
228 ELECTRICAL PAPERS.
which brings (164) to
065)
which is the complete solution. By integration with respect to z we
find the effect due to a steady arbitrary distribution of e put on at
= 0; thus
Ji
ewdz
a 66 )
where / = d(J>/dp, and w is the normal current-function X-t-qY. To
express the F"-solution, turn the first w into u. The extension to e
variable with t, as in Part III., is obvious. But as the only practical
case of e variable with t is the case of periodic e, whose solution can be
got immediate!} 7 from the equations (162) by putting p 2 = - ?i 2 , constant,
the extension is useless. Note that q Q and q l are not equal in (162),
and therefore in the periodic solution obtained from (162) direct they
must be both used.
The quantity - <$>' which occurs here is identical with the former
complete '2(U - T) of the line and terminal apparatus of (157) or (!58).
Let C be the finally-reached steady current. By (166) it is
(167)
To this apply (163), with p Q = 0. Then a finite expression for C Q is
C Q = ew dz, ........................... (168)
Q are what w and > become when p = in them. Or,
rather, it would be so if q Q and q 1 taken as identical could be consistent
with /> = 0. But this is not generally true, so that (168) is wrong. To
suit our present purpose, we must write, by (162),
( 169 )
the q being used in w , and the q l in w r Now we can take p = 0, and
get the correct formula to replace (168), viz.
o\
the second meaning that p = in w and w r
If there is no leakage (K= in S"), C becomes a constant, given by
C = edz-r Rdz + R Q + R, .................. (171)
o
ON THE SELF-INDUCTION OF WIRES. PART IV. 229
where the numerator is the total impressed force, and the denominator
the total steady resistance ; R, H , and R^ being what E", - Z , and Z l
become when^? = in them.
But when there is leakage (170) must be used; it would require a
very special distribution of impressed force to make C the same every-
where. To find the corresponding distribution of F", say V& in the
steady state, we have then
so that a single differentiation applied to (170) finds
Knowing thus C finitely, we may write (166) thus,
where C is given in (170). The summation here, with / = 0, is there-
fore the expansion of C Q .
The internal state of the wire is to be got by multiplying the first w
by such a function of r, distance from the axis, and of whatever other
variables may be necessary, as satisfies the conditions relating to inward
propagation of magnetic force, and whose value at the boundary is
unity. In the simple case of a round solid wire, (172) becomes, by
(87), Part II,
w\ewdz
r ^ sr
This gives C n the current through the circle of radius r, less than a t
the radius of the wire, C 0r being the final value. The value of s l is
( - 47iY* 1 & 1 ^)i. Here of course we give to /x a , &j, and a-^ their proper
values for the particular value of z. As before remarked, they must
only vary slowly along z.
In the case of a wire of elliptical section it is naturally suggested
that the closed curves taking the place of the concentric circles defined
by r = constant in (173) are also ellipses; and that in a wire of square
section they vary between the square at the boundary and the circle at
the axis. The propagation of current into a wire of rectangular sec-
tion, to be considered later, may easily be investigated by means of
Fourier-series, at least when the return-current closely envelops it.
Explicit Example of a Circuit of Varying Resistance, etc. Bessel Functions.
As an explicit example of the previous, let us, to avoid introducing
new functions, choose the electrical data so that the current-functions
A' and Y are the / and K functions. This can be done by letting R"
be proportional and S" inversely proportional to the distance from
one end of the line. Let there be no leakage, and
where S is a constant, and R'f a function of d/dt, but not of z. The
electromagnetic and electrostatic time-constants do not vary from one
230 ELECTRICAL PAPERS.
part of the line to another. The equation of the current-function is
-!&-**" ....................... (152a)
from which we see that
X= JJ(fz), Y= K (fz), where /= ( -
But, owing to the infinite conductivity a* the 2 = end of the line,
making K Q (fz) = GO there, we shall only be concerned with the J Q
function, that is, on the left side of the impressed force, in the first
place. Since V is made permanently zero at z 0, the terminal condi-
tion there is nugatory. So
w = J Q (fz), and w = J Q (fz) + q&tfz) ;
and u
on the left and right sides of an impressed force, say at z = z 2 . The
value of q v got from the V= Zfi condition at z = l, is
We have also
and the (7-solution (166) becomes *
l -*), ......... (166a)
where = - qJSop, and q 1 is given by
If we short-circuit at z = l, making ^i = 0, we introduce peculiarities
connected with the presence of the series of j^'s belonging to /= 0.
The expression of ^ is then, by (160a), q^ = J^fl) / K^fl). It seems
rather singular that we should have anything to do with the K-^
function, seeing that C and V are expanded in series of the / and
/! functions. But" on performing the differentiation of with respect
it turns out to be all right, the denominator in (166a) becoming
in general ; whilst in the / = case, which makes > = J-KJ' / 2 , we have
The value of when p = in it is, by inspection of the expansions of
J l and K lt simply %RJP, the steady resistance of the line ; E Q being the
* [In accordance with the remarks in the footnote on page 226, we should write
the equation (166a) thus :
where O is the expression for the steady current at z due to e.]
ON THE SELF-INDUCTION OF WIRES. PART IV. 231
constant that fig becomes with p = 0. We may therefore write (166a)
thus :
***
dp dp^
where the first term is (7 , the finally-reached current ; the following
summation, extending over them's belonging to/=0, is its expansion,
and therefore cancels the first term at the first moment ; and the third
part is a double summation, extending over all the/s except /= 0, each
/-term having its following infinite series of p-terms. This quantity
(the third part) is zero initially as well as finally. If there were no
elastic displacement permitted (S = 0), the solution would be repre-
sented by the remainder of (172a), for we should then have C inde-
pendent of z, and
P edz=\ R"dz.C=lR'W.C
[
Jo
for the differential equation of (7, whose solution is plainly given by the
first two terms. The third part of (Ilia} is therefore entirely due
to the combined action of the electrostatic and magnetic induction.
When the impressed force is entirely at z - 1, and of such strength as
to produce the steady current (7 , and if we take R" = R + Lp, where R
and L are constants, there will be only two ^>'s to each /, given by
/ 2 ^= -S p(E + Lp). The subsidence from the steady state, on removal
of the impressed force, is represented by
V J (fi) S7p
where the summations range over the ^'s, not counting the p = - RjL
whose G'-term is exhibited separately; there is no corresponding F-term.
A comparatively simple solution of this nature may be of course inde-
pendently obtained in a more elementary manner. On the other hand,
great power is gained by the use of more advanced symbolical methods,
which, besides, seem to give us some view of the inner meaning of the
expansions and of the operations producing them, that is wanting in the
treatment of a special problem on its own merits, by the easiest way
that presents itself.
Homogeneous Circuit. Fourier Functions. Expansion of Initial State to
suit the Terminal Conditions.
Leaving, now, the question of variable electrical constants, let the line
homogeneous from beginning to end, so that R" and S" are functions
p, but not of z. The normal current-functions are then simply
X = cos mz, Y= sin mz,
'here ra is the function of p given by - m? = R"S", so that
w = cos mz + q sin mz, u = (m/S") (sin mz - q cos mz). ( 1 74)
232 ELECTRICAL PAPERS.
Let there be a single impressed force e 2 at z = z 2 ; then the differen-
tial equations of the currents on the left and right sides of the same,
corresponding to (162), will be
where g and q l are given by
=-^X (
m ' Zl (
As before, in the case of an arbitrary distribution of e we are led to the
solution (165), wherein for w (and for u in the corresponding F'-formula)
use the expressions (174), in which q is to be the common value of the
q Q and q l of (1606), and
= values. They are, if
= g say,
if .R is the steady resistance of line (both conductors), and K is the
conductance of the insulator, both per unit length of line ;
if R Q = effective steady resistance at the z = terminals, and
_ gi sin gli - KR-^ cos gli
1 gi cos gli + KR l sin gli 9
if R l = effective steady resistance at the z = I terminals.
The expression on the right side of (176) is, of course, real in the
exponential form, and the steady distribution of V is got by
KF = - dCJdz.
Using the thus-obtained expressions, we reach the (172) form of C-
solution, and the corresponding
The value of <' here, got by differentiation with respect to p, may be
written in many ways, of which one of the most useful, for expansions
in Fourier series, is the following. Let
w = (l+qrf cos(mz+0);
ON THE SELF-INDUCTION OF WIRES. PART IV. 233
d m d f, ,/m
then
( m Z ^- Z \-l\ (177)
dp d(ml)\S" (m/S")* + Z^~J J
Corresponding to this,
finds the angles ml ; it is got by the union of
tan(9 = %/w, tan (ml + 6) = Sf'ZJm, .......... (179)
which are equivalent to (160&).
For example, if we take R ff = R, constant, thus abolishing inertia,
and S" = Sp, no leakage, and S constant (R and S not containing p, that
is to say), the expansion of F (an arbitrary function of z) is [see also
vol. i., p. 123, and p. 152]
sin (mz + 0) I V^ sin (mz + 6)dz
Jo _, ....... (180)
L m Z I~ Z Q \
d(ml) ty (m/Spf + Z.zJ
subject to (178). Here p = - m?/ES, so that the state of the line at
time t after it was V^ when left to itself, is got by multiplying each
term in the expansion by ^- r ^ t i RS . The corresponding current is given
by RC = - dF/dz. But the solution thus got will usually only be
correct, although (180) is correct, when there is, initially, no energy in
the terminal apparatus. If there be, additional terms in the numerator
of (180) are required, to be found by the energy-difference method of
Part III. They will not alter the value of the right member of (180) at
all ; they only come into effect after the subsidence has commenced.
Similar remarks apply whatever be the nature of the line. It is,
however, easy to arrange matters so that the energy in the terminal
apparatus shall produce no effect in the line. For example, join the
two conductors at one end of the line through two equal coils in
parallel ; if the currents in these coils be equal and similarly directed
in the circuit they form by themselves, they will not, in subsiding,
affect the line at all.
Returning to (177), or other equivalent expression, it is to be
observed that particular attention must be paid to the roots ml = 0,
which may occur, or to the series of roots p belonging to the m = case,
when we are working down from the general to the special, and happen
to bring in m = 0. Take ^ = for instance, making, by (175) and
(1606),
<= -Z Q -^
Up
rhere m 2 = - SpR". Then
_-
dp dp 2m \dp p ) 2
234 ELECTRICAL PAPERS.
Now, as long as Z^ is finite, m cannot vanish ; but when Z is zero,
giving ml = any integral multiple of IT, m = is one case. Then we
have, when m is finite,
and
dp 2\dp p *dp 2 dp^
but when m is zero the middle term on the right of the preceding
equation becomes finite, making
The result is that the current-solution contains a term, or infinite
series, apparently following a different law to the rest, with no corre-
sponding terms in the F-solution. This merely means that the mean
current subsides without causing any electric displacement across the
dielectric, when the ends are short-circuited (z?=0); so that if, in the
first place, the current is steady, and there is no displacement, there
will be none during the subsidence.
Transition from tJie Case of Resistance, Inertia, and Elastic Yielding
to the same without Inertia.
The transition from the combined inertia-and-elasticity solutions to
elasticity alone is very curious. Thus, let Z=0 at both ends, and
R" = R + Lp, where R and L are constants not containing p. The rise
of current due to e is shown by
the m's in the summation being ir/l t 2ir/l, etc. ; and each having two
's, given by
The m = part is exhibited separately, and is what the solution would
be if e were a constant (owing to the constancy of R). But, whatever e
be, as a function of z, the summation comes to nothing initially, on
account of the doubleness of the p's t just as in (I72a) the double
summation vanishes by reason of every ^-summation vanishing when
Now, in (183), let L be exceedingly small. The two p's approximate
to - m?/BS, the electrostatic one, and to - B/L, the magnetic one,
which goes up to oo , the storehouse for roots. The current then rises
thus :
C-
Fed? H -f- m / L \ 2 ^ , C l
: Mz '^ * J + ^Vcoswwl
Jo R* M Jo
--Ri2j mz ] o e(
But the first line on the right side is equivalent to
Vcosrascosm^^l-e-"^) (184)
ON THE SELF-INDUCTION OF WIRES. PART IV. 235
and here the exponential term vanishes instantly, on L being made
exactly zero, so that (184) becomes
I _ -' 2 /fls\ (185}
R Rl J
except at the very first moment, when it gives C=e/Ii, which is quite
wrong, although the preceding formula, giving 0=0 at the first
moment, is correct. Or, (185) is equivalent to
from which inertia has disappeared. Here V is given by (188) below.
The process amounts to taking one half the terms of the summation in
(183), and joining them on to the preceding term to make up e/fi,
which is quite arbitrary. An alternative form of (185) is
C= 7 edz + y]cosmzecosmzdz.- m2t / s ........ (186)
jKijo m*- J
On the other hand, there is no such peculiarity connected with the
^-solution in the act of abolishing inertia. The m = term is
- (sin mz I edz\ which =0,
ffl\8p Jo /
because m is zero and p finite. Therefore V rises thus,
inwzl ecosmzdz
-
before abolition of inertia. But as L is made zero, the denominator
becomes m 2 for the electrostatic p, and oo for the other ; thus one half
the terms vanish, leaving
-*/), ............ (188)
" Jo
when = 0, without any of the curious manipulation to which the
current-formula was subjected.
Transition from the Case of Resistance, Inertia, and Elastic Yielding
to the same without Elastic Yielding.
Next, let us consider the transition from the combined elasticity-and-
inertia solution to inertia alone (of course with resistance in both cases,
as in the preceding transition). It is usual to wholly ignore electro-
static induction in investigations relating to linear circuits. This is
equivalent to taking $ = 0, stopping elastic displacement, and compelling
the current to keep in the wires always, i.e. when the insulation is
perfect, as will be here assumed. We then have, by (145),
-as- ' -Ts-^ .................. (189 >
236 ELECTRICAL PAPERS.
By integrating the second of these with respect to z we get rid of F",
and obtain the differential equation of (7,
say, ........... (190)
whence follows this manner of rise of the current, when e is steady
and put on everywhere at the time t = 0, reaching the final value (7 ,
............... (191)
f England commenced, to the conclusion that long-distance signalling
(i.e. hundreds of miles) was possible by induction, a conclusion which
las been somewhat supported by results, so far as the experiments have
238 ELECTRICAL PAPERS.
yet gone. Recognising the great complexity of the problem, and the
difficulty of hitting the exact conditions, I made no special calculations,
but preferred to be guided by general considerations; for, in the en-
deavour to be precise when the data are uncertain and very variable,
one is in great danger of swallowing the camel.
One may be fairly well acquainted with electromagnetism, and also
with the capabilities of the telephone, and yet receive the idea of
signalling by induction long distances with utter incredulity, or at
least in the same way as one might accept the truth of the statement,
that when one stamps one's foot the universe is shaken to its founda-
tions. Quite true, but insensible a few yards away. The incredulity
will probably be based upon the notion of rapid decrease with distance
of inductive effects. This, however, leaves out of consideration an im-
portant element, namely the size of the circuits.
The coefficients of electromagnetic induction of linear circuits are
proportional to their linear dimensions. If, then, we increase the size
of two circuits n times, and also their distance apart n times, the mutual
inductance M is increased n times. Let R^ and R. 2 be the resistances
of primary and secondary. The induced current (integral) in the
secondary due to starting or stopping a current C\ in the primary is
MCJUft or Me^R^Ry if e l be the impressed force in the primary. Now
increasing the linear dimensions, and the distance, in the ratio n (with
the same kind of wire) increases M, R ly and R. 2 all n times. So only
e l remains to be increased n times to get the same secondary-current
impulse. We can therefore ensure success in long-distance experiments
on the basis of the success of short-distance experiments, with elements
of uncertainty arising from new conditions coming into operation at the
long distances.
But practically the result must be far more favourable to the long
than to the short distances than the above asserts. For no one, when
multiplying the distance and size of circuits, say ten times, would think
of putting ten telephones in circuit to keep rigidly to the rule. Thus
it may be that only a slight increase of e 1 is required, on account of M
being multiplied in a far greater ratio than the resistances, or the self-
inductances. Thus, it is not uncommon for the R and L of a telephone
to be 100 ohms and 12 million centim. These form the principal parts
of the R and L of a circuit of moderate size, and of course do not in-
crease when we enlarge the circuit. It is therefore certain that we can
signal long distances on the above basis, with a margin in favour of the
long distances, which will be large or small according as the circuits are
small or large.
Again, if e l in the primary be periodic, of frequency W/^TT, the ratio of
the amplitude of the current in the secondary to that in the primary
willbe '
Now, without any statement of the magnitude of the current in the
primary, if it be largely in excess of requirements for signalling in the
primary, so that -^ part, say, would be sufficient for the purpose, then
we shall have enough current in the secondary if the above ratio is only
ON THE SELF-INDUCTION OF WIRES. PART IV. 239
y^. But, without going to precise formulae, it may be easily seen that
the above ratio may be made quite a considerable fraction, in com-
parison with T J^, with closed metallic circuits whose linear dimensions
and distance are increased in the same ratio. But we should expect a
rapid decrease of effect when the mean distance between the circuits
exceeds their diameter, keeping the circuits unchanged. (It should be
understood that squares, circles, etc., are referred to.)
The theory seems so very clear (though it is only the first approxi-
mation to the theory), that it would be matter for wonder and special
inquiry if we found that we could not signal long distances by induction
between closed metallic circuits, starting on the basis of a short-distance
experiment, and following up the theory.
As a matter of fact, my brother found it was possible to speak by
telephone between two metallic circuits of J mile square, J mile between
centres, using two bichros with the microphone.
Now, coming to metallic lines whose circuits are closed through the
earth, the theory is rendered far more difficult on account of there
being a conduction-current from the primary to the secondary due
to the earth's imperfect conductivity. We therefore have, to say
nothing of electrostatic induction, a superposition of effects due to
induction and conduction, the latter being far more difficult to theo-
retically estimate than the former. But the reasoning regarding the
magnetic induction is not very greatly changed, although not so
favourable to long-distance signalling. If the return-currents diffused
themselves uniformly in all directions from the ends of the line, the
same property of n-fold increase of M with %-fold lengthening of the
lines and their distance would still be true. But the diffusion is one-
sided only, and is even then only partial, especially when exceedingly
rapid alternations of current take place. But we have the power of
counterbalancing this by the multiplication of the variations of current
in the primary that we can get by making and breaking the circuit,
with a considerable battery-power if necessary, getting something
enormous compared with the feeble variations of current in the micro-
phonic circuit, or that can work a telephone. Electrostatic induction
also comes in to assist, as it increases the activity of the battery, and
therefore the current in the secondary also.
But, as regards wires connected to earth, this does not profess to be
more than the very roughest reasoning, though in my opinion quite
plain enough to show that we may ascribe the signalling across 40 miles
of country between lines about 50 miles long mainly to induction, as we
should be necessitated to do if we carried the experiment further and
closed the circuits metallically by roundabout courses, for then the
plain arguments relating to induction will become valid. Experiments
of this kind are of the greatest value from the theoretical point of view,
and it is to be hoped that they will be greatly extended.
240 ELECTRICAL PAPERS.
PART V.
St. tenant's Solutions relating to the Torsion of Prisms . applied to the
Problem of Magnetic Induction in Metal Rods, with the Electric
Current longitudinal, and with close-fitting Return-Current.
The mathematical difficulties in the way of the discovery of exact
solutions of problems concerning the propagation of electromagnetic
disturbances into wires of other than circular section or, even^ if of
circular section, when the return-current is not equidistantly distributed
as regards the wire, or is not so distant that its influence on the dis-
tribution of the wire-current throughout its section may be disregarded
are very considerable. As soon as we depart from the simple type
of magnetic field which occurs in the case of a straight wire of circular
section, we require at least two geometrical variables in place of the
one, distance from the axis of the wire, which served before ; and we
may have to supplement the magnetic force " of the current," as usually
understood, by a polar force, or a force which is the space-variation of
a single-valued scalar, the magnetic potential, in order to make up the
real magnetic force.
There are, however, some simplified cases which can be fully solved,
viz., when the external magnetic field, that in the dielectric, is abolished,
by enclosing the wire in a sheath of infinite conductivity. It is true
that we must practically separate the wire from the sheath by some
thickness of dielectric, in order to be able to set up current in the
circuit by means of impressed force, so that we cannot entirely abolish
the external magnetic field ; but we may approximate in a great
measure to the state of things we want for purposes of investigation.
The wire, of course, need not be a wire in the ordinary sense, but a
large bar or prism. The electrostatic induction will be ignored,
requiring the wire to be not of great length ; thus making the problem
a magnetic one.
Consider, then, a straight wire or rod or prism of any symmetrical
form of section, so that, when a uniformly distributed current passes
through it, its axis is the axis of the magnetic field, where the intensity
offeree is zero. Let a steady current exist in the wire, longitudinal
of course, and let the return-conductor be a close-fitting infinitely-
conducting sheath. This stops the magnetic field at the boundary of
the wire. The sudden discontinuity of the boundary magnetic-force is
then the measure and representative of the return-current.
The magnetic energy per unit length is JLC 2 , where C is the current
in the wire and L the inductance per unit length. As regards the
diminution of the L of a circuit in general, by . spreading out the
current, as in a strip, instead of concentrating it in a wire, that is a
matter of elementary reasoning founded on the general structure of L.
If we draw apart currents, keeping the currents constant, thus doing
work against their mutual attraction, we diminish their energy at the
same time by the amount of work done against the attraction. Thus
the quantity ^LC* of a circuit is the amount of work that must be done
ON THE SELF-INDUCTION OF WIRES. PART V. 241
to take a current to pieces, so to speak ; that is, supposing it divided
into infinitely fine filamentary closed currents, to separate them against
their attractions to an infinite distance from one another. We do not
need, therefore, any examination of special formulae to see that the
inductance of a flat strip is far less than that of a round wire of the
same sectional area; their difference being proportional to the differ-
ence of the amounts of the magnetic energy per unit current in the
two cases. The inductance of a circuit can, similarly, be indefinitely
increased by fining the wire ; that of a mere line being infinitely great.
But we can no more have a finite current in an infinitely thin wire
than we can have a finite charge of electricity at a point, in which case
the electrostatic energy would also be infinitely great, for a similar
reason ; although by a useful and almost necessary convention we may
regard fine-wire circuits as linear, whilst their inductances are finite.
Now, as regards our enclosed rod with no external magnetic field, we
can in several cases estimate L exactly, as the work is already done, in
a different field of Physics. The nature of the problem is most simply
stated in terms of vectors. Thus, let h be the vector magnetic force
when the boundary of the section perpendicular to the length is circular,
and H what it becomes with another form of boundary ; then
H = h + F, and F=-Vfi ......................... (la)
That is, the field of magnetic force differs from the simple circular type
by a polar force F, whose potential is ft. This must be so because the
curl of H and of h are identical, requiring the curl of F to be zero. To
find F we have the datum that the magnetic force must be tangential
to the boundary, and therefore have no normal component ; or, if N be
the unit vector-normal drawn outward,
-FN = hN ................................. (2a)
is the boundary-condition. This gives F, when it is remembered that
F must have no convergence within the wire.
In another form, since we have h circular about the axis, and of
intensity 27nT at distance r from it, the current-density being F ; or
h = 27rr o Vkr, .... ............................. (3o)
if r is the vector distance from the axis in a plane perpendicular to it,
and k a unit vector parallel to the current ; we have
CI
if s be length measured along the bounding curve, in the direction of
the magnetic force. The boundary-condition (2a) therefore becomes,
in terms of the magnetic potential,
which, with V 2 12 = 0, finds the magnetic potential. Here p l is length
measured outward along the normal to the boundary.
H.E.P. VOL. ii. Q
242 ELECTRICAL PAPERS.
Or, we may use the vector-potential A. It is parallel to the current,
and consists of two parts ; thus,
where the second part on the right side is, except as regards a constant,
what it would be if the boundary were circular, its curl being /xh. To
find A', let its tensor be A' ; then
V 2 -^' = 0, and A f = fj.irT r 2 t ..................... (7a)
the latter being the boundary-condition, expressing that A is zero at
the boundary. Comparing with (5a), we see that (7 a) is the simpler.
The magnetic energy per unit length of rod, say I 7 , is
the summation extending over the section. But 2 FH = 0, because F is
polar and H is closed ; so that
T= 2 /zh 2 /87r - 2 /xF 2 /87r = 2 /xh 2 /87r + 2 /xhF/87r ............. (9a)
Or, in Cartesian coordinates, let H^ and H 2 be the x and y com-
ponents of the magnetic force H, z being parallel to the current ; then
express (la), and (Sa) is represented by
the latter form expressing
It will be observed that the mathematical conditions are identical
with those existing in St. Tenant's torsion problems. Thus, if a and ft
are the y and x tangential strain-components in the plane x, y in a
twisted prism, and y the longitudinal displacement along z, parallel to
the length of the prism, we have
where T is the twist (Thomson and Tait, Part II., 706, equation (9) ).
The corresponding forces are n times as great, if n is the rigidity (loc. cit.
equation (10) ) ; so that the energy per unit length is
fc 2 ) over section ...................... (13a)
Also, to find y, we have
(loc. cit. equations (12) and (18)). Comparing (14a) with (5a), (1
with (10a), and (13a) with the first of (lift), we see that there is
perfect correspondence, except, of course, as regards the constan
concerned. The lines of tangential stress in the torsion-problem and
the lines of magnetic force in our problem are identical, and the energy
is similarly reckoned. We may therefore make use of all St. Venant'
results.
ON THE SELF-INDUCTION OF WIRES. PART V. 243
It will be sufficient here to point out that the ratio of the inductance
of wires of different sections is the same as the ratio of their torsional
rigidities. Thus, as L \^ in the case of a round wire, that of a wire
of elliptical section, semiaxes a and b, is L = {jab/ (a 2 + b 2 ) ; when the
section is a square, it is -4417/x; when it is an equilateral triangle,
3627/x, etc. [Remember the limitation of close-fitting return, above
mentioned.] That of a rectangle will be given later in the course of
the following subsidence-solution.
Subsidence of Initially Uniform Current in a Rod of Rectangular Section,
with close-fitting Return-Current.
Consider the subsidence from the initial state of steady flow to zero,
when the impressed force that supported the current is removed, in a
prism of rectangular section. Let 2a and 26 be its sides, parallel to x
and y respectively, the origin being taken at the centre. Let H-^ and
H 2 be the x and y components of the magnetic force at the time /. Let
E be the intensity of the magnetic-force vector E, which is parallel to
z; then the two equations of induction ( (6), (7), Part L), or
curl H = 47rF, - curl E = /^H,
are reduced to
(15a)
_
dx dy
if F is the current-density, lc the conductivity, //. the inductivity. (I
speak of the intensity of a "force" and of the "density" of a flux,
believing a distinction desirable.) The equation of F is therefore
of which an elementary solution is
F = cosm cosny **, ........................ (I8a)
if 4:7r^Jcp= -(m 2 + rc 2 ) ......................... (190)
At the boundary we have, during the subsidence, E = 0, or F = 0;
therefore
cos mx cos ny = Q at the boundary,
or cos ma = 0, cosnb = Q, ........... ......... (20a)
or ma = \TT, f TT, |TT, etc. ; nb = ditto. The general solution is therefore
the double summation over m and n,
F = 22 A cos mx cos ny t pt ,
if we find A to make the right member represent the initial state.
This has to be F = F , a constant. Now
1 = 2 (2 /ma) sin ma cos mx, from x= -a to + #,
1 = 2(2/7i&) siunb cosny, from y= -b to +6.
244 ELECTRICAL PAPERS
Hence the required solution is
.
ab ~ m *-J n
or T = ir o V V gj n ^ 8in ^ cos mx cos ny 4*. . ..(21a)
ab <^^ mn
From this derive the magnetic force by (15a). Thus
m
- sin Wft sin mx cos *y
n y
The total current in the prism, say C, is given by
4*7= 2* ff/^ rt - 2
.^ .
by line-integration round the boundary. Or
4
if CQ = 4^&r o , the initial current in the prism.
Since the current is longitudinal, and there is no potential-difference,
the vector-potential is given by E = - A ; or, A being the tensor of A,
A is got by dividing the general term in the F-solution (21a) by -pk;
giving
A 167ru^-v>:-\sinmasm nb nf
A = J~y V 7 -^ r-cosTwacoswye^ .......... (24a)
^^ 2
Since the magnetic energy is to be got by summing up the product
F over the section, we find, by integrating the square of F, that the
amount per unit length is
2 **
'
By the square-of-the-force method the same result is reached, of
course. We may also verify that Q + f=0 during the subsidence, Q
being the dissipativity per unit length of prism.
The steady inductance per unit length is the L in T=^LC^ which
(25a) becomes when t = ; this gives
(26a)
-y
-(ma) 2 1
The lines of magnetic current are also the lines of equal electric
current-density. That is, a line drawn in the plane x, y through the
points where F has the same value is a line of magnetic current. For,
if s be any line in the plane x, y,
= component of /xH perpendicular to s,
ON THE SELF-INDUCTION OF WIRES. PART V. 245
so that H is parallel to 8, when dE/ds = 0. The transfer of energy is,
as usual, perpendicular to the lines of magnetic force and electric force.
The above expression (26a) for L may be summed up either with
respect to ma or to nb, but not to both, by any way I know. Thus,
writing it
r(nb) 2 + -(ma) 2
we may effect the second summation, with respect to nb, regarding ma
as constant in every term. Use the identity
l-x_ P-* - 6-** 1 -*' _ 2^ cos(tmg/2Q
""
where i has the values 1, 3, 5, etc. Take x = Q, iirl'2l = nb, h = (b/a)(ma),
1 = 1, and apply to (27a), giving
...(28a)
where the quantity in the {} is the value of the second 2 in (27a).
The first part of (28a) is again easily summed up, and the result is
in which summation, we may repeat, ma has the values JTT, |TT, |TT, etc.
The quantities a and b may be exchanged ; that is, a/b changed to b/a,
without altering the value of L. This follows by effecting the ma
summation in (2Qa) instead of the nb, as was done.
When the rod is made a flat sheet, or a/b is very small, we have
L = ^7Tfj,(a/b).
Compare (29ft) with Thomson and Tait's equation (46) 707, Part
II. Turn the nab 2 outside the [ ] to nab B , and multiply the 2 by 2.
These corrections have been pointed out by Ayrton and Perry. When
made, the result is in agreement with the above (29a), allowing, of
course, for changed multiplier. (I also observe that the - T in their
equation (44) should be +T, and the +T in (45), (the second T) should
be -T.) Such little errors will find their way into mathematical
treatises ; there is nothing astonishing in that ; but a certain collateral
circumstance renders the errors in their equation (46) worthy of being
long remembered. For the distinguished authors pointedly called
attention to the astonishing theorems in pure mathematics to be got by
the exchange of a and b, such as rarely fall to the lot of pure mathe-
maticians. They were miraculous.
Effect of a Periodic Impressed Force acting at one end of a Telegraph Circuit
with any Terminal Conditions. The General Solution.
I now pass to a different problem, viz., the solution in the case of a
periodic impressed force situated at one end of a homogeneous line,
246 ELECTRICAL PAPERS.
when subjected to any terminal conditions of the kind arising from the
attachment of apparatus. The conditions that obtain in practice are
very various, but valuable information may be arrived at from the
study of the comparatively simple problem of a periodic impressed
force, of which the full solution may always be found. In Part II. I
gave the fully developed solution when the line has the three electrical
constants E, L, and S (resistance, inductance, and electric capacity), of
which the first two may be functions of the frequency, but without any
allowance for the effect of terminal apparatus. If we take L = we
get the submarine-cable formula of Sir W. Thomson's theorj 7 ; but,
although the effect of L on the amplitude of the current at the distant
end becomes insignificant when the line is an Atlantic cable, its omis-
sion would in general give quite misleading results.
There are some & priori reasons against formulating the effect of the
terminal apparatus. They complicate the formulas considerably in the
first place ; next, they are various in arrangement, so that it might
seem impracticable to formulate generally ; and, again, in the case of a
very long submarine cable, we may divide the expression of the current-
amplitude into factors, one for the line and two more for the terminal
apparatus, of which the first, for the line, is always the same, whilst
the apparatus-factors vary, and are less important than the line-factor.
But in other cases the terminal apparatus may be of far greater import-
ance than the line, in their influence on the current-amplitude, whilst
the resolution into independent factors is no longer possible.
The only serious attempt to formulate the effect of the terminal
apparatus with which I am acquainted is that of the late Mr. C. Hockin
(Journal S. T. E. and E., vol. v. p. 432). His apparatus arrangement
resembled that usually occurring then in connection with long sub-
marine cables, including, of course, many derived simpler arrange-
ments ; and from his results much interesting information is obtainable.
But the results are only applicable to long submarine cables, on account
of the omission of the influence of the self-induction of the line. The
work must, therefore, be done again in a more general manner. It is,
besides, independently of this, not easy to adapt his formulae, in so far
as they show the influence of terminal apparatus, to cases that cannot
be derived from his. For instance, the effect of magnetic induction in
the terminal arrangements was omitted. I have therefore thought it
worth while to take a far more general case as regards the line, and at
the same time have endeavoured to put it in such a form that it can be
readily reduced to simpler cases, whilst at the same time the results
apply to any terminal arrangements we choose to use.
The general statement of the problem is this. A homogeneous line,
of length I, whose steady resistance is R, inductance L, electric capacity
S, and conductance of insulator K, all per unit length of line, is acted
upon by an impressed force F Q sin nt at one end, or in the wire attached
to it ; whilst any terminal arrangements exist. Find the effect pro-
duced; in particular, the amplitude of the current at the end remote
from the impressed force. If the line consists of two parallel wires, R
must be the sum of their resistances per unit length.
ON THE SELF-INDUCTION OF WIRES. PART V. 247
Let C be the current in the line and V the potential-difference at
distance z from the end where the impressed force is situated. Then
are our fundamental line-equations. Here R" = B + L(d/dt) to a first
approximation, and =R' + L'(d/dt) in the periodic case, where R f and
U are what R and L become at the given frequency. Let the terminal
conditions be
V=Z at z = l end,\ ,
-F sin^+F=^ <7 at * = end,/
so that P"= ^ (7 would be the z = terminal condition if there were no
impressed force.
The solution is a special case of the second of (1626), Part IV., which
we may quote. In it take
S" = K+Sp, ll'' = R' + I/p t ................... (36)
meaning d/dt so far. Also put z 2 = 0, 2 = F sin nt, and
-m* = F* = (K+Sp)(B' + I/p), ................... (46)
ind put the equation referred to in the exponential form. Thus,
_ - T
"" " Z ) - -"(F/ff' - ZJ (F/S" + Z ) ^ SI
This is the differential equation of C in the line. Now in F^ S", Z ,
and Z lt let d 2 /dt' 2 = - n*. It is then reducible to
d (A'P f + B'Q>n*) + (A'Qf - B'
giving the amplitude and phase-difference anywhere ; and the ampli-
ide is
(76)
Here P r and Q f are functions of z, whilst A' and B f are constants.
Put
=
wh r :l- m
The values of P and Q are
P =
,
, ..... .................. ; ........... .................. M"
ssing the following properties, to be used later,
................ (106)
I
248 ELECTRICAL PAPERS.
The expressions of Mb R(, L' , L{ can only be stated when the terminal
conditions are fully given. Their structure will be considered later.
P and Q depend only upon the line.
Let
A=R f - Sn\R&{ + I%Ll) + K(RW - UJW\ ]
B = Un + Sn(R',R{ - L' Q L(n*) + Kn(R f JL{ -P R(L f ,\ I
The effect of making the substitutions (86) in (56) is to express C in
terms of the P, Q of (96) and the A, B, a, b of (116); thus :
'{-(... + ........ - ...... ) .............. +(.- ....... - ....... ) .............. }-'->]
........... -(5-6) ......... \... + (A + a) .......... + (A-a) ............
The dots indicate repetition of what is immediately above them. Here
we see the expressions for the four quantities A', B f , P f , Q f of (66),
which we require. (126) therefore fully serves to find the phase-differ-
ence, if required. I shall only develope the amplitude-expression (76).
It becomes, by (126),
+ 2 cos 2Q(l-
.......................... (136)
in terms of A, B, a, b of (116).
Derivation of the General Formula for the Amplitude of Current
at the End remote from the Impressed Force.
This referring to any point between 2 = and I, a very important
simplification occurs when we take z = l. It reduces the numerator to
It only remains to simplify the denominator as far as
ON THE SELF-INDUCTION OF WIRES. PART V. 249
possible, to show as explicitly as we can the effect of the terminal
apparatus, which is at present buried away in the functions of A, J5,
a, 6 occurring in (136).
First of all, we may show that the product of the coefficients of 2W
and ~' 2PI equals one-fourth the square of the amplitude of the circular
part in the denominator. This is an identity, independent of what
A, B, a, b are. (136) therefore takes the form
= 2 F (P 2 + g 2 )* -f \_G Pl + Ht-~ pl - 2(GH)l cos 2(Ql + 0)J. (146)
The following are the expansions of the quantities occurring in the
denominator of (136) :
Let
P = R f * + LV, / 2 = ^ 2 + ZV, l{ = R? + Lfn* (156)
Then
A 2 + & = P + (K* + &tf)I*I? + 2(R f R{ - L f L{n 2 )(KR f + L'Stf)
+ 2(R{L' Q + R' Q L{)n\KL' - R'S),
a 2 + 6 2 = (P 2 + Q 2 ) { (R' Q + Rff + (L' Q + Z()V } ,
Aa + 3b = (R> + R()(R'P + L'nQ) + (U, + L()n(UnP - R'Q) \...( 1 66)
+ (R' I? + R(I$)(KP + SnQ) + (LJ/ + L(I%)n(KQ - SnP),
Ab-aB= (R' + R()(R f Q - L f nP) + (ZJ + L{)n(R'P + L'nQ}
+ (R' I? + R{1%)(KQ - SnP) - (L' Q I? + L(I*)n(KP + SnQ). >
These may be used direct in the denominator of (146), which is the
same as that of (136). But G and H may be each resolved into the
product of two factors, each containing the apparatus-constants of one
end only. Noting therefore that the B in (146) is given by
whose numerator and denominator are given in (166) [the numerator
being (GH)* sin 20, and the denominator (GH)* cos 201 it will clearly be
of advantage to develop these factors. First observe that the expansion
of H is to be got from that of G, using (166), by merely turning P to
- P and Q to - Q. We have therefore merely to split up one of them,
say G. If we put R{ = 0, L{ = in G it becomes
/ 2 + (P 2 + Q- 2 )/ 2 + 2P(RIR' + LiL'n*) + 2 Q(L'nRf> - R'nLQ. ( 1 86)
If, on the other hand, we put .# = 0, Z = in G, it becomes the same
function of R{, L{ as (186) is of jR, L' Q . It is then suggested that G is
really the product of (186) into the similar function of R{, L(; when
the result is divided by I 2 . This may be verified by carrying out the
operation described. But I should mention that it is not immediately
evident, and requires some laborious transformations to establish it,
making use of the three equations (106). When done, the final result
is that (146) becomes
(196)
250 ELECTRICAL PAPERS.
wherein 6? and H contain only constants belonging to the apparatus
at z = Q, and 6^ and H l those belonging to z = l, besides the line-
constants. Only one of the four need be written ; thus
(206)
From this get H by changing the signs of P and Q. Then, to obtain
G 1 and H lt the corresponding functions for the z = I end, change R' Q to
E{ and L f Q to L{. These functions have the value unity when the line
is short-circuited at the ends, (Z = 0, Z l = 0). They may therefore be
referred to as the terminal functions. Their form is invariable. We
only require to find the R f and L', or the effective resistance and
inductance of the terminal arrangements, and insert in (206) and its
companions.
The Effective Resistance and Inductance of the Terminal Arrangements.
Thus, let the two conductors at the z = I end be joined through a
coil. Then R{ is its resistance, L( its inductance, the steady values,
and the accents may be dropped, except under very unusual circum-
stances, and 7j is its impedance at the given frequency, when on short-
circuit. But if the coil contain a core, especially if it be of iron,
neither R l nor .Lj can have the steady values, on account of the
induction of currents in the core. Their approximate values at a given
frequency may be experimentally determined by means of the Wheat-
stone Bridge. Of course R^ and L^ are really somewhat changed in a
similar manner by allowing any induction between the coil and external
conductors, the brass parts of a galvanometer, for instance ; L going
down and R going up, though this does not materially affect I.
If, instead of a coil, it be a condenser of capacity S 1 that is inserted
at z = I ; then, since
'
we have Z l =
Therefore take R{ = 0, and L{ = - (fy* 2 )" 1 -
The condenser behaves, so far as the current is concerned, as a coil of
no resistance and negative inductance, the latter decreasing as the
frequency is raised, and as the capacity is increased; tending to become
equivalent to a short-circuit, though this would require a great fre-
quency in general, as the gwsi-negative inductance is large. (Thus,
^=100, =10- 15 = one microfarad, make L(= - 10 11 . To make the
inductance of a coil be 10 11 it must contain a very large number of
turns of fine wire.) Thus, whilst the condenser stops slowly periodic
or steady currents, it tends to readily pass rapidly periodic currents, a
property which is very useful in telephony, as in V"an Rysselberghe's
system.
On the other hand, the coil passes the slowly periodic, and tends to
stop the rapidly periodic, a property which is also very useful in tele-
phony. A very extensive application of this principle occurs in the
system of telephonic intercommunication invented and carried out by
ON THE SELF-INDUCTION OF WIRES. PART V. 251
Mr. A. W. Heaviside, known as the Bridge System, from the telephones
at the various offices being connected up as bridges across from one to
the other of the two conductors which form the line. Whilst all
stations are in direct communication with one another, one important
desideratum, there is no overhearing, which is another. For all
stations except the two which are in correspondence at a certain time
have electromagnets of high inductance inserted in their bridges, which
electromagnets will not pass the rapid telephonic currents in appreci-
able strength, so that it is nearly as if the non-working bridges were
non-existent ; and, in consequence, a far greater length of buried wire
can be worked through than on the Sequence system, wherein the
various stations have their apparatus in sequence with the line ; whilst
at the same time (in the Bridge system) a balance is preserved against
inductive interferences. When the two stations have finished corre-
spondence, they insert their own electromagnets in their bridges. As
these electromagnets are used as call-instruments, responding to slowly
periodic currents, we have the direct intercommunication. Of course
there are various other details, but the above sufficiently describes the
principle.
As regards the property of the self-induction of a coil in stopping or
greatly decreasing the amplitude of rapidly periodic currents, or acting
as an insulation at the first moment of starting a current, its influence
was entirely overlooked by most writers on telegraphic technics before
1878, when I wrote on the subject [vol. I., p. 95J. A knowledge of the
important quantity (A >2 4- L 2 n 2 )$, which is now the common property of
all electrical schoolboys (especially by reason of the great impetus
given to the spread of a scientific knowledge of electromagnetism by
the commercial importance of the dynamo), was, before then, confined
to a few theorists.
If the coil R, L, and the condenser S l be in parallel, we have
TV-*- - f-ia^^f.
Tri-r. F THE "
U3STIV EH
or _= i~ -i^- ' tJ. - X /^ OF
C (\-LS l n 2 Y + (RS l nf '
which show the expressions of R{ and L{, the second being the co-
efficient of p, the first the rest.
Similarly in other simple cases. And, in general, from the detailed
nature of the combination inserted at the end of the line, write out the
connections between the current and potential-difference in each branch,
id eliminate the intermediates so as to arrive at F=Z 1 C t the differ-
ential equation of the combination, wherein Z l is a function of p or djdt.
~^ 2 = -n 2 , and it takes the form Z l = R{-\-L(p, wherein R{ and L{
functions of the electrical constants and of n\ and are the required
effective R[ and L{ of the combination, to be used in (206), or rather, in
its z = l equivalent 6r r
As regards the z = end, it is to be remarked that, owing to the
current being reckoned positive the same way at both ends, when we
252 ELECTRICAL PAPERS.
write V Z C as the terminal equation, it is - Z Q that corresponds to
Z r Thus - Z Q = RQ + L' p, where, in the simplest case, E' Q and L[ are
the resistance and inductance of a coil.
Special Details concerning the above. Quickening Effect of Leakage. The
Long-Cable Solution, with Magnetic Induction ignored.
So far sufficiently describing how to develope the effective resistance
and inductance expressions to be used in the terminal functions G and
H, we may now notice some other peculiarities in connection with the
solution (19&). First short-circuit the line at both ends, making the
terminal functions unity, and = 0. The solution then differs from
that given in Part II. , equation (82), in the presence of the quantity K,
the former Sn now becoming (K' 2 + S' 2 ri 2 )%, whilst P and Q differ from
the former P and Q of (78), Part II., by reason of K, whose evan-
escence makes them identical. If we compare the old with the new
P and ft we find that
U becomes U-KR'l&tf,\
E' becomes R' + KL'/S, J"
in passing from the old to the new. Then the function
E'* + L'W , ao (R'
-W~~
or is unaltered by the leakage. It follows that the equation (85),
Part II., is still true, with leakage, if we make the changes (216) just
mentioned in it, or put
instead of using the 1/ and h expressions of Part II.
At the particular frequency given by n 2 = KE'/L'S, we shall have
P = Q = (%)*(R'* + U*n*)l(K* + S* 2 )* = Q)*(R'S + KU}n, ... (23b)
making
... (246)
If we should regard the leakage as merely affecting the amplitude of
the current at the distant end of a line, we should be overlooking an
important thing, viz., its remarkable effect in accelerating changes in
the current, and thereby lessening the distortion that a group of signals
suffers in its transmission along the line. If there is only a sufficient
strength of current received for signalling purposes, the signals can be
far more distinct and rapid than with perfect insulation, as I have
pointed out and illustrated in previous papers. Thus the theoretical
desideratum for an Atlantic cable is not high, but low insulation the
lowest possible consistent with having enough current to work with.
Any practical difficulties in the way form a separate question.
Eegarding this quickening effect, or partial abolition of electrostatic
retardation, I have [vol. I., pp. 531 and 536] pushed it to its extreme
ON THE SELF-INDUCTION OF WIRES. PART V. 253
in the electromagnetic scheme of Maxwell. In a medium whose con-
ductivity varies in any manner from point to point, possessed of
dielectric capacity which varies in the same manner (so that their ratio,
or the electrostatic time-constant, is everywhere the same), but destitute
of magnetic inertia (^ = 0, no magnetic energy), I have shown that
electrostatic retardation is entirely done away with, except as regards
imaginable preexisting electrification, which subsides everywhere accord-
ing to the common time-constant, without true electric current, by the
discharge of every elementary condenser through its own resistance.
This being over, if any impressed force act, varying in any manner in
distribution and with the time, the corresponding current will every-
where have the steady distribution appropriate to the impressed force
at any moment, in spite of the electric displacement and energy ; and,
on removal of the impressed force, there will be instantaneous dis-
appearance of the current and the displacement. This seems impossible ;
but the same theory applies to combinations of shunted condensers,
arranged in a suitable manner, as described in the paper referred to.
Of course this extreme state of things is quite imaginary, as we
cannot really overlook the magnetic induction in such a case. If we
regard it as the limiting form of a real problem, in which inertia occurs,
to be afterwards made zero, we find that the instantaneous subsidence
of the electrostatic problem becomes [with reflecting barriers] an
oscillatory subsidence of infinite frequency but finite time-constant,
about the mean value zero ; which is mathematically equivalent to
instantaneous non-oscillatory subsidence.
The following will serve to show the relative importance of E, S, K,
and L in determining the amplitude of periodic currents at the distant
end of a long submarine cable, of fairly high insulation-resistance :
4 ohms per kilom. makes 72 = 40 4 ,
imicrof. S
100 megohms,, #=10- 22 .
Here, it should be remembered, K is the conductance of the insulator
per centim. The least possible value of L would be such that LS = v~ 2 ,
where v = 30 10 ; this would make L = only. But it is really much
greater, requiring to be multiplied by the dielectric constant of the
insulator in the first place, making L=2say. It is still further
increased by the wire, and considerably by the sheath and by the
extension of the magnetic field beyond the sheath, to an extent which
is very difficult to estimate, especially as it is a variable quantity ; but
it would seem never to become a very large number, as of course an
iron wire for the conductor is out of the question. But leaving it
unstated, we have, by (96), taking R f = B, L' = L,
Yl*
VJ
254 ELECTRICAL PAPERS.
Now 71/27T is the frequency, necessarily very low on an Atlantic cable.
We see then that the first L 2 n 2 is quite negligible in its effect upon P,
even when we allow L to increase greatly from the above L = 2. The
high insulation also makes the (BK-LSn 2 ) part negligible, making
approximately
P= = (1^.1 0- 8 ,
P being a little greater than Q, at least when L is small. Now this is
equivalent to taking L = Q, K=Q, when
P=e = (pS)i, (256)
reducing (196) to
C = 2F (*/ J B)J * {0
say
256 ELECTRICAL PAPERS.
if R{, L{ are the effective resistance and inductance, to be used in
making
,7-2 2 , o- , ,.
++ .......... (3
Variation of L^ alone makes G l a minimum when
T &-,n . R
and if we take K^ = (condenser non-leaky, and not shunted), we have
the value of G 1 given by (30&) again, independent of the condenser.
Similarly we can come round to the same G l = 2 again. These rela-
tions are singular enough, but it is difficult to give them more than a
very limited practical application to the question of making the mag-
netic force of the coil a maximum, although the (305) relation is not
subject to any indefiniteness.
PART VI.
General Remarks on the Christie considered as an Induction Balance.
Full-Sized and Reduced Copies.
The most important as well as most frequent application of Mr. S. H.
Christie's differential arrangement, known at various times under the
names of Wheatstone's parallelogram, lozenge, balance, bridge, quad-
rangle, and quadrilateral, is to balance the resistances of four conductors,
when supporting steady currents due to an impressed force in a fifth,
and this is done by observing the absence of steady current in a sixth.
But its use in other ways and for other purposes has not been neglected.
Thus, Maxwell described three ways of using the Christie to obtain
exact balances with transient currents (these will be mentioned later in
connection with other methods) ; Sir W. Thomson has used it for
balancing the capacities of condensers* ; and it has been used for other
purposes. But the most extensive additional use has been probably in
connection with duplex telegraphy ; and here, along with the Christie,
we may include the analogous differential-coil system of balancing, which
is in many respects a simplified form of the Christie.
On the revival of duplex telegraphy some fifteen years ago, it was
soon recognised that " the line " required to be balanced by a similar
line, or artificial line, not merely as regards its resistance, but also as
regards its electrostatic capacity approximately by a single condenser;
better by a series of smaller condensers separated by resistances ; and,
best of all, by a more continuous distribution of electrostatic capacity
along the artificial line. The effect of the unbalanced self-induction
was also observed. This general principle also became clearly recog-
nised, at least by some, that no matter how complex a line may be,
* Journal S. T. E. and E., vol. I., p. 394.
ON THE SELF-INDUCTION OF WIRES. PART VI. 257
considered as an electrostatic and magnetic arrangement, it could be per-
fectly balanced by means of a precisely similar independent arrange-
ment ; that, in fact, the complex condition of a perfect balance is
identity of the two lines throughout. The great comprehensiveness of
this principle, together with its extreme simplicity, furnish a strong
reason why it does not require formal demonstration. It is sufficient
to merely state the nature of the case to see, from the absence of all
reason to the contrary, that the principle is correct.
Thus, if ABjC and AB 2 C [see figure on p. 263] be two identically
similar independent lines (which of course includes similarity of
environment in the electrical sense in similar parts), joined in parallel,
having the A ends connected, and also the C ends, and we join A to C
by an external independent conductor in which is an impressed force e,
the two lines must, from their similarity, be equally influenced by it, so
that similar parts, as B x in one line and B 2 in the other, must be in the
same state at the same moment. In particular, their potentials must
always be equal, so that, if the points B T and B 2 be joined by another
conductor, there will be no current in it at any moment, so far as the
above-mentioned impressed force is concerned, however it vary. The
same applies when it is not mere variation of the impressed force e, but
of the resistance of the branch in which it is placed. And, more gener-
ally, Bj and B 2 will be always at the same potential as regards disturb-
ances originating in the independent electrical arrangement joining
A to C externally, however complex it may be.
There is, however, this point to be attended to, that might be over-
looked at first. Connecting the bridge-conductor from B x to B 2 must
not produce current in it from other causes than difference of potential ;
for instance, there should be, at least in general, no induction between
the bridge-wire and the lines, or some special relation will be required
to keep a balance. This case might perhaps be virtually included under
similarity of environment.
If we had sufficiently sensitive methods of observation, the statement
that one line must be an exact copy of the other would sometimes have
to be taken literally. But the word copy may practically be often used
to mean copy only as regards certain properties, either owing to the
balance being independent of other properties, or owing to our inability
to recognise the effects of differences in other properties. Thus, in the
steady resistance-balance we only require AB X and AB 2 to have equal
total resistances, and likewise BjC and B 2 C ; resistances in sequence
being additive. But evidently, if the balance is to be kept whilst B x
and B 2 are shifted together from end to end of the two lines, the resist-
ance must be similarly distributed along them.
If, now, condensers be attached to the lines, imitating a submarine
cable, though of discontinuous capacity, we require that the resistance
of corresponding sections shall be equal, as well as the capacities of
corresponding condensers, in order that we shall have balance in the
variable period as well as in the steady state ; and the two properties,
resistance and capacity, are the elements involved in making one line a
copy of the other.
H.E.P. VOL. n. R
258 ELECTRICAL PAPERS.
In case of magnetic induction again, if AB X C and AB 2 C each consist
of a number of coils in sequence, they will balance if the" coils are alike,
each for each, in the two lines, and are similarly placed with respect to
one another. But the lines will easily balance under simpler conditions,
inductances being additive, like resistances ; and it is only necessary
that the total self-inductions of AB X and AB 9 (including mutual induc-
tion of their parts) be equal, and likewise of B X C and B 2 C. Again, if a
coil a l in the branch AB X have another coil ^ in its neighbourhood (not
in either line, but independent), and a 2 , in the branch AB 2 , be a copy of
!, we can complete the balance by placing a coil 6 2 (which is a copy of
&j) in the neighbourhood of the coil 2 , so that the action between a l
and &j is the same as that between 2 and b 2 . But it is not necessary
for 'b l and 6 2 to be copies of one another except in the two particulars
of resistance and inductance ; whilst as regards their positions with
respect to a-^ and # 2 , we only require the mutual inductance of a l and 6 X
to equal that of a 2 and b y
On the other hand, if 6 X be not a coil of fine wire, but a piece of
metal that is placed near the coil 15 many more specifications are
required to make a copy of it. The piece of metal is not a linear
conductor ; and, although no doubt only a small number (instead of an
infinite number) of degrees of freedom allowed for, would be sufficient
to make a practical balance, yet, as we have not the means of simply
analyzing pieces of metal (like coils) into a few distinct elements, we
must generally make a copy of 6 X by means of a similar piece, 5 2 , of the
same metal, and place it with respect to a 2 as \ is to a v to secure a
good balance. But very near balances may be sometimes obtained by
using quite dissimilar pieces of metal, dissimilarly placed.
So far, copy signifies equality in certain properties. But one line
need be merely a reduced copy of the other. It is only when we
inquire into what makes one line a reduced copy of another, that we
require to examine fully the mathematical conditions of the case in
question. In the state of steady flow the matter is simple enough. If
AB X has n times the resistance of AB 2 , then must BjC have n times the
resistance of B 2 C to keep the potentials of B x and B 2 equal. If con-
densers be connected to the lines, as before mentioned, we require,
first, the resistance-balance of the last sentence applied to every section
between a pair of condensers; and next, that the capacity of a condenser
in the line AB X C shall be, not n times (as patented by Mr. Muirhead, I
believe), but l/n of the capacity of the corresponding condenser in the
line AB 2 C [vol. I., p. 25]. If the lines are representable by resistance,
inductance, electrostatic capacity, and leakage-conductance (E, L, S, K of
Parts IV. and V., per unit length), one line will be a reduced copy of the
other if, when R and L in the first line are n times those in the second,
S and K in the second are n times those in the first, in similar parts.
Conjugacy of Two Conductors in a Connected System. The Characteristic
Function and its Properties.
After these general remarks, and preliminary to a closer consideration
of the Christie, let us briefly consider the general theory of the conjugacy
ON THE SELF-INDUCTION OF WIRES. PART VI. 259
of a pair of conductors in a connected system, when an impressed force
in either can cause no current in the other, either transient or per-
manent. The direct way is to seek the full differential equation of the
current in either, when under the influence of impressed force in the
other alone. Let V=ZG be the differential equation of any one branch,
C being the current in it, V the fall of potential in the direction of (7,
and Z the differential operator concerned, according to the notation of
Parts III., IV., and V. If there be impressed force e in the branch, it
becomes e+ V=ZG. We have 2 F~=0 in any circuit, by the potential-
property ; therefore 2e = 2ZC in any circuit. Also the currents are
connected by conditions of continuity at the junctions. These, together
with the former circuit-equations, lead us to a set of equations :
C v C' 2 , ..., being the currents, and e lt e. 2 , ... the impressed forces in
branches 1, 2, etc. ; F being common to all, and it and the /'s being
differential operators. We arrive at similar equations when the
differential equation of a branch is not merely between the V and C of
that branch, but between those of many branches ; for instance, when
is the form of the differential equation of branch 1.
Now let there be impressed force e in one branch only, and C be the
current in a second, dropping the numbers as no longer necessary. We
then have
FC=fe ................................... (3c)
Conjugacy is therefore secured by fe = 0, making C independent of
e. Therefore fe is the complex condition of conjugacy. If, for
example,
fe = a e + a^e + a 2 e + . . . , ........................ (4c)
where the a' s are constants, functions of the electrical constants con-
cerned, then, to ensure conjugacy, we require
ff = 0, i = 0, ^2 = 0, etc., ............... (5c)
separately ; and if these a's cannot all vanish together we cannot have
conjugacy.
What C may be then depends only upon the initial state of the
system in subsiding, or upon other impressed forces that we have nothing
to do with. As depending upon the initial state, the solution is
C^Ac**; ................................ (6c)
the summation being with respect to the p's which are the roots of
F(j>) = 0, p being put for d/dt in F ; and the A belonging to a certain p
is to be obtained by the conjugate property of the equality of the
mutual electric to the mutual magnetic energy of the normal systems of
any pair of p's.
As depending upon , the impressed force in the conductor which is
260 ELECTRICAL PAPERS.
to be conjugate to the one in which the current is (7, let e be zero before
time = 0, and constant after. Then, by (3c),
...(7e)
if C is the final steady current, and F / = dF/dp, the summation being
with respect to the p's.*
If there is a resistance-balance, = 0, C' = 0, and
Now, subject to (4c), calculate the integral transient current :
= value of f(p)e/pF(p) when ^> = 0,
if jF is the p = value of F. If then 04 = also, we prove that the
integral transient current is zero.
Supposing both a Q = 0, 0^ = 0, then
therefore
andtherefore
When
As regards Maxwell's previous formula (22), art. 678, however, there is
disagreement still.
References to authors who have written on the subject of induction
of currents in cores other than, and unknown to, and less comprehen-
sively than, myself, are contained in Lord Rayleigh's recent paper.* So
far as the effect on an induction-balance is concerned, when oscillatory
currents are employed, it is to be found, as he remarks, by calculating
the reaction of the core on the coil-current. This I have fully done in
my article on the subject. Another method is to calculate the heat in
the core, to obtain the increased resistance. This I have also done.
When the diffusion-effect is small, its influence on the amplitude and
* Phil. Mag., December, 1886.
278 ELECTRICAL PAPERS.
phase of the coil-current is the same as if the resistance of the coil-
circuit were increased from the steady value R to [vol. I., p. 369]
= E + 2/7r(7r^VcV) 2 = R + BI say.
" Many phenomena which may be experimentally observed when rods
are inserted in coils may be usefully explained in this manner." Here
H and k are the inductivity and conductivity of the core, of length I,
the same as that of the coil, n/2ir the frequency, c the core's radius,
and N the number of turns of wire in the coil per unit length ; whilst
is that part of the steady inductance of the coil-circuit which is con-
tributed by the core.
The full expression for the increased resistance due to the dissipation
of energy in the core is to be got by multiplying the above R l by Y t
which is given by [vol. I., p. 364]
_ _
2.6.8* V 3.10. 4.14.
where y = (lirpknc*)*. The value of R' is therefore R + R^. The
series being convergent, the formula is generally applicable. The law
of the coefficients is obvious. I have slightly changed the arrangement
of the figures in the original to show it. We may easily make the
core-heat a large multiple of the coil-heat, especially in the case of iron,
in which the induced currents are so strong. When y is small enough,
we may use the series obtained by division of the numerator by the
denominator in (49c), which is
16.24 15. 16 3 . 9
Corresponding to this, I find from my investigation [vol. I., p. 370]
of the phase-difference, that the decrease of the effective inductance
from the steady value is expressed by
y / 19?/ , 229w 2 , \
-
When the same core is used as a wire with current longitudinal, and
again as core in a solenoid with induction longitudinal, the effects are
thus connected. Let L l be the above steady inductance of the coil so
far as is due to the core, and L{ its value at frequency ?i/27r, when it
also adds resistance R{ to the coil. Also let E 2 be the steady resistance
of the same when used as a wire, and R( and Li its resistance and
inductance at frequency w/2w, the latter being what ^ then becomes.
Then
J TITOT.1 /7 T"fc T T~>/ T t . T\l T t "\
(52o)
ON THE SELF-INDUCTION OF WIRES. PART VI. 279
I did not give any separate development of the L( of the core, cor-
responding to (48c) and (49c) above for B f , but merged it in the ex-
pression for the tangent of the difference in phase between the impressed
force and the current in the coil-circuit. The full development of L{ is
same denominator as in (49c)
The high-frequency formulae for E{ and L{ are
(2*)*'
if y = IQz 2 . When z is as large as 10, this gives
#( = ^=2234 L&,
whereas the correct values by the complete formulae are
#{ = 198^, {=-225 L r
It is therefore clear that we may advantageously use the high-
frequency formulae when z is over 10, which is easily reached with iron
cores at moderate frequencies.
The corresponding fully developed formulae for R( and L f 2 , when the
current is longitudinal, are
_ __
6.16V 23.10.16 3*. 14. 16
_
2.6.16V 3. 2 2 . 10.16V 4.3 2 .14.16
showing the laws of formation of the terms, and
I4 = + 2 2 .6.16V 1 + 2.3 2 .10.16V 1+ 3.4 2 .'l4.r6V
5" ............................................................... '
the denominator being as in the preceding formula. At z =10, or
y=1600, these give
whereas Lord Rayleigh's high-frequency formulae, which are
^ = 2-234^2, 5 = J/*x -447.
is particular frequency makes the amplitude of the magnetic force in
e case of the core, and of the electric current in the other case,
fourteen times as great at the boundary as at the axis of the wire or
core (see Part I.). As, however, we do not ordinarily have very thick
wires for use with the current longitudinal, the high-frequency formulae
are not so generally applicable as in the case of cores, which may be as
280 ELECTRICAL PAPERS.
thick as we please, whilst by also increasing the number of windings
the core-heating per unit amplitude of coil-current may be greatly
increased.
If the core is hollow, of inner radius c , else the same, the equation
of the coil-current is, if e be the impressed force and G the current in
the coil-circuit whose complete steady resistance and inductance are R
and L, whilst L^ is the part of L due to the core and contained hollow
(dielectric current in it ignored),
........... (53c)
sc J (sc)-qK (sc)
when q depends upon the inner radius, being given by
(whose value is zero when the core is solid), and
There may be a tubular space between the core and coil, and E, L may
include the whole circuit. In reference to this equation (53c), how-
ever, it is to be remarked that there is considerable labour involved in
working it out to obtain what may be termed practical formulae,
admitting of immediate numerical calculation. The same applies to a
considerable number of unpublished investigations concerning coils and
cores that I made, including the effects of dielectric displacement ; the
analysis is all very well, and is interesting enough for educational pur-
poses, but the interpretations are so difficult in general that it is
questionable whether it is worth while publishing the investigations, or
even making them.
The Christie Balance of Resistance, Permittance, and Inductance.
Leaving now the question of cores and the balance of purely magnetic
self-induction, and returning to the general condition of a self-induction
balance, Z 1 Z 4t = Z 2 Z 3) equation (23e), let the four sides of the quadri-
lateral consist of coils shunted by condensers. Then R, L, and S
denoting the resistance, inductance, and capacity of a branch, we have
Z={Sp + (R + Ip)- 1 }- 1 ; ..................... (55c)
so that the conjugacy of branches 5 and 6 requires that
{S l
}, ............ (56c)
wherein the coefficient of every power of p must vanish, giving seven
conditions, of which two are identical by having a common factor. It
is unnecessary to write them out, as such a complex balance would be
useless ; but some simpler cases may be derived. Thus, if all the Z/s
ON THE SELF-INDUCTION OF WIRES. PART VI. 281
vanish, leaving condensers shunted by mere resistances, we have the
three conditions
I (57c)
which may be compared with the three self-induction conditions (25c)
to (27 c).
If we put ES=y, the time-constant, the second of (57c) may be
written
which corresponds to (26c). If $ 2 = = $ 4 , the single condition in
addition to the resistance-balance is i/ 1 = y y If S l = = /SL it is y 3 = y 4 .
Next, let each side consist of a condenser and coil in sequence.
Then the expression for Z is
Z=R + Lp + (Sp)-\ ......................... (59c)
which gives rise to five conditions,
~ L 1 S 1 ~
1111
+ = +-,
1
fO = L 4 -i s + Af 14 (l+JJ 4 /^) 1 , 94 .
' ' \o = 4 - 4 + jif 14 (i + 4/4) - jif /4 f '
(0=4-4-^(1+^) \ , 95 .
' ' |o = 4 - 4 - irji + 4/4) + jft/ij'
If we compare the two general conditions (83c), (84c), we shall see
that whenever
we may obtain the reduced forms of the conditions by adding together
the values of L% L 4 given by every one of the M' s concerned. We
may therefore bracket together certain sets of the M' s. To illustrate
this, suppose that M 13 and M 24 are existent together, and all the other
IT's are zero. Then (92c) and (93c) give, by addition,
which are the conditions required.
Similarly M 12 and Jf 34 may be bracketed. Also M 6V M 62 , M 63 , M^ t
and M 65 . Also M 51 , M^, M^, M 54 , and M 56 . But M u and M 23 will
not bracket.
Miscellaneous Arrangements. Effects of Mutual Induction between the
Branches.
As already observed, the self-induction balance (28c), (29c) is inde-
pendent of M 12 and 1T 34 , when these are the sole mutual inductances
concerned; that is, when R^R^, L^L^, R^ = R^ L 3 = L. By (92c)
and (93c) we see that independence of M 13 and M 24 is secured by
making all four branches 1, 2, 3, 4 equal in resistance and inductance.
But it is unsafe to draw conclusions relating to independence when
several coils mutually influence, from the conditions securing balance
when only two of the coils at a time influence one another. Let us
examine what (83c) and (84c) reduce to when there is induction between
ON THE SELF-INDUCTION OF WIRES. PART VI. 287
all the four branches 1, 2, 3, 4, but none between 5 and the rest or
between 6 and the rest. Put all M' s = which have either 5 or 6 in
their double suffixes, and put L = L y Then we may write the con-
ditions thus :
-M^ .................. (96c)
= (L, + L,)(M U - M 2 ,) + (L, - L,)(M 2 , - M 13 ) + M% 3 - M\,
+ (M 2 , - Jf 13 )( J/34 - M 12 ) + (M u - M 23 )(M 2 , + M IB - M 12 - M,,}. (97c)
The simplest way of satisfying these is by making
M U = M 23 and M 24 = M 13 ................. (98c)
If these equalities be satisfied, we have independence of M 12 and M M .
Now, if we make the four branches 1, 2, 3, 4 equal in resistance and
inductance, so that in (96c) and (97c) we have R^ = R^ and L^ = L^ the
first reduces to
Q = M 14 -M 23 , ............................ (99c)
so that it is first of all absolutely necessary that M U = M 2 # if the
balance is to be preserved ; whilst, subject to this, the second condition
reduces to
= (l/ 24 -Jf 13 )(Jf 34 -Jlf 12 ), .................. (lOOc)
so that either M Z4 = M , 3 , or else M M = M 12 . Thus there are two ways
of preserving the balance when all four branches are equal, viz.,
Jf 14 = l/ 23 and M 24: = M 13 , independent of the values of M 12 and If 34 ;
and M u = 3/ 23 and Af 34 = M 12 , independent of the values of Jf 24 and M 13 .
The verification of these properties, (98c) and later, makes some very
pretty experiments, especially when the four branches consist, not
merely of one coil each, but of two or more. The meanings of some of
the simpler balances are easily reasoned out without mathematical
examination "of the theory ; but this is not the case when there is
simultaneous induction between many coils, and their resultant action
on the telephone-branch is required.
Returning to (96c) and (97c), the nearest approach we can possibly
make to independence of the self-induction balance of the values of all
the W s therein concerned, consistent with keeping wires 3 and 4 away
from one another for experimental purposes, is by winding the equal
wires 1 and 2 together. Then, whether they be joined up straight,
which makes M 13 = M 23 and M u = M<, identically, or reversed, making
M^ - M 23 and M u = - Jf 24 , we shall find that
MU-M*
is the necessary and sufficient condition of preservation of balance.
At first sight it looks as if M 3l and M 32 must cancel one another
when wires 1 and 2 are reversed. But although 1 and 2 cancel on 3,
yet 3 does not cancel on 1 and 2 as regards the telephone in 5. The
effects are added. On the other hand, when wires 1 and 2 are straight,
3 cancels on them as regards the telephone, but 1 and 2 add their-
effects on 3. Similar remarks apply to the action between 4 and the
equal wires 1 and 2 when straight or reversed ; hence the necessity of
the condition represented by the last equation.
288 ELECTRICAL PAPERS.
On the other hand, M 6l and M 62 cancel when 1 and 2 are straight,
and add their effects when they are reversed: whilst M 6l and M 52
cancel when 1 and 2 are reversed, and add their effects when they are
straight, results which are immediately evident. But wires 1 and 2
must be thoroughly well twisted, before being wound into a coil, if it is
desired to get rid of the influence of, say, M G1 and M 62 , when it is a coil
that operates in 6, and this coil is brought near to 1 and 2.
This leads me to remark that a simple way of proving that the
mutual induction between iron and copper (fine wires) is the same as
between copper and copper, which is immensely more sensitive than
the comparison of separate measurements of the induction in the two
cases, is to take two fine wires of equal length, one of iron, the other of
copper, twist them together carefully, wind into a coil, and connect up
with a telephone differentially. On exposure of the double coil to the
action of an external coil in which strong intermittent currents or
reversals are passing, there will be hardly the slightest sound in the
telephone, if the twisting be well done, with several twists in every
turn. But if it be not well done, there will be a residual sound,
which can be cancelled by allowing induction between the external or
primary coil and a turn of wire in the telephone-circuit. A rather
curious effect takes place when we exaggerate the differential action by
winding the wires into a coil without twists, in a certain short part of
its length. The now comparatively loud sound in the telephone may
be cancelled by inserting a nonconducting iron core in the secondary
coil, provided it be not pushed in too far, or go too near or into the
primary coil. This paradoxical result appears to arise from the secondary
coil being equivalent to two coils close together, so that insertion
of the iron core does not increase the mutual inductance of the primary
and secondary in the first place, but first decreases it to a minimum,
which may be zero, and later increases it, when the core is further
inserted. Reversing the secondary coil with respect to the primary
makes no difference. Of course insertion of the core into the primary
always increases the mutual inductance and multiplies the sound. The
fact that one of the wires in the secondary happens to be iron has
nothing to do with the effect.
Another way of getting unions of the two conditions of the induction-
balance is by having branches 1 and 3 equal, instead of 1 and 2. Thus,
if we take E 1 = E 3 , L^ = L^ R 2 = R in A^ and A 2 , (73c) and (74c), we
obtain fifteen sets of double conditions similar to those already given,
out of which just four (as before) unite the two conditions. Thus,
using M IB only, we have
L 2 =L (lOlc)
and the same if we use M 24: only, and the same when both M IB and M 24
are operative. That is, the self-induction balance is independent of M 13
and M 24 . This corresponds to (81c) and (82c).
The other two are M Z5 and M 45 . With M 2b we have
= L 2 -Lt-2M 25 (102e)
and with ^T 45 , = L 2 - Z 4 - 27lf 45 (103c)
ON THE SELF-INDUCTION OF WIRES. PART VII. 289
The remaining eleven double conditions corresponding to (85c) to
(95c) need not be written down.
Several special balances of a comparatively simple kind can be
obtained from the preceding by means of inductionless resistances,
double-wound coils whose self-induction is negligible under certain cir-
cumstances, allowing us to put the L's of one, two, or three of the four
branches 1, 2, 3, 4 equal to zero. We may then usefully remove the
ratio-of-equality restriction if required. This vanishing of the L of a
branch of course also makes the induction between it and any other
branch vanish.
For instance, let L l = L% = L 4 = ; then
= # 2 L 3 + 71/36(^ + ^2) (104c)
gives the induction-balance when M 3(] is used, subject to R^R^R^Ry
And
= R 2 L 3 -M Bb (R 2 + ^) (105c)
is the corresponding condition when M^ is used. But M 56 will not give
balance, except in the special case of S.H. currents, with a false resist-
ance-balance. The method (104c) is one of Maxwell's. His other two
have been already described.
In the general theory of reciprocity, it is a force at one place that
produces the same flux at a second as the same force at the second place
does at the first. That the reciprocity is between the force and the
flux, it is sometimes useful to remember in induction-balances. Thus
the above-mentioned second way of having a ratio of equality is merely
equivalent to exchanging the places of the force and the vanishing flux.
We must not, in making the exchange, transfer a coil that is operative.
For example, in the M 6i method (79c), there is induction between
branches 6 and 4; M 45 (equation (88c)), on the other hand, fails to give
balance. But if we exchange the branches 5 and 6, it is the battery
and telephone that have to be exchanged ; so that we now use M^
which gives silence, whilst M 6i will not.
I have also employed the differential telephone sometimes, having
had one made some five years ago. But it is not so adaptable as the
quadrilateral to various circumstances. I need say nothing as to its
theory, that having been, I understand, treated by Prof. Chrystal.
Using a pair of equal coils, it is very similar to that of the equal-ratio
quadrilateral.
PART VII.
Some Notes on Part VI. , (1). Condenser and Coil Balance.
After my statement [p. 260, vol. n.] of the general condition of con-
jugacy of a pair of conductors, and the interpretation of the set of
equations into which it breaks up, I stated that in cases where, by the
presence of inverse powers of p, there could not be any steady current
in either of the to-be conjugate conductors due to impressed voltage in
the other, a true resistance-balance was still wanted to ensure con-
H.E.P. VOL. ii. T
290 ELECTRICAL PAPERS.
jugacy when the currents vary. I am unable to maintain this hasty
generalisation. In the example I gave, equations (59c) to (61c), in
which each side of the quadrilateral consists of a condenser and a coil
in sequence, so that there can be no steady current in the bridge-wire,
it is true that the obvious simple way of getting conjugacy is to have a
true resistance-balance. The conditions may then be written
23 '
and either
?:? and ;>:j'. ore.se ^ and *-*>}
X B ~ X & y%y> X 2~ ^ y}
where R stands for the resistance and L for the inductance of a coil.
S for the permittance of the corresponding condenser, x for the coil
time-constant L/fi, and y for the condenser time-constant ES ; that is,
we require either vertical or else horizontal equality of time-constants,
electrostatic and magnetic, subject to certain exceptional peculiarities
similar to those mentioned in connection with the self-induction balance.
It is also the case that on first testing the power of evanescence of the
other factor on the right of equation (61c), it seemed to always require
negative values to be given to some of the necessarily positive quanti-
ties concerned. But a closer examination shows that this is not neces-
sary. As an example, choose
^ = 1, 2 = 2, R. = 3, tf^HVj
A = -\, 2 = 5, L 3 = ^ Z 4 = f, I ...... (3d)
S 1= =7, S 2 = 5, S 3 = f|, fli-trJ
It will be found that these values satisfy the whole of equations (61c),
and yet the resistance-balance is not established. No doubt simpler
illustrations can be found. We must therefore remove the requirement
of a resistance-balance when there can be no steady current, although
the condition of a resistance-balance, when fulfilled, leads to the simple
way of satisfying all the conditions.
(2). Similar Systems.
If V=Zfi be the characteristic equation of one system and F=Z 2 C
that of a second, V being the voltage and C the current at the terminals,
they are similar when
Z-JZ 2 = n, any numeric
Here Z is the symbol of the generalised resistance of a system between
its terminals, when it is, save for its terminal connexions, independent
of all other systems ; a condition which is necessary to allow of the
form V= ZC being the full expression of the relation between V and C,
Z being a function of constants and of p,p 2 , p B , etc., and p being d/dt.
To ensure the possession of the property (46?), we require first of all
that one system should have the same arrangement as the other, as a
coil for a coil, a condenser for a condenser, or equivalence (as, for
instance, by two condensers in sequence being equivalent to one) ; and,
ON THE SELF-INDUCTION OF WIRES. PART VII. 291
next, that every resistance and inductance in the first system be n
times the corresponding resistance and inductance in the second
system, and every permittance in the second system be n times the
corresponding one in the first.
Then, if the two systems be joined in parallel, and exposed to the
same external impressed voltage at the terminals, the potentials and
voltages will be equal in corresponding parts, whilst the current in any
part of the second system will be n times that in the corresponding
part of the first. Also the electric energy, the magnetic energy, the
dissipativity, and the energy-current in any part of the second system
are n times those in the corresponding part of the first.
The induction-balance got by joining together corresponding points
through a telephone is, of course, far more general than the Christie
balance, limited to four branches, each subject to V=ZG\ at the same
time, however, it is less general than the conditions which result when
the full differential equation is worked out.*
By the above, any number of similar systems may be joined in
parallel, having then equal voltages, and their currents in the ratio of
the conductances. They will behave as a single similar system, the
conductance of any part of which is the sum of the conductances of the
corresponding parts in the real systems; and similarly for the per-
mittances and for the reciprocals of the inductances. If, on the other
hand, they be put in sequence, the resultant Z is the sum of the separate
Z's, the current in all is the same, and the voltages are proportional to
the resistances.
When the systems are not independent the above simplicity is lost ;
and I have not formulated the necessary conditions of similarity in an
extended sense except in some simple cases, of which a very simple
one will occur later in connexion with another matter.
(3). The Christie Balance of Resistance, Self and Mutual Induction.
The three general conditions of this are given in equations (72c) to
(74c). If, now, we introduce the following abbreviations,
m 3 = S + I
m 6 = Z, 2 + Z 4 +L 6
m 13 = - L 5 + M 13 - M
the conditions mentioned reduce simply to
R\R = ^^
(mj + m l3 + m 16 )# 4 - 7%^ = (m sl
(m, + m ]3 )m 36 =
* This general property is, it will be seen, of great value in enabling us to avoid
useless and lengthy mathematical investigations. In another place [p. 115, vol.
II.], I have shown how to apply it to the at first sight impossible feat of balancing
iron against copper.
292
ELECTRICAL PAPERS.
The interpretation is, that as there are only three independent
currents in the Christie arrangement, there can be only six independent
inductances, viz., three self and three mutual ; and these maybe chosen
to be the above ra's, whose meanings are as follows. Let the three
circuits be AB^A, CB 2 B,C, and
AB 2 CA in the figure, so that the
currents in them are C v C B , and C 6 .
Then m p ra 3 , and m 6 are the self,
and m 13 , m
m 6l the mutual induct-
ances of the three circuits.
Now if the four sides of the
quadrilateral consist merely of short
pieces of wire, which are not bent
into nearly closed curves, it is clear
that (Qd) are the true conditions, to
which alone can definite meaning be
attached ; the inductance of a short
wire being an indefinite quantity,
depending upon the position of other wires. We may therefore start
ab initio with only these six inductances, and immediately deduce
[p. 107, vol. II.] the conditions (Qd), saving a great deal of preliminary
work. But, on coming to practical cases, in which the inductances do
admit of being definitely localised in and between the six branches of
the Christie, we have to expand the m's properly, using (5d) or as
much of them as may be wanted, and so obtain the various results in
Part VI. Therefore equations (6d) are only useful as a short registra-
tion of results, subject to (5d), and in the remarkably short way in
which they may be got; a method which is, of course, applicable to
an) 7 network, which can only have as many independent inductances
as there are independent circuits, plus the number of pairs of the same.
(4). Reduction of Coils in Parallel to a Single Coil.
In Part VI. [p. 267, vol. II.], in speaking of the inductometer, I
referred to the most useful property that a pair of equal coils in parallel
behave as one coil to external voltage, whatever be the amount of
mutual induction between them ; a property which, excepting in the
mention of mutual induction, I had pointed out in 1878 [p. Ill, vol. I.].
But, although there appears to be no other case in which this property
is true for any value of the mutual inductance, which is the property
wanted, yet, if a special value be given to it, any two coils in parallel
will be made equivalent to one.
The condition required is obviously that Z, the generalized resistance
of the two coils in parallel, should reduce to the form R + Lp. Equa-
tion (30c) gives Z\ to make the reduction possible, on dividing the
denominator into the numerator, the second remainder must vanish.
Performing this work, we find
LL-m 2 /^7 X
-- -'
z
- 2m J
ON THE SELF-INDUCTION OF WIRES. PART VII. 293
which shows the effective resistance and inductance of the coils in
parallel, r l and r 2 being their resistances, and l lt / 2 , m the inductances \
subject to
giving a special value to m, which, if it be possible, will allow the coils
to behave as one coil, so that, when put in one side of the Christie, the
self-induction balance can be made. This equation (Sd) is the expression
of the making of coils 1 and 2 similar, in the extended sense, being the
simple case to which I referred above. Let a unit current flow in the
circuit of the two coils. Then ^ - m and l 2 -m are the inductions
through them, and these must be proportional to the resistances,
making therefore the actual inductions through them always the same.
Similarly, if any number of coils be in parallel, exposed to the same
impressed voltage V, with the equations
we have, by solution,
if D be the determinant of the coefficients of the C"s in (9rf), and N rs the
coefficient of m n in D. So, if C = C l + C 2 + . . . be the total current, we
have
<7=r(2Jv~)/; therefore Z=D/(2N), .......... (lid)
where the summation includes all the JV's. To reduce Z to the single-
coil form, we require the satisfaction of a set of conditions whose num-
ber is one less than the number of coils.
The simplest way to obtain these conditions is to take advantage of
the fact that, if any number of coils in parallel behave as one, the
currents in them must at any moment be in the ratio of their conduct-
ances. Then, since by
V- T
F- r A=P( m 31 C l
are the equations of voltage, when we introduce
into them, we obtain the required conditions :
The induction through every coil at any moment is the same in amount;
also the voltage due to its variation, and the voltage supporting current,
and the impressed voltage.
294 ELECTRICAL PAPERS.
(5). Impressed Voltage in the Quadrilateral. General Property of a
Linear Network
In my remarks on [p. 271, vol. II.], relating to the behaviour of
batteries when put in the quadrilateral, I, for brevity in an already
long article, left out any reference to the theory. As is well known, in
the usual Christie arrangement (see figure, above) the steady current in
5, due to an impressed voltage in any one of 1, 2, 3, 4, is the same
whether 6 be open or closed, if a steady impressed voltage in 6 give no
current in 5. But the distribution of current is not the same in the
two cases ; so that, when we change from one to the other, the current
in 5 changes temporarily ; as may be seen in making Mance's test of
the resistance of a battery, or by simply measuring the resistance of the
battery in the same way as if it had no E.M.F., using another battery in
6, but taking the galvanometer-zero differently. We, in either case,
have not to observe the absence of a deflection ; or, which is similar,
the absence of any change in the deflection ; but the equivalence of two
deflections at different moments of time, between which the deflection
changes. Hence Mance's method is not a true mil method, unless it be
made one by having an induction-balance as well as one of resistance ;
in which case, if the battery behave as a mere coil or resistance, which
is sometimes nearly true, especially if the battery be fresh, we may
employ the telephone instead of the galvanometer.
The proof that the complete self-induction condition, Z 1 Z 4 = Z 2 Z 3 ,
where the Z's stand for the generalised resistances of the four sides of
the quadrilateral, when satisfied, makes the current in the bridge-wire
due to impressed force in, for example, side 1, the same whether branch
6 be opened or closed, without any transient disturbance, is, formally,
a mere reproduction of the proof in the problem relating to steady
currents. Thus, suppose
B
where ^ is a steady impressed force in side 1, and A and B the proper
functions of the resistances, in the case of the common Christie, but
without the special condition E^ = R 2 R 3 which makes a resistance-
balance. Then we know that if we introduce this condition into A
and B, the resistance R Q can be altogether eliminated from the quotient
A/B, making C 5 due to e l independent of J? 6 .
Now, in the extended problem, in which it is still possible to repre-
sent the equation of a branch by V=ZC, wherein Z is no longer a
resistance, we have merely to write Z for R in the expansion of A/B to
obtain the differential equation of (7 5 ; and consequently, on making
Z^ 4 = Z^Z B , we make A/B independent of Z G . Hence, the current in
the bridge-wire is independent of branch 6 altogether when the general
condition of an induction-balance is satisfied, making branches 5 and 6
conjugate.
But, as is known to all who have had occasion to work out problems
concerning the steady distribution of current in a network, there is a
great deal of labour involved, which, when it is the special state
ON THE SELF-INDUCTION OF WIRES. PART VII. 295
involved in a resistance-balance, is wholly unnecessary. This remark
applies with immensely greater force when the balance is to be a uni-
versal one, for transient as well as permanent currents ; so that the
proper course is either to assume the existence of the property required
at the beginning, and so avoid the reductions from the complex general
to the simple special state, or else to purposely arrange so that the
reductions shall be of the simplest character. Thus, to show that C 5 is
independent of branch 6, when there is an impressed voltage in (say)
side 1, making no assumptions concerning the nature of branch 6, we
may ask this question, Under what circumstances is C 5 independent of
C 6 ? And, to answer it, solve for C 5 in terms of e 1 and (7 C , and equate
the coefficient of C 6 to zero.
Thus, writing down the equations of voltage in the circuits
and BjCBgBj in the above figure, we have
e l = Z 1 C 1
/ , g ,v
when there is no mutual induction between different branches, but not
restricting Z to a particular form ; and now putting
<7 4 =
we obtain
which give
C -
(Z,
making C' 5 independent of C Q when the condition of conjugacy of
branches 5 and 6 is satisfied.
If there are impressed voltages in all four sides of the quadrilateral,
then (ISd) obviously becomes
c - (
which makes C 5 always zero if e^ = e^ e 3 = e^ and Z^Z Z^Zy As an
example, let e 2 = Q, 4 = 0; then, if there is conjugacy of 5 and 6, and
also
the impressed forces are also balanced. Putting, therefore, batteries in
sides 1 and 3, and letting them work an intermitter in branch 6, we
obtain a simultaneous balance of their resistances and voltages, and
know the ratio of the latter. If self-induction be negligible, we may
take Z as E, the resistance ; if not negligible, it must be separately
balanced.
But should there be mutual induction between different branches,
this working-out of problems relating to transient states by merely
turning R to Z partly fails. We may then proceed thus : As before,
write down the equations of voltage in the circuits AB 1 B 2 A and
296 ELECTRICAL PAPEES.
CB^C, using the six independent inductances of these and of the
circuit CAB 2 C. Thus,
b - R 2 C 2 +p(m 1 C 1
if there is an impressed voltage in side 1. As before, eliminate C 2 ,
and (7 4 by (16d), and we obtain
[Hi + E 2 +p(m l
which, by solution for C 5 , gives its differential equation at once in
terms of e l and C 6 . To be independent of (7 6 , we require
(^ 2 -^m 16 ){ 3 + 4 +Xm 3 +m 13 )} = (^-pm 36 ){E l + E 2 +p(m 1 +m. 13 )}, (23d)
which, expanded, gives us the three equations (6d) again, showing that
C 5 depends upon e 1 and the nature of sides 1, 2, 3, and 4, subject to
(23d), and of 5, but is independent of the nature of (7 6 altogether,
except in the fact that the mutual induction between branch 6 and
other parts of the system must be of the proper amounts to satisfy
(23d) or (6d).
The extension that is naturally suggested of this property to any
network whose branches may be complex, and not independent, is
briefly as follows. The equations of voltage of the branches will be of
the form
wherein the Z's are differentiation-operators.
Suppose branches ra and n are to be conjugate, so that a voltage in
m can cause no current in n. First exclude m's equation from (24d)
altogether, and, with it, Z mm . Then write down the equations of
voltage in all the independent circuits of the remaining branches, by
adding together equations (24d) in the proper order ; this excludes the
Ps, and leaves us equations between the e's and all the independent
(7's, but one fewer in number than them. Put the C m terms on the left
side, then we can solve for all the currents (except C m ) in terms of C m
and the e's. That the coefficient of C m in the C n solution shall vanish
is the condition of conjugacy, and when this happens, C n is not merely
independent of e m but also of Z mm , though not of Z ml , Z m ^ etc.
I have dwelt somewhat upon this property, and how to prove it for
transient states, because, although it is easy enough to understand how
the current in one of the conjugate branches, say n, is independent of
current arising from causes in the other conjugate branch, m, yet it is
far less easy to understand how, when m is varied in its nature, and
therefore wholly changes the distribution of current in all the branches
(except one of the conjugate ones) due to impressed forces in them, it
does not also change the current in the excepted branch n. Conscien-
tious learners always need to work out the full results in a problem
ON THE SELF-INDUCTION OF WIRES. PART VII. 297
relating to the steady-flow of current before they can completely satisfy
themselves that the property is true.
Note on Part III. Example of Treatment of Terminal Conditions.
Induction-Coil and Condenser.
One of the side-matters left over for separate examination when
giving the main investigation of Parts I. to IV. was the manner of
treatment of terminal conditions when normal solutions are in question,
especially with reference to the finding of the terms in the complete
solution arising from an arbitrary initial state which are due to the
terminal apparatus, concerning which I remarked in Part III. that the
matter was best studied in the concrete application. There is also the
question of finding the nature of the terminal arbitraries from the mere
form of the terminal equation, without knowledge of the nature of the
arrangement in detail, except what can be derived from the terminal
equation.
Let, for example, in the figure, the thick line to the right be the
beginning of the telegraph-line, and what is to the left of it the terminal
apparatus, consisting of an induction-coil and a shunted condenser.
The line is joined through the primary of the induction-coil, of resist-
ance R v to the condenser of permittance S QJ whose shunt has the con-
ductance /t0, and whose further side is connected to earth, as symbolised
by the arrow-head.* Let R% be the resistance of the secondary coil,
and L v L^ M the inductances, self and mutual, of the primary and the
secondary. At the distant end of the line, where z = l, we may have
r\AAA/\A
another arrangement of apparatus, also joined through to earth, though
this is not necessary. The line and the two terminal arrangements
form the complete system, supposed to be independent of all other
systems.
Now suppose there to be no impressed voltage in any part of the
system, so that its state at a given moment depends entirely upon its
initial state at the time of removal of the impressed voltage ; after
which, owing to the existence of resistance, it must subside to a state
of zero electric force and zero magnetic force everywhere (with some
* It is not altogether improbable that the arrangement shown in the figure,
with the receiving instrument placed in the secondary circuit, would be of advan-
tage. A preliminary examination of the form of the arrival-curve when this
arrangement is used for receiving at the end of a long cable, with K =0, yields a
favourable result. But the examination did not wholly include the influence of
the resistances on the form of the curve.
298 ELECTRICAL PAPERS.
exceptional cases in which there is ultimately electric force, though not
magnetic force), the manner of the subsidence to the final state
depending upon the connexions of the system. The course of events
at any place depends upon the initial state of every part, including the
terminal apparatus, which may be arbitrary, since any values may be
given to the electrical variables which serve to fully specify the amount
and distribution of the electric and magnetic energies.
Suppose that F", the transverse voltage, and U, the current in the
line, are sufficient to define its state, i.e. as electrical variables, when
the nature of the line is given, and that u and w are the normal
functions of V and C in a normal system of subsidence. Then, at time
t, we have
r=VAue*, C=2Awe pt , ..................... (le)
wherein the p'& are known from the connexions of the whole system ;
each normal system having its own p, and also a constant A to fix its
magnitude. The value of A is thus what depends upon the initial
state, and is to be found by an integration extending over every part
of the system. In one case, viz., when the initial state is what could
be set up finally by any distribution of steadily acting impressed force,
we do not need to perform this complex integration, since we may
obtain what we want by solving the inverse problem of the setting up
of the final state due to the impressed force, as done by one method in
Part III., and by another in Part IV. If also the initial state of the
apparatus be neutral, so that it is the state of the line only that
determines the subsequent state, we can pretty easily represent matters,
viz., by giving to A the value
wherein U and W are the initial V and G in the line, whose per-
mittance and inductance per unit length are S and L; so that the
numerator of A is the excess of the mutual electric over the mutual
magnetic energy of the initial and a normal state, whilst the
denominator A is twice the excess of the electric over the magnetic
energy of the normal state itself, which quantity may be either
expressed in the form of an integration extending over the whole
system, or, more simply, and without any of the labour this in-
volves, in the form of a differentiation with respect to p of the deter-
minantal equation. For instance, when we assume L = 0, and we make
the line-constants to be simply Pi and S, its resistance and permittance
per unit length (constants), as we may approximately do in the case of
a submarine cable that is worked sufficiently slowly to make the effects
of inertia insensible, in which case we have
-?-* --*
so that we may take
- cos
XI
ON THE SELF-INDUCTION OF WIRES. PART VII. 299
if m 2 RSp] then equation (2e) becomes
where the undefined terms F and Y l in the numerator depend upon
the terminal apparatus, and F in the denominator is defined by
which is the determinantal equation arising out of the terminal
conditions
F=Z Q C at 2 = 0, and V=Z at z = l ..... ....(7e)
(See equations (177) to (180), Part IV.) We have now to add on to
the numerator of A the terms corresponding to the initial state of the
terminal apparatus, when it is not neutral. As the process is the
same at both ends of the line, -we may confine ourselves to the 2 =
apparatus, according to the figure. First we require the form of Z Qt the
negative of the generalized resistance of the terminal apparatus. It
consists of three parts, one due to the condenser, a second to the
primary coil, and a third to the presence of the secondary ; thus,
-Z^(K Q + S () p)^ + (R l +L 1 p)-MY(R 2 + L 2 p)-\ ......... (Se)
showing the three parts in the order stated. Now as shown in Part
III., dZ Q /dp expresses twice the excess of the electric over the magnetic
energy in a normal system (when jp becomes a constant), per unit square-
of-current. Performing the differentiation, we have
o_ o r /O.A
dp' (ffo l
Here we may at once recognise that the first term represents twice the
electric energy of the condenser per unit square-of-current, that the
second term is the negative of twice the magnetic energy of the
unit primary current, and that the fourth is similarly the negative of
twice the magnetic energy of the secondary current per unit primary
current; whilst the third, which at first sight appears anomalous, is
the negative of twice the mutual magnetic energy of the unit primary
and corresponding secondary current. Thus, if w be the normal
current-function, that is, by (4e), W Q - (m/R) cos 6, we have
............... <">
as the expressions for the normal voltage of the condenser, for the
primary current, and for the secondary current. If then V Q , C v and
C 2 are the initial quite arbitrary values of the voltage of the condenser,
and of the primary and secondary currents, their expansions must be
C -^Aw - y-o (\\ e \
fci-M* 2-
300 ELECTRICAL PAPERS.
Also, the excess of the mutual electric over the mutual magnetic energy
of the initial state F , C v (7 2 , and the normal state represented by
(We) is
and this is what must be added to the numerator in (5e) to obtain the
complete value of A, if we also add the corresponding expression Y l
for the apparatus at the other end, if it be not initially neutral. Using
this value of A in (\e) and in (lie) with the time-factor e** attached,
and in the corresponding expansions for the other end, we thus express
the state of the whole system at any time.
Since, initially, V is U, and independent of the state of the terminal
apparatus, it follows that in the expansion
the parts of A depending on the apparatus contribute nothing to U, so
that, by (5e) and (12e), we have the identities
for all the values of z from to I.
It may have been observed in the above that the use of (9e) was
quite unnecessary, owing to the forms of the normal functions in (100)
being independently obtainable from our h-priori knowledge of the
terminal apparatus in detail, from which knowledge the form of Z Q in
(Se) was deduced ; so that, without using (9e), we could form (lie) and
(12e). I have, however, introduced (9e) in order to illustrate how we
can find the complete solution, without knowing the detailed terminal
connexions, from a given form of Z. We must either decompose
dZ /dp into the sum of squares of admissible functions of p, multiplied
by constants, say,
where a v a 2 , etc., are the constants, and /], / 2 , ... the functions of p- y
or else into the form of the sum of squares and products, thus
When this is done, we know that the terminal arbitraries are
F^VAfw, F 2 = ?Af 2 w F 3 = ?Af B w , ...(16*)
and that r o = w o{ fl i^i/i + ^2/2 + ^3/8+ } ( 17e )
in the case (I4e) of sums of squares, wherein the F's may have any
values, assuming that we have satisfied ourselves that they are all
independent ; with the identities
= 2X/>, = 2,>, etc (ISe)
Thus, in the case (9e), the first, second, and fourth terms are of the
proper form for reduction to (14e), but the third is not. We are
ON THE SELF-INDUCTION OF WIRES. PART VII. 301
certain, therefore, that there cannot be more than three arbitraries, if
there be so many. Now, if we do not recognise the connection between
the third term and those which precede and follow it (as may easily
happen in some other case), we should rearrange the terms to bring it
to the form (14e); for instance, thus :
, lg ,
which is what we require. We may then take
Further, we can certainly conclude, provided a x is positive, and a 2 and
fl 3 are negative, that the first term on the right of (19e) stands for
electric (or potential) energy, and the remainder for magnetic (or
kinetic). It is clear that we may assume any form of Z that we please
of an admissible kind (e.g., there must be no such thing as|>*), find the
arbitraries, and fully solve the problem that our data represent, whether
it be or be not capable of a real physical interpretation on electrical
principles. I have pursued this subject in some detail for the sake of
verifications ; it is an enormous and endless subject, admitting of in-
finite development. Owing, however, to the abstractly mathematical
nature of the investigations to say nothing of the length to which
they expand, although when carried on upon electrical principles they
are much simplified, and made to have meaning I merely propose to
give later one or two examples in which circular functions of p are
taken to represent Z.
Although, however, the state of the line at any moment is fully
determinable for any form of the terminal Z's, when they alone are
given, from the initial state of the line, provided the initial values of
the terminal arbitraries be taken to be zero, and although it is similarly
determinable when particular values are given to the arbitraries, whose
later values also are determinable by affixing the time-factor, it does
not appear that this determinateness of the later values of the terminal
arbitraries is always of a complete character, when the sole data relating
to them are the form of Z and their initial values. For it is possible
for a terminal arrangement to have a certain portion conjugate with
respect to the line ; and although the state of the line will not be
affected by initial energy in that portion, yet it will influence the later
values of the other terminal arbitraries. This might wholly escape
notice in an investigation founded upon a given form of Z with un-
detailed connections, owing to the disappearance from Z of terms
depending upon the conjugate portion. In such a case the reduced
form of Z cannot give us the least information concerning the influence
of the portion conjugate to the line. It is as if it were non-existent.
If, however, Z be made more general, so as to contain terms depending
upon the conjugate portion, although they be capable of immediate
elimination from Z, it would seem that the indeterminateness must be
removed.
302
ELECTRICAL PAPERS.
Some Notes on Part IV. Looped Metallic Circuits. Interferences due to
Inequalities, and consequent Limitations of Application.
It is scarcely necessary to remark that, in the investigation of Parts
I. and II. , the choice of a round wire or tube surrounded by a coaxial
tube for return-conductor was practically necessitated in order to allow
of the use of the well-known J and J L functions and their complements,
because it was not merely the total current in the wire with which we
were concerned, but also with its distribution. Next, in order that it
should be a question of self-induction, and not one of mutual induction
also, with fearful complications, it was necessary to impose the con-
dition that the wire, tubular dielectric, and outer tube should be a
self-contained system, making the magnetic force zero at the outer
boundary. It is true that no external inductive effect is observable
when the double-tube circuit is of moderate length. But electrostatic
induction is cumulative ; and it is certain that, by sufficiently lengthen-
ing the double tube, we should ultimately obtain observable inductive
interferences. Our investigation, then, only applies strictly when the
double tube is surrounded on all sides, to an infinite distance, by a
medium of infinite elastivity and resistivity.
(Maxwell termed 4?r/c, when c is the dielectric constant, the electric
elasticity. I make this the elastivity : first, to have one word for two ;
next, to avoid confusion with mechanical elasticity; and, thirdly, to
harmonise with the nomenclature I have used for some time past.
Thus :
Flux.
Conduction-Current
Induction . . .
Displacement . .
Resistivity.
Conductivity.
Inductivity.
Elastivity.
Permittivity.
Resistance.
Conductance.
Inductance.
Elastance.
Permittance.
Force.
Electric.
Magnetic.
Electric.
The elastance of a condenser is the reciprocal of its permittance, and
elastivity is the elastance of unit volume, as resistivity is the resistance
of unit volume, and conductivity the conductance of unit volume.
As for "permittivity" and "permittance," there are not wanting
reasons for their use instead of " specific inductive capacity " (electric),
and " electrostatic capacity." The word capacity alone is too general ;
it must be capacity for something, as electrostatic capacity. It is an
essential part of my scheme to always use single and unmistakable
words, because people will abbreviate. Again, capacity is an unadapt-
able word, and is altogether out of harmony with the rest of the
scheme. Now the flux concerned is the electric displacement, involving
elastic resistance to yielding from one point of view, and a capacity for
permitting the yielding from the inverse ; hence elastance and permit-
tance, the latter being the electrostatic capacity of a condenser. There
are now only two gaps left, viz. for the reciprocals of inductivity and
inductance. " Resistance to lines of force " and " magnetic resistance "
will obviously not do for permanent use.)
ON THE SELF-INDUCTION OF WIRES. PART VII. 303
If this restriction be removed, we have self- and mutual-induction
concerned, and interferences ; or, even if there be no external con-
ductors, we have still the electric current of elastic displacement, and
with it electric and magnetic energy outside the double tube. But,
ignoring these, we have the following striking peculiarities : Putting
on one side the question of the propagation of disturbances into the
conductors, which is so interesting a one in itself, we find that the
electrical constants are three in number the resistance, permittance,
and inductance of the double-tube per unit of its length ; whilst the
electrical variables are two the current in each conductor, and the
transverse voltage. The effective resistance per unit length is the sum
of their resistances, which may be divided between the two conductors
in any ratio ; the permittance is that of the dielectric between them ;
and the inductance is the sum of that of the dielectric, inner, and outer
conductors. Another remarkable peculiarity is, that equal impressed
forces, similarly directed in the two conductors at corresponding places,
can do nothing; from which it follows that the effective impressed
force may, like the effective resistance, be divided between the con-
ductors in any proportion we please.
In Part IV., having in view the rapidly extending use of metallic
circuits of double wires looped, excluding the earth, consequent upon
the development of telephonic communication in a manner to eliminate
inductive interferences, I extended the above-described method to a
looped circuit consisting of a pair of parallel wires. So far as propaga-
tion into the wires is concerned, it is merely necessary that they should
not be too close to one another, to allow of the application of the J Q and
/! functions to them separately. Now suspended wires are usually of
iron, and are not set too close, so that the application is justified. On
the other hand, buried twin wires, though very near one another, are
of copper, and also considerably smaller than the iron suspended wires;
so that the diffusion-effect, though not so well representable \>y the
above-named functions, is made insignificant. Dismissing, as before,
this question of inward propagation, we have, just as in the tubular
case, two electrical variables and three constants, viz. the transverse
voltage, the current in each wire, and the effective resistance, permit-
tance, and inductance.
First of all, let the wires be alone in an infinite dielectric. Then we
have similar results to those concerning the double-tube. The effective
resistance, which is the sum of the resistances of the wires, may be
divided between them in any proportions ; and so may be the effective
impressed voltage. The effective permittance is that of the condenser
consisting of the dielectric bounded by the two wires, the surface of one
being the positive, and that of the other the negative coating. Or, in
another form, the effective permittance is the reciprocal of the elastance
from one wire to the other. In the standard medium, this elastance is,
in electrostatic units, the same as the inductance of the dielectric in
electromagnetic units. Thus,
-, (If)
304 ELECTRICAL PAPERS.
if ?\ and r 2 be the radii of the wires, and r 12 their distance apart
(between axes), and ^ the inductivity of the dielectric. And
Their product, when in the same units, is v~ 2 , the reciprocal of the
square of the speed of undissipated waves through the dielectric. The
two variables, transverse voltage and current, fully define the state of
the wires, except as regards the diffusion-effect in them, of course, and
an effect due to outward propagation into the unbounded dielectric from
the seat of impressed force, which is made insignificant by the limitation
of the magnetic field (in sensible intensity) due to the nearness of the
wires as compared with their length. To LQ has to be added a variable
quantity, whose greatest value is J/^ + |/x 2 , if /^ and /x 2 are the inducti-
vities of the wires, to obtain the complete inductance per unit length.
So far, then, there is a perfect correspondence between the double-
tube and the double-wire problem. But when we proceed to make
allowance for the presence of neighbouring conductors, as, for instance,
the earth, although there is a formal resemblance between the results in
the two cases, when proper values are given to the constants concerned,
yet the fact that in one case the outer conductor encloses the inner,
whilst in the other this is not so, causes practical differences to exist.
For example, there are two constants of permittance concerned in the
coaxial tube case, that of the dielectric between them, and that of the
dielectric outside the outer tube. But in the case of looped wires there
are three, which may be chosen to be the permittance of each wire with
respect to earth including the other wire, and a coefficient of mutual
permittance. There are, similarly, three constants of inductance, and
two of resistance, and at least two of leakage, viz. from each wire to
earth, with a possible third direct from wire to wire. This is when the
wires are treated in a quite general manner, and arbitrarily operated
upon ; so that there must be four electrical variables, viz., two currents
and two potential-differences or voltages. I have somewhat developed
this matter in my paper " On Induction between Parallel Wires "
[p. 116, vol. I.]; and as regards the values of the constants of capacity
concerned, in my paper " On the Electrostatic Capacity of Suspended
Wires" [p. 42, vol. I.]. As may be expected, the solutions tend to
become very complex, except in certain simple cases. If, then, we can
abolish this complexity, and treat the double wire as if it were a single
one, having special electrical constants, we make a very important
improvement. I have at present to point out certain peculiarities
connected with the looped-wire problem in addition to those described
in Part IV., and to make the necessary limitations of application of the
method and the results which are required by the presence of the earth.
First of all, even though the wires be not connected to earth, if they
be charged and currented in the most arbitrary manner possible, we
must employ the four electrical variables and the ten or eleven electrical
constants as above mentioned. On the other hand, going back to the
looped wires far removed from other conductors, there are but two
ON THE SELF-INDUCTION OF WIRES. PART VII. 305
electrical variables and four constants (counting one for leakage). Now
bring these parallel wires to a distance above the earth which is a large
multiple of their distance apart. The constant S of permittance is a
little increased. The method of images gives
where r lt r. 2 are the radii of the wires, r 12 their distance apart, s v s 2 their
distances from their images, and s 12 the distance from either to the
image of the other; but, owing to s^/s^ being nearly unity, the per-
mittance S does not sensibly differ from the value in an infinite
dielectric, or the earth has scarcely anything to do with the matter.*
If, however, the wires be brought close to the earth, the increase of
permittance will become considerable ; this is also the case when the
wires are buried. The extreme is reached when each wire is surrounded
by dielectric to a certain distance, and the space between and surround-
ing the two dielectrics is wholly filled up with well-conducting matter.
Then the permittance S becomes the reciprocal of the sum of the
elastances of the two wires with respect to the enveloping conductive
matter; in another form, the effective elastance is the sum of the
elastances of the two dielectrics. Returning to the suspended wires, if
the earth were infinitely conducting, the effective inductance would be
the reciprocal of S in (3/) with //, written for c, in electromagnetic units,
with ^(/Xj + /x 2 ) added ; whilst, allowing for the full extension of the
magnetic field into the earth, we should have the formula (I/), giving a
slightly greater value. The effective resistance is of course the sum of
the resistances, and the effective leakage-resistance would be the sum of
the leakage-resistances of the two wires with respect to earth, if that were
the only way of getting leakage between the wires, but it must be
modified in its measure by leakage being mostly from wire to wire over
the insulators, arms, and only a part of the poles.
But if there be any inequalities between the wires, differential effects
will result, due to the presence of the earth, in spite of its little influence
on the value of the effective permittance ; whereby the current in one
wire is made not of the same strength as in the other, and the
charge on one wire not the negative of that on the other. The
propagation of signals from end to end of the looped-circuit will not
then take place exactly in the same manner as in a single wire. To
allow for this, we may either bring in the full, comprehensive system of
electrical constants and variables; or, perhaps better, exhibit the
differential effects separately by taking for variables the sum of the
* On the other hand, Mr. W. H. Preece, F.R.S., assures us that the capacity is
half that of either wire (Proc. Roy. Soc. March 3, 1887, and Journal S. T. E. and
E., Jan. 27 and Febr. 10, 1887). This is simple, but inaccurate. It is, however,
a mere trifle in comparison with Mr. Preece's other errors ; he does not fairly
appreciate the theory of the transmission of signals, even keeping to the quite
special case of a long and slowly worked submarine cable, whose theory, or what
he imagines it to be, he applies, in the most confident manner possible, universally.
There is hardly any resemblance between the manner of transmission of currents
of great frequency and slow signals. [See also p. 160, vol. n.]
H.K.P. VOL. n. u
306 ELECTRICAL PAPERS.
potentials of the wires (taking earth at zero potential) and half the
difference of the strength of current in them, in addition to the differ-
ence of potential of the wires and half the sum of the current-strengths,
which last are the sole variables when the wires are in an infinite
dielectric, or else are quite equal. By adopting the latter course our
solutions will consist of two parts, one expressing very nearly the same
results as if the differential effects did not exist, the other the differ-
ential effects by themselves.
Another result of inequalities is to produce inductive interferences
from parallel wires which would not exist were the wires equal. As
an example, let an iron and a parallel copper wire be looped, and tele-
phones be placed at the ends of the circuit. Even if the wires be well
twisted, there is current in the telephones caused by rapid reversals in
a parallel wire whose circuit is completed through the earth. Again, if
two precisely equal wires be twisted, and telephones placed at the ends
as before, the insertion of a resistance into either wire intermediately
will upset the induction-balance and cause current in the terminal tele-
phones when exposed to interference from a parallel wire. This inter-
ference can be removed by the insertion of an equal resistance in the
companion-wire at the same place. In the working of telephone
metallic circuits with intermediate stations and apparatus, we not only
introduce great impedance by the insertion of the intermediate apparatus,
thus greatly shortening the length of line that can be worked through,
but we produce inductive interferences from parallel wires, unless the
intermediate apparatus be double, one part being in circuit with one
wire, the other part (quite similar) in circuit with the other. In
mentioning my brother's system of bridge-working of telephones (in
Part V.), whereby the intermediate impedance is wholly removed, I
mentioned, without explanation, the cancelling of inductive interfer-
ences. The present and preceding paragraphs supply the needed
explanation of that remark. The intermediate apparatus, being in
bridges across from one wire to the other, do not in the least disturb
the induction-balance, so that transmission of speech is not interfered
with by foreign sounds.
But theory goes much further than the above in predicting inter-
ferences than practice up to the present time verifies. For instance, if
two perfectly equal wires be suspended at the same height above the
ground and be looped at the ends, terminal telephones will not be
interfered with by variations of current in a parallel wire equidistant
from both wires of the loop-circuit, having its own circuit completed
through the earth. But if the loop-circuit be in a vertical plane, so
that one wire is at a greater height above the ground than the other,
there must be terminal disturbance produced, even when the disturbing
wire is equidistant. Similarly in the many other cases of inequality
that can be mentioned.
The two matters, preservation of the induction-balance, and trans-
mission of signals in the same manner as on a single wire, are intimately
connected. If we have one, we also have the other. The limitations
of application of the method of Part IY. may be summed up in saying
ON THE SELF-INDUCTION OF WIRES. PART VIII. 307
that the loop-circuit must either be far removed from all conductors, in
which case equivalence of the wires is quite needless ; or else they must
be equal in their electrical constants. In the latter case the effective
resistance R is the double of that of either wire, and the effective
permittance, inductance, and leakage are to be measured as before
described, whilst the variables are the transverse voltage from wire to
wire and the current in each. But the four electrical constants may
vary in any (not too rapid) manner along the line. And the impressed
force (in the investigations of Part IV.) may also be an arbitrary func-
tion of the distance, provided it be put, half in one wire, half in the
other, oppositely directed in space. For, although equal, similarly
directed impressed forces will cause no terminal disturbance (and none
anywhere if other conductors be sufficiently distant), yet disturbances
at intermediate parts of the line will result. It is true that the most
practical case of impressed voltage is when it is situated at one end
only of the circuit, when it is of course equally in both wires, or not in
them at all ; but there is such a great gain in the theoretical treatment
of these problems by generalising, that it is worth while to point out
the above restriction.
Besides this case of equality of wires, which is precisely the one that
obtains in practice, there are other cases in which, by proper propor-
tioning of the electrical constants of the two looped wires, the induction-
balance is preserved ; and, simultaneously, we obtain transmission of
signals as on a single wire. [But this is not an invariable rule.] Their
investigation is a matter of scientific interest, though scarcely of prac-
tical importance.
I have yet to add investigations by* the method of waves (mentioned
in Part IV.), by which I have reached interesting results in a simple
manner.
PART VIII.
The Transmission of Electromagnetic Waves along Wires without Distortion.
One feature of solutions of physical problems by expansions in
infinite series of normal solutions is the very artificial nature of the
process. If it be a case of subsidence towards a state of equilibrium,
then, if a sufficient time has elapsed since the commencement of the
subsidence to allow the great mass of (singly) insignificant systems to
nearly vanish, leaving only two or three important systems, which may
be readily examined or merely one, the most important then the
process is natural enough. It is the early stage of the subsidence
that is so artificially represented, when the resultant of a very large
number of normal solutions must be found before we come to what we
want. Sometimes, too, the full investigation of the normal systems in
detail is prevented by mathematical difficulties connected with the
roots of transcendental equations. This goes very far to neutralise the
advantage presented by the ease with which solutions in terms of
normal functions may be obtained.
308 ELECTRICAL PAPERS.
In some respects these difficulties are evaded by the consideration of
the solution due to a sinusoidal impressed force. The method is very
powerful ; and, by considering the nature of the results through a
sufficiently wide range of frequencies, we may indirectly gain, with
comparatively little trouble, knowledge that is unattainable by the
method of normal systems.
But the real desideratum, which, if it can be reached, is of paramount
importance, is to get solutions which can be understood and appreciated
at first sight, and followed into detail with ease, presenting to us, as
nearly as possible, the effects as they really occur in the physical
problem, disconnected from the often unavoidable complications due to
the form of mathematical expression. To illustrate this, it is sufficient
to refer to the elementary theory of the transmission of waves without
dissipation along a stretched flexible cord. If we employ Fourier-series,
we are doing mathematical exercises. But only use the other method,
in which arbitrary disturbances are transferred bodily in either direction
at constant speed, e.g., u= ,, _ ^
and we get rid of the mathematical complications, and can interpret
results as we see their physical representatives in reality for instance,
when we agitate one end of a long cord.
Now there is one case, and, so far as I know at present, only one, in
the many-sided question of the transmission of electromagnetic disturb-
ances along wires, which admits of this simple and straightforward
method of treatment. Singularly enough, it is not by the simplifying
process of equating to zero certain constants, and so ignoring certain
effects, that we reach this unique state of things, but rather the other
way, generalising to some extent. It is usual to ignore the leakage of
conductors, sometimes also the inductance, and sometimes the per-
mittance. But we must take all the four properties into account which
are symbolised by resistance, leakage-conductance, inductance, and per-
mittance, to reach the much-desired result. Briefly stated, the effects
are these, roughly speaking. If there be only resistance and per-
mittance, there is, when disturbances of an irregular character are sent
along a long circuit, both very great attenuation and very great dis-
tortion produced. The distortion at the end of an Atlantic cable is
enormous. Now if we introduce leakage, we shall lessen the distortion
considerably, but at the same time increase the attenuation. On the
other hand, if we introduce inductance (instead of leakage) we shall
lessen the attenuation as well as the distortion. And, finally, if we
have both leakage and inductance, in addition to resistance and per-
mittance, we may so adjust matters, by the effects of inductance and of
leakage being opposite as regards distortion, as to annihilate the dis-
tortion altogether, leaving only attenuation. The solutions can now be
followed into detail in various cases without any laborious and round-
about calculations. Besides this, they cast much light upon the more
difficult problems which occur when not so many physical actions are
in question.
In my usual notation, let E, L, S, and K be the resistance, inductance,
ON THE SELF-INDUCTION OF WIRES. PART VIII. 309
permittance, and leakage-conductance of a circuit, per unit length, all
to be treated, in the present theory, as constants ; and let V and C be
the transverse voltage and the current at distance z. The fundamental
equations are
............. (Ig)
p standing for d/dt. Here C is related to the space-variation of Vir\
the same formal manner as is V to the space-variation of C. This
property allows us to translate solutions in an obvious manner, and
gives rise to the distortionless state of things. Let
LStf = l, and E/L = K/S=q ................ (20
The equation of Fis then
and the complete solution consists of waves travelling at speed v with
attenuation but without distortion. Thus, if the wave be positive, or
travel in the direction of increasing z, we shall have, iff^z) be the state
of V initially,
(5g)
If Vy C 2 be a negative wave, travelling the other way,
(70
Thus, any initial state being the sum of V^ and V^ to make V t and of
C l and (?2 to make (7, the decomposition of an arbitrarily given initial
state of V and C into the waves is effected by
r^Mr+LvC), V^\(V-LvV) ................ (80
We have now merely to move V^ bodily to the right at speed v, and F" 2
bodily to the left at speed 0, and attenuate them to the extent e~ gt , to
obtain the state at time t later, provided no changes of conditions have
occurred. The solution is therefore true for all future time in an
infinitely long circuit. But when the end of a circuit is reached,
a reflected wave usually results, which must be added on to obtain the
real result.
In any portion of a solitary wave, positive or negative, the electric
and magnetic energies are equal, thus
|iC? = JffF? ............................... (90
The dissipation of energy is half in the wires and half without, thus
(100
When a positive and a negative wave coexist, and energies are added,
cross-products disappear. Thus the total energy is always
or L(C! + CS); ............... (110
310 ELECTRICAL PAPERS.
the total dissipativity is always
, or
and the total energy-flux is always
The relation V^LvC^ is equivalent to C 1 = SvF l ; i.e., a charge SF
moving at speed v is the equivalent of a current C of strength equal to
their product. But it is practically best to employ Lv t the ratio of the
force V to the flux C being then at once expressible or measurable in
ohms. For v is 30 ohms, and L is a convenient numeric, say from 2 up
to 100, according to circumstances. Z = 20 is a convenient rough
measure in the case of a pair of suspended copper wires. This makes
our critical impedance 600 ohms. It must not be confounded with
resistance, of course, though measurable in ohms. The electric and
magnetic forces are perpendicular. It is the total flux of energy which
is expressed by the product VG, not the dissipativity.
Regarding v, its possible greatest value is the speed of light in wcuo.
AY lien there is distortion also, making the apparent speed variable, it
does not appear that under any circumstances the speed can exceed v.
Now the classical experiments of Wheatstone indicated a speed half as
great again as that of light. Would it not be of scientific interest to
have these important experiments carefully repeated, on a straight
circuit (as well as of other forms), to ascertain whether, on the straight
circuit, the speed is not always less than, rather than greater than, that
of light, and whether there was any difference made by curving the
circuit ]
The following remark may be useful. In treatises on electro-
magnetism by the German methods, a current-element and its properties
of attraction, repulsion, etc., occupy an important place. It is, how-
ever, quite an abstraction, and devoid of physical significance when by
itself. But the current-element in our theory above, ssLy.V=V 1 con-
stant through unit distance, C= V^Lv through the same unit distance,
F'and C zero everywhere else, is a physical reality (with limitations to
be mentioned). It is a complete electromagnetic system of itself, with
the electric currents closed. To fix ideas most simply, the two con-
ductors may be a wire with an enveloping tube separated by a dielectric,
and by our current-element we imply a definite electric field, magnetic
field, and dissipation of energy, which can exist apart from all other
current-elements. It is only an abstraction in this quite different
sense, that we could not really terminate the element quite suddenly,
and that in the process of travelling it must be distorted from causes
not considered in our fundamental equations, one cause being the
diffusion of current in the conductors in time, which alone serves
to prevent the propagation of an abrupt wave-front, either in our
distortionless system, or when there is marked distortion. Even
assuming that Maxwell's representation of the electromagnetic field is
not correct, there seems to me to be very marked advantage in assum-
ing its correctness, even as a working hypothesis, from its exceeding
physical explicitness in dynamical interpretation, without specifying a
ON THE SELF-INDUCTION OF WIRES. PART VIII. 31 1
special mechanism to correspond. AVe have also the inimitable advan-
tage of abolishing once for all the speculations about unclosed currents,
and the insoluble problems they present. In Maxwell's scheme currents
always close themselves, and cannot help it.
It will be seen that our waves, in the above, do not in any way differ
from plane waves of light (in Maxwell's theory), save in being attenuated
by dissipation of energy in the dielectric (when it is a tubular conducting
dielectric bounded by a pair of conductors that is in question), and also
in the bounding conductors, and in being practically of quite a different
order of wave-length. The lines of energy-flux are parallel to the wires,
(a wave simply carries its energy with it, less the amount dissipated) ;
these are also the lines of pressure, for the electrostatic attraction equals
and cancels the electromagnetic repulsion. The variation of the pres-
sure constitutes a mechanical force, half derived from the electro-
magnetic force, half from the magneto-electric force. Here, however,
I am bound to say I cannot follow readily. If this mechanical force
exist, there must be corresponding acceleration of momentum ; if it do
not exist, or be balanced, the stress supposed is not the real stress,
though it may be a part of it. Again, if it be the real stress, and there
be the corresponding acceleration of momentum, this is equivalent to
introducing an impressed force (mechanical), and it must be allowed for.
The matter is difficult all round. Yet Maxwell's stresses, assumed to
exist in the fluid dielectric between conductors, account perfectly for
the forces between them, when the electric and magnetic fields are
stationary. But when they vary, then the region of mechanical force
due to stress-variation extends into the dielectric medium. As for
Maxwell's stress in a magnetised medium, there are so many different
arrangements of stress that will serve equally well, that I cannot have
any faith whatever in the special form given by Maxwell
It is also well to remember that we are not exactly representing
Maxwell's scheme, but a working simplification thereof. The lines of
energy-transfer are not quite parallel to the conductors, but converge
upon them at a very acute angle on both sides of the dielectric. Only
by having conductors to bound it of infinite conductivity can we make
truly plane waves. Then they will be greatly distorted, unless we at
the same time remove the leakage by making the dielectric a non-
conductor instead of a feeble conductor ; when we have undissipated
waves without attentuation or distortion.
Properties of the Distortionless Circuit itself, and Effect of Terminal
Reflection and Absorption.
Now to mention some properties of the distortionless circuit. A pair
of equal disturbances, travelling opposite ways, on coincidence, double V
and cancel C. But if the electrifications be opposite, /^is annulled and
C doubled on coincidence.
On arrival of a disturbance at the end of a circuit, what happens
depends upon the connections there. One case is uniquely simple.
Let there be a resistance inserted of amount Lv. It introduces the
312 ELECTRICAL PAPERS.
condition V=LvC if at say B, the positive end of the circuit, and
V - LvC if at the negative end A, or beginning. These are the
characteristics of a positive and of a negative wave respectively ; it
follows that any disturbance arriving at the resistance is at once
absorbed. Thus, if the circuit be given in any state whatever, without
impressed force, it is wholly cleared of electrification and current in the
time l/v at the most, if I be the length of the circuit, by the complete
absorption of the two waves into which the initial state may be
decomposed.
But let the resistance be of amount 7^ at say B ; and let V-^ and V 2
be corresponding elements in the incident and reflected wave. Since
we have
we have the reflected wave given by
T<
(15?)
If M l be greater than the critical resistance of complete absorption, the
current is negatived by reflection, whilst the electrification does not
change sign. If it be less, the electrification is negatived, whilst the
current does not reverse.
Two cases are specially notable. They are those in which there is
no absorption of energy. If 7^ = 0, meaning a short-circuit, the
reflected wave of V is a perverted and inverted copy of the incident.
But if R l = oo , representing insulation, it is C that is inverted and
perverted.
After reflection, of course, we have the original wave travelling to
the absorber or absorbing reflector, or pure reflector, and the reflected
wave coming from it. Let p be the coefficient of attenuation at A,
and p : at B, these being the values of the ratio of the reflected to the
incident waves at A and at B, which may be + or - , due to terminal
resistances (without self-induction or other cause to produce a modified
reflected wave ; some of these will come later) : and let p be the
attenuation from end to end of the circuit (A to B or B to A), viz.,
Then an elementary positive disturbance F starting from A becomes
attenuated to pV Q on reaching B; becomes p^V^ by reflection at B;
travels to A, when it becomes pVi^oJ * s reflected, becoming p^p-j^V^}
and so on, over and over again, until it becomes infinitesimal, by the
continuous dissipation of energy in the circuit, and the periodic losses
on reflection. But if the circuit have no resistance and no leakage,
and the terminal resistances be either zero or infinity, there is no
subsidence, and the to-aud-fro passages with the reversals at A and
B continue for ever.
If an impressed force e be inserted anywhere, say at distance z v it
causes a difference of potential of amount e there, which travels both
ways ( + \e, to the right, and - \e to the left) at speed v, with the
ON THE SELF-INDUCTION OF WIRES. PART VIII. 313
proper attenuation as the waves progress. That is, taking for simplicity
the zero of z at the seat of impressed force, we set up a positive wave
F 1= J"-^', (170)
and a negative wave V 2 = - \e c + * ZILv ; (1&7)
these being true when z is less than vt in the first, and - z is less than
vt in the second. On arrival at A and B these waves are reflected in
the manner before described. It will be understood that the original
waves still keep pouring in, so long as e is kept on. By successive
attenuations we at length arrive at a steady state, which is that cal-
culable by Ohm's law, allowing for leakage.
If the impressed force be at A, and the circuit be short-circuited
there, making /o = - 1, the two initial waves are converted into one,
thus,
77" _ a f-fo/Lv /I Q~\
ri-et , ( iy #7
true when z is not greater than vt. On arrival at B, if the resistance
there be Lv, nothing more happens, i.e., (190) is the complete solution.
This is something quite unique in its way. If e at A vary in any
manner with the time, the current at B varies in the same manner at
a time l/v later. Thus, if e=f(t), the current at B is
But if we short-circuit at B, we superimpose first a negative wave
F 2 = -ep.- R(l - t)ILv = -ep^.^ ILv , (210)
beginning at time l/v and travelling towards A ; then at time 2l/v add a
positive wave
F 3 =V. -^>, : (220)
and so on, ad inf., settling down to the steady state.
The Fourier-series solution in this case (got by the method of
Part IV.) is
Rz Rz
p- p~ l " T^ V J q 2 + vV
This includes the whole process of setting up the final state, but
requires laborious examination to extract its real meaning, which we
have already described, (m goes from TT, 27r, STT, ... , up to .) When
the summation vanishes, we have left the term independent of t, of
which the positive part is the sum of the positive waves Fj, Fg, etc..
and the negative is the sum of the negative waves F" , etc., above
((190), (210, (220>
The uniquely simple case of complete absorption at B of the first
wave is much more troublesome by Fourier-series than is the really
more complex (230) case. In some other cases in which we can by the
method of waves solve completely, and in a rational manner, the
Fourier-series are difficult to interpret.
Let us construct the complete solution when the terminal resistances
314 ELECTRICAL PAPERS.
have any values ; by (150) we know p and p lt and by (160) we express
p. First of all we have the positive wave
^ = ^(1-^)6-^, ............................ (240)
true when z is not greater than vt. When t = l/v it is complete, and
remains on. Then begins
travelling towards A, when it is complete and remains on. The third
wave then begins :
which reaches B at time t = 3//0, and remains on. The fourth wave
then starts :
reaching A at time il/v ; and so on. We thus follow the whole history
of the establishment of the final state. The resultant positive wave is
the sum of V-^ V^ ... , and the resultant negative wave the sum of V^
V^ ... , which are in geometrical progression ; so that finally we have
In the positive component-waves the current is got by dividing V by
Lv, and in the negative waves by - Lv, so that we get the resultant
final current by dividing Fin (280) by Lv and changing the sign of the
second term, Expressing the negative waves of V.
Should L and S have their values changed in any way, the final state
(280) will be unaltered, but the manner in which it is established will
not be the same, of course. We can, however, form a very fair idea of
the process from the above, when RjL is not greatly different from K/S,
especially if the circuit be sufficiently short to make the attenuation
p be not great.
The case of no resistance is peculiar. There is no steady state if
there be no resistance to make the to-and-fro waves (which may be
regarded as a single wave overlapping itself) attenuate. Thus, if there
be short-circuits at A and B, and also R = 0, K= 0, the first wave due
to e at z = is
^ = from 2 = to z = vt.
Then, when this is completed, we have to add on the reflected wave
F" 2 = - e from z = I to z = 21 - vt,
so that when B is reached, there is no electrification left. This is a
period, and the state of electrification repeats itself in the same way.
But the current doubles itself the moment the first wave reaches B, and
the region of doubled current then extends itself to A, where it is at
once increased to a trebled value ; and so on, ad inf., every reflection
adding e/Lv to the current. Thus the current in time mounts up
infinitely, though never becoming permanently steady at any spot.
The least resistance anywhere inserted will cause a settling down to (or
mounting up to) a final steady current.
ON THE SELF-INDUCTION OF WIRES. PART VIII. 315
Effect of Resistances and Conducting Bridges Intermediately Inserted.
Let us now examine the effect of an intermediately inserted
resistance r. (If the circuit be a double wire, then, in accordance
with the Section on Interferences in Part VII., half the resistance
should be put in one wire, and half in the other, just opposite.)
Let a wave be going towards r, and let V v V^ and V B be corre-
sponding elements in the incident, reflected, and transmitted waves.
As we have
3 ,
and Fj and V z are positive waves, whilst V 2 is a negative wave, there-
fore
rjr^l+r/ZLv)-*, ....................... (300)
and ^=^+^
From (310) we see that an element of the original wave, on arriving at
the resistance, is divided into two parts, both of the same sign as regards
electrification, of which one goes forward, the other backward, increasing
the electrification behind. The attenuation caused by the resistance
is expressed by (300). If there be n resistances r, such that nr = Rz,
equidistantly arranged, the attenuation produced in the distance z will
be the n th power of the right member of (300), and in the limit, when
the resistances are packed infinitely closely, each being infinitely small,
the attenuation in distance z becomes
(320)
This, it will be observed, is when there is no leakage. R is the resist-
ance per unit length, uniformly distributed.
Now consider the effect of a bridge of conductance k t in the absence
of resistance in the wires, or of uniform leakage. We now have
v + v = y , i
c\ + c 2 =cl'+kr 3 , r
if Fj, F 2 , F 3 be corresponding incident, reflected, and transmitted
elements. Consequently
r a /r 1 -(i+*/2,&)-i, ..................... (340)
and C =
Compare with (300), (310). Observe the changes from voltage to
current, inductance to permittance, and resistance to conductance. It
is the current that now splits without loss, (like the charge before), so
that the reflected electrification is negative, if the incident be positive.
The attenuation in distance z due to uniformly distributed leakage-
conductance K per unit length is therefore
We may infer from this opposite behaviour of a resistance in the
main circuit, and of a bridge across it, that if r/L = k/S, there will be
316 ELECTRICAL PAPERS.
no reflected wave. We must, however, see whether combining the
resistance and bridge does not alter the nature of the result. When
the resistance r and the bridge of conductance k coexist at the same
spot, we shall have
F 1+ r 2 =(i+r/i.)r 3 , \ , 36 .
fi -r,-r,+ (F, + ?-,)*/,&,/
whence r - -(*/&)(- + ) , 37( .x
T-r + ZLv + ( -
So the reflected wave is annulled when
or by r/L = k/S when r and k are infinitely small. When this happens,
the attenuation is
^^-(l+r/Z*)-!, ........................ (390
and, therefore, when R and K are uniformly distributed,
is the attenuation in distance z. We have thus a complete electrical
explanation of the distortionless system ; reflection due to conductance
in the dielectric itself is annulled by reflection due to the boundary
resistance (of the wires). If there be no leakage, any travelling
isolated disturbance will cast a slender tail behind it, whose electrifica-
tion is similarly signed to that of the nucleus, whilst the current in the
tail points to its tip. On the other hand, if there be leakage, but no
resistance in the wires, the travelling disturbance will cast off a tail of
a different kind, viz., of the opposite electrification to the nucleus, and
of the same current as in the nucleus. And when the resistances in the
wires and in the dielectric are properly balanced, the formation of tails
is prevented altogether.
From this manner of viewing the matter we can get hints as to the
solution of other and more difficult partial differential equations than
the one we are concerned with. Keeping to it, however, we may
somewhat generalise it by making the attenuation-rate a function of
the distance, and also the speed, but managing so that there shall be no
tailing. Thus, it is clear that if L and S be constant, whilst R and K
are functions of z such that their ratio is constant, the speed will be
constant, and there will be no tailing, whilst the attenuation in distance
z Z Q will be
exp ~
Now if we make the speed also variable, we must inquire how to
prevent tailing due to what is equivalent to a change of medium, as
when light goes from air into glass perpendicularly. The condition
that there be no reflected ray is yu,^ = fj, 2 v 2 in that case, ^ and /* 2 being
the inductivities, and v^ and v 2 the speeds. In our present case it is
Lft = L z v 2 when the wires and the dielectric have no resistance and no
ON THE SELF-INDUCTION OF WIRES. PART VIII. 317
conductance respectively; L v v l being the values on one side, L 2 , v 2
those on the other side of the discontinuity. That is, the quantity Lv
must not vary with z, if there is to be no tailing.
We should, however, make sure that this is the condition when we
have simultaneously L, S, R, and K in operation. Let, then, r and k
be the resistance in the main circuit, and the conductance of a bridge
across it, at a place where the main circuit changes in inductance and
permittance from L, S to Z/, S f t the main circuit being supposed to
have itself no resistance or leakage. Let V^, V^ and F" 3 be corre-
sponding elements of an incident, reflected and transmitted wave.
We have, by common electrical principles, united with the properties
T- LvC,
from which
There is no reflected wave when the numerator on the right of
20) vanishes, or when
r -, k . Lv , rk
and then Fg/Fi^O +r/W)- 1 ......... : ..................... (450)
So, if we take Lv = L'v f , we secure the desired result, because the product
rk ultimately vanishes when we distribute resistance and conductance
continuously. That is to say, if Lv does not vary, and ft/L = K/S
always, there will be no tailing, the speed will be a function of z, viz. :
and the attenuation-rate will be a function of z, as indicated by (400).
To verify, observe that our fundamental equations (1) may be
written, if E/L = KjS,
hence, if Lv be constant, we have
which become identical if V= LvC, indicating a complete satisfaction
when q and v are functions of z. Then
are the equations of positive or negative waves.
318 ELECTRICAL PAPERS
Approximate Method of following the Growth of Tails, and the
Transmission of Distorted Waves.
The substitution of isolated resistances and conducting bridges for
continuously distributed resistance and leakage leads to a very easy
way of following the course of events when there is distortion by
a want of the balance between the resistance in the main circuit and
the leakage which is required to wholly remove the distortion. As
may be expected, the results are only rough approximations, but the
method is so easy to follow, and gives so much information of a rough
kind, that it is worthy of attention. The subject is quite a large one
in itself, and would need a large number of diagrams to fully illustrate.
I shall therefore only briefly indicate the nature of the process.
Suppose there is no leakage whatever. Then, unless the resistance
in the main circuit be low, there will usually be much distortion due to
tailing, unless the waves be of great frequency, making E/Ln small.
The smaller this quantity is, by either reducing R, or increasing L or
the frequency, the nearer do we approximate to a state of little
distortion, and to attenuation represented by
-Xz/2Lv
in the distance z. In fact, in long-distance telephony we do not need
any excessive leakage to bring about an approximation to the state of
things which prevails in our distortionless system (where, however,
disturbances of any kind, not merely waves of veiy great frequency, are
propagated without distortion), and the attenuation is of course less
than when there is leakage. As this, however, would require us to
examine the sinusoidal solutions of Parts II. and V., we may now keep
to the question of tailing and its approximate representation.
Let it be required to find how a charge, initially given existent in a
small portion of the circuit, and at rest, divides, when left to itself.
We know that if there were no resistance, it would immediately
separate into equal halves, which would travel with speed v in opposite
directions without attenuation or distortion. And, if there be resist-
ance, but accompanied by proper leakage to match, the same thing will
happen, with attenuation. Now there is to be no leakage ; this keeps
the total charge unchanged. If then there were no tailing there would
be no attenuation. But the charges, on separation, cast out slepder
tails behind them, so that they are joined by a band (the two tails
superimposed). The heads, therefore, or nuclei, are attenuated, besides
being distorted; the loss of charge from them is to be found in the tails.
It is sufficient to consider the progress of one of the two halves of the
initial disturbance, say that which moves to the right, and the tail it
casts behind it.
Localise the resistance at points, between which there is no resist-
ance, and let the attenuation in passing each resistance (equidistantly
placed) be any convenient large proper fraction, say -^ ; though this is
scarcely large enough it is convenient, as all operations will consist in
multiplications by 9 and simple additions. Let the initial charge,
moving to the right, be 10,000, extending uniformly over a complete
ON THE SELF-INDUCTION OF WIRES. PART VIII. 319
section between two resistances, and let a be the time taken to travel
one section. Then first we have
->
10,000 ;
-<- ->
1,000, 9,000 ;
-<- -> ' -- ->
900, 100, 900, 8,100;
<- ->-<--> - ->
810, 90, 820, 180, 810, 7,290.
The figures in the successive lines show the distribution of the charge
in the consecutive sections to right and left, initially and after intervals
a, 2a, 3a, etc. First of all ~$ of the initial charge passes into the next
section to the right, and the other y 1 ^ is reflected back by the resistance
to where it was at the beginning. Then these two charges similarly
divide, -^ of each going forward, the other y 1 ^ backward. The arrows
indicate the direction of motion of a charge. All subsequent operations
consist in pairing the charges which are moving towards one another in
the proportions T 9 ^ and y 1 ^. After seven operations we have this
result :
<- ->
531, 59, 566, 120, 583, 184, 591, 245, 583, 302, 565, 371, 530, 4773;
so that more than half the original charge is in the tail. The directions
of motion are alternately to left and to right, so that it is only necessary
to know this, and not to continue drawing the arrow-heads. The
currents are alternately + and - .
But we should, to approach reality, extend the original charge at
least over two sections, instead of one only. To do this, we have
merely to add each of the numbers to the one following it. After
seven operations, therefore, an initial charge of 20,000 extending over
two sections, and moving to the right, becomes distributed thus :
531, 590, 625, 686, 703, 767, 775, 836, 828, 885,867, 936,901,5303,4773;
which is really something like its distribution when the resistances are
uniformly spread. The corresponding current is not represented by
these figures, of course, owing to the opposite direction of current in
alternate segments when the original charge extended over only one
segment. Allowing for this fact, the current, after seven operations,
due to 20,000 over two sections initially, is represented by
-4- -<-.--.'-*
531, 472, 507, 446, 463, 399, 407, 346, 338, 281, 263, 194,169,4243,4773.
In the head the current is positive. In the whole of the tail (repre-
sented by the small numbers) the current is negative. We see that
the division of the initial charge over two sections has not been
sufficient to remove the fluctuations wholly, though the reversals have
disappeared.
In course of time, if the circuit be sufficiently long, the nucleus is so
attenuated as to practically make the charge one long tail stretching
320 ELECTRICAL PAPERS.
out both ways, and tending to do so equally, so that the greatest V-
disturbance is at or near the origin to the right of it. The current is
then negative in the hinder part and also in a portion of the forward
part, and positive in the rest. That is, the region of positive current
extends gradually from the nucleus into the tail.
Now pass to the other kind of tail, due to reflection by leakage.
If there be no resistance in the circuit, but uniform leakage instead, we
have tailing and distortion of a distinct kind. It is the current-element
that splits into two parts, one going forward, the other backward on
passing a bridge, whilst the electrification in the reflected wave is the
negative of that in the incident. If, then, the attenuation be ^ as
before (ratio of transmitted to incident wave), at every one of the
isolated conducting bridges which we use to replace uniformly dis-
tributed leakage-conductance, we shall have the same results as above
precisely, except that current takes the place of transverse voltage.
Thus the first row of figures (after seven operations) shows the current
distribution (everywhere positive) due to an initial charge 10,000 (with
corresponding current as before) extending over one section ; the second
row that due to 20,000 over two sections; and the third row the
corresponding distribution of electrification, positive in the head, and
negative in all the rest. Observe that as, when there was no leakage, the
line-integral of V remained constant, so now that there is leakage, the
line integral of C remains constant. In one case it is really conserva-
tion or persistence of the electrification l&Pft*; in the other, of the
momentum \LCdz. In the one case the momentum-integral subsides,
the time-factor being e~ Kt!L in the other the electrification-integral
subsides, the time-factor being c~ Xil8 . In both cases the energy sub-
sides towards zero, in spite of the persistence of electrification or of
momentum.
When we have both resistance in the conductors and leakage, the
tail is positive or negative (referring to the electrification), according as
RjL is greater or less than K/S. The latter case is quite out of
ordinary practice, which aims at high insulation; the results are con-
sequently very singular, when considered in more detail, which cannot
be done now.
In a somewhat similar manner to that in which we have roughly
followed the growth of tails, we may follow the progress of signals
through a circuit, and obtain the arrival-curves of the current at the
distant end, or rather, we may obtain curves resembling the real ones
somewhat by drawing curves through the zigzags which result. The
method has no recommendation whatever in point of accuracy : its real
recommendation lies in the facility with which a general knowledge of
the whole course of events may be obtained, and I daresay some
people may think that of not insignificant moment.
To make the method intelligible, without going into detail elaborately,
let the circuit be perfectly insulated, and in only seven sections, at each
of the six junctions of which is concentrated one-sixth part of the
ON THE SELF-INDUCTION OF WIRES. PART VIII. 321
resistance of the real circuit. The results will now depend materially
upon the ratio RljLv, whether it be a large number, or small. First,
let it be small, say Rl = ^Lv. The attenuation at each resistance (Rl/Q)
is then T 9 g- as before. Let us also insert resistances of amount Lv at
both ends, to stop reflections and complications. Then, starting with
10,000 in the first section, we proceed thus :
->
A. 10,000;
<- >
1,000, 9,000 ;
->
0, 900, 8,100;
<- ->
810, 90, 810, 7,290;
0, 738, 162, 729, 6,561 ;
-<- ->
664, 74, 672, 219, 656, 5,905;
0, 612, 134, 612, 262, 590, 5,314; B.
+
551, 61, 564, 181, 557, 295, 0;
0, 514, 112, 520, 219, 29, 266.
If a = time of going one section, this gives the whole history of the
circuit from the moment of putting on a steady impressed force at A up
to 9a, or 2a after commencement of arrival of the current at B. The
calculation is precisely that by which we should calculate (by the
previously described method) the progress of a charge 10,000 initially
in the first section and moving to the right. In time a, 9,000 goes
forward to the second section, 1,000 is reflected back. After another
step the 1,000 is absorbed, whilst T 9 ^ of the 9,000 goes forward, and 1 \ ) -
is reflected back. This brings us to the third line. The first arrival at
B is of 5,314, the second of 266, and so on (not carried further). The
sum total of all the arrivals at B when carried further is 5,999, which
really means 6,000. That is, -^ of the charge would go out at B and
y 4 ^ at A. Now the same figures serve with the impressed force,
which we have to imagine continuously sending into the first section
the 10,000 wave. The real state of electrification of the line at any
stage is to be found by summing up the columns, and the real state of
current by summing up the columns with allowance made for the fact
that all charges moving to the left mean negative currents. Thus the
current at A falls to its final strength, whilst at B it rises to it. Of
course the current would not really arrive at B in a perfectly sudden
manner to |-| of its final strength, though it would arrive far more
suddenly than the current arrives at the end of an Atlantic cable. The
final current is (e/2Lv) x -6. If we increase the number of sections so
greatly that the first arrival at B is insensible, then the arrival-curve
will resemble that at the end of an Atlantic cable (or even much shorter
cables). The value of e~ Kl/Lv is exceedingly small in such a case.
H.E.P. VOL. ii. x
322 ELECTRICAL PAPERS.
Now if we short-circuit at A and B the process is essentially the
same, although we must not absorb all reflected waves arriving at A,
and all transmitted waves arriving at B, but reflect them properly.
This causes there to be a sort of bore running to and fro, in addition to
the regular action, so that the arriving current at B gives a sudden
jump at regular intervals 2l/v apart; these jumps get smaller and
smaller rapidly at each repetition, of course. But should the circuit be
so long that the first increment of current at B is insensible, this jump-
ing cannot occur. It is also to be remarked that the insertion of
terminal resistances stops the oscillatory action.
It was my intention to have given the equations of the tails, positive
or negative, or mixed, but as the investigation would unduly extend
the length of the present communication, I propose to consider the tails
in the next Part IX. At present I may remark that the equation is in
the form of a series of rising powers of (vt + z), true when z( course, since we
require the lowest possible resistance to reduce attenuation and dis-
tortion. It is possible, therefore, that such an insulator might be of
great service in cables for telephony and telegraphy, especially as its
insulation-resistance could not be so high as is ordinarily the case. The
changed permittance must also be allowed for, though.
As regards open wires, if of copper, and of low resistance, good
telephony is possible to ridiculously great distances, further than any
one wants to speak, without troubling about getting the leakage to be
large.
There is a value of L which gives the least attenuation. For since, in
the distortionless system, the received current is
if short-circuited at A, but with resistance Lv at B ; or one half this
amount, if there be resistance Lv both at A and at B, we see that
fil = Lv, ................................. (51?)
makes C B a maximum. But the attenuation is then so trifling that to
carry this out (by increasing L) would be, if possible, quite unnecessary
in the case of a long circuit.
Again, in the case of no leakage at all, it may be shown by an
examination of the sinusoidal solution in Part V., that if RjLn be small,
we approximate towards the same formula but with the index - Rl/'2Lv,
sothat J8-2Z. ............................... (620)
gives the value of Lv which makes the current received at B a maximum
to suit a given resistance of circuit. It may also be shown by the
same formula that if the receiver have small inductance, the resistance
it should have (when of a given size and shape) to make the magnetic
force a maximum approximates to Lv, which is the critical resistance
that absorbs all arriving disturbances.
May 7, 1887.
XLI. ON TELEGRAPH AND TELEPHONE CIRCUITS.*
[February, 1887 ; but now first published.]
APP. A. On the Measure of the Permittance and Retardation of Closed
Metallic Circuits.
OWING to the fact that most of the circuits of which mention is made in
my brother's paper consist of or contain a considerable amount of
* [This article consists of the three appendices that I wrote to the paper of Mr.
sts of the three appendices that I wrote to the paper o
myself on " The Bridge System of Telephony," which
A. W. Heaviside and myself on " The Bridge System of Telephony," which paper
324 ELECTRICAL PAPERS.
buried wires, and therefore possess considerable permittance, combined
with the fact that these buried wires have very high resistance, as much
as 45 ohms per mile, and with the further fact that the self-induction of
these lines is small, we may, leaving on one side the question of the
apparatus (which is no unimportant one in itself), regard the transmis-
sion of telephonic currents through the lines as being governed mainly
by the three factors resistance, permittance, and length of line.
Take, therefore, for starting-point the now well-known theory of the
submarine cable promulgated by Sir W. Thomson in 1855, which was
so curiously foreshadowed by Ohm in 1827, in his celebrated memoir on
the galvanic circuit, when guided by an analogy between the flow of
electricity and the flow of heat, which is now known to be entirely
erroneous.
A translation of Ohm's memoir is contained in vol. II. of Taylor's
"Scientific Memoirs," and Sir W: Thomson's writings on the subject of
the submarine cable are collected in vol. n. of his " Mathematical and
Physical Papers."
Electromagnetic induction is wholly ignored. The line is a single
wire, fully defined by the three data its length, and its resistance and
permittance per unit length. The circuit is completed through the
" earth," supposed to have no resistance, and to extend right up to the
dielectric material which envelops it, whose outer boundary is therefore
taken to be permanently at potential zero. On these suppositions, a
single quantity F", the potential of the wire, when given along it, fully
expresses its state at a given moment, and we may exactly calculate the
effect at the distant end of the line (or at any other part), due to
arbitrarily varying the potential by a battery at the beginning; the
periods of time concerned being, in lines of different lengths, governed
by the important law of the squares. Thus if E be the resistance, and
$ the permittance per mile of a cable of length /, the retardation is pro-
portional to ESI 2 , a certain interval of time, which, if R be in ohms, and
S in microfarads, is expressed in millionths of a second, owing to the
ohm being 10 9 and the microfarad 10~ 15 c.g.s. electromagnetic units.
If there be two cables, with constants fi v S ls l v and R 2 , $ 2 , 1 2 , and we
operate similarly upon them, the time required to set up a given state
in the first will be to that required to set up the corresponding state in
the second, as R^lf is to R^S^- For instance, if it take 1 second to
bring the current at the distant end to y 9 ^ of its full strength due to a
steady impressed voltage at the beginning of the first cable, and the
was intended for presentation to the Soc. Tel. Eng. and Electricians, but which
never got so far, owing to the objections of the official censor. I have omitted the
portion of Appendix C relating to the distortionless circuit, as the matter is more
fully treated elsewhere in this volume. The portions of the obnoxious paper
contributed by myself (about 20 pages) are also omitted, for a similar reason. I
was given to understand that the official censor ordered it all to be left out,
because he considered that the Society was saturated with self-induction, and
should be given credit for knowing all about it. See, however, Art. xxxvin.,
p. 160, in this volume for evidence to the contrary. The present article may now
usefully serve as appendices to the preceding one "On the Self-induction of
Wires," since it consists mainly of practical applications of the theory contained
therein. ]
ON TELEGRAPH AND TELEPHONE CIRCUITS. 325
retardation Ii^S^ of the second cable be 5 times that of the first, it
will take 5 seconds to bring the current at the distant end of the second
line to T 9
-rS t p + Mp(K, + S t p),
From these we may deduce (12) by taking
When, instead of two pairs of condensers only, as in fig. 2, we have
a large number of pairs, the earth-wire r must run on and join the
middles of every pair. We see from this that the equal KM.F.'S in
R l and R 2 will cause currents in them similarly directed which will not
return immediately by the wire r in the figure, but only partly there,
the rest going further and getting to the auxiliary wire through other
condensers. Supposing, then, we have the condensers, etc., uniformly
distributed, if the impressed forces be also uniformly distributed along
the two wires, there would be, by their mutual cancelling, little if any
effect produced (not referring to the balance at the terminals, which is
independent of uniformity of distribution of the equal E.M.F.'S). But,
generally, the E.M.F.'S will not be thus uniformly distributed.
The general equations of self- and mutual-induction of parallel wires,
ON TELEGRAPH AND TELEPHONE CIRCUITS. 339
given in "Induction between Parallel Wires" [vol. I., p. 116], show
that if we start with a pair of equal wires looped, and then introduce
some inequality, we cause the induction-balance to be a little upset, and
simultaneously we cause the circuit to behave not quite the same as a
single wire, as described in App. A. Thus, if the wires be equal in all
respects, and be at the same height above the ground, they behave as
one ; and also, if exposed to the interference of a parallel wire equi-
distant from them, the balance will not be upset. But if the paired
wires be in a vertical plane, and therefore at different heights above
the ground, we cause a small departure from behaviour as a single wire,
and also slightly upset the balance, even although the interfering wire
be equidistant from the paired two. Both effects will be small, and it
is questionable whether they would be observable. But I am informed
by my brother that the interference arising from one wire being of iron
and the other of copper has been observed in his district.
When the circuit is completed by a concentric tube, the external
permittance of the tube will give rise to interference, if the circuit be
long enough. This has not yet been observed.
Practical telephonists who keep their eyes open have unusual oppor-
tunities of observing very curious and interesting electrostatic and
magnetic effects. Unfortunately, however, the demands of business, to
say nothing of other reasons, usually prevent their careful examination,
record, and explanation.
APP. C. On the Propagation of Signals along Wires of Low Resistance,
especially in reference to Long-Distance Telephony.
A WHOLLY exaggerated importance has been attached by some writers
to electrostatic retardation. I do not desire to underrate its import-
ance in the least its influence is sometimes paramount, but the
application of reasoning based solely upon electrostatic considerations
should certainly be limited to such cases where the application is legiti-
mate. Now some writers, without any justification, take Sir W.
Thomson's theory of the submarine cable to be the theory for universal
(or almost universal) application, supposing that magnetic induction is
merely a disturbing cause, introducing additional retardation, but only
to an extent which is practically negligible in copper circuits. This is
very wide of the truth. What has yet to be distinctly recognised by
practicians, is that the theory of the transmission of signals along wires
is a many-sided one, and that the electrostatic theory shows only one
side a very important one, but having only a limited application in
some of the more modern developments of commercial electricity,
notably in telephony, especially through wires of low resistance. Some-
times magnetic inertia itself becomes a main controlling factor.
In my paper "On the Extra Current" [Art. xiv., vol. I., p. 53] I
brought the consideration of magnetic induction into the theory of the
propagation of disturbances along a wire, by the introduction of the
E.M.F. of inertia, according to Maxwell's system, in accordance with
which the inductance per unit length of wire is twice the magnetic
340 ELECTRICAL PAPERS.
energy of the unit current in the wire. Calling this L, the momentum
is LC and the E.M.F. due to its variation is - LC per unit length.
In my paper "On Induction between Parallel Wires" [Art. xix.,
vol. I., p. 116] I have further considered the question; and more
recently, 1885-6-7, in the course of my articles "Electromagnetic
Induction and its Propagation," and "The Self-induction of Wires," I
have given a tolerably comprehensive theory of the propagation of dis-
turbances, and have worked out certain important parts of it in detailed
solutions suitable for numerical calculation. In the present place I pro-
pose to give some practical applications of the formulae, in addition to
what I have already given, to be followed by an account of the principal
properties of a distortionless circuit, which casts considerable light on
the subject by reason of the simplicity of treatment it allows.
Roughly speaking, we may divide circuits into five classes :
(1). Circuits of considerable permittance, to be regarded as submarine
cables in general, according to the electrostatic theory, unless the wave-
frequency be great or the resistance very low. Long overhead wires of
comparatively small permittance may sometimes be included, especially
if the resistance be high.
(2). Short lines which may be treated by disregarding the electro-
static permittance altogether, and considering only the resistance and
inductance, provided the frequency be not too great. Ordinary short
telephone-circuits usually come under this class.
(3). An intermediate class, in which both the electrostatic and
magnetic sides have to be considered simultaneously. This class is
rather troublesome to manage in general.
(4). Yet another class brought into existence by the late extensions
of the telephone in America and on the Continent, and of rapidly
increasing importance, in which wires of small resistance and small
permittance are used combined with high frequencies, and in which the
permittance (though small) must not be ignored, since, in combination
with the inductance it produces an approximation towards the trans-
mission of signals without distortion. The theory is then, even when
the line is thousands of miles long, quite unlike the electrostatic theory.
(5). Distortionless circuits, now to be first described, in which, by
means of a suitable amount of leakage, the distortion of waves is
abolished. Though rather outside practice, except that extreme cases
of the last class resemble it, this class is very important in the compre-
hensive theory, because it supplies a sort of royal road to the more
difficult parts of the subject.
There may also be sub-classes derived from the above. For instance,
a leaky submarine cable, in which resistance, permittance and leakage-
conductance control matters, whilst inertia may be of insensible influence.
The peculiarity that is brought in by magnetic inertia (symbolised
by the inductance) combined with electric displacement, is propagation
by elastic waves (similar to the waves that may be sent along a flexible
cord, or perhaps better, a common clothes-line, though even then there
is not usually enough resistance), as distinguished from the waves of
diffusion (as of heat in metals) which is the main characteristic of the
ON TELEGRAPH AND TELEPHONE CIRCUITS. 341
slow signalling through an Atlantic cable. The two features are always
both present, but sometimes one is paramount, as in class (1), and
sometimes the other, as in classes (4) and (5). [The Americans who
went in for wires of low resistance had, I think, no idea of the import-
ant theoretical significance of the step they took, but did it because
they wanted long-distance telephony, and because wires of high resist-
ance would not go a characteristically American way of doing things.
Yet their action led the way to a rapid recognition of the sound
practical merits of Maxwell's theory of the dielectric.]
Let E, S, L, K be the resistance, permittance, inductance, and leakage-
conductance respectively, per unit length of circuit, which may be a
single wire with earth-return, or a pair of wires in loop, in which case
the wires should generally be equal, to avoid the interferences which
would remain in spite of the twisting by which the greater part of the
interferences from other circuits may be eliminated. Also, let Fand
C be the potential-difference and current at distance z ; then
-VF=(E + L P )C, -VC=(K+Sp)r, ............. (1)
where V stands for d/dz and p for d/dt, are the fundamental equations.
Now suppose that an oscillatory impressed force acts at the beginning
of the line. Let p denote the ratio of its amplitude to that of the
current. At z = Q, p is plainly the impedance of the circuit to the
impressed force. If the line were perfectly insulated, and had no
permittance, p would be a constant for the whole circuit, at a given
frequency. But the range of the current is not everywhere the same
(besides varying in phase), so that p is a function of z. The term
impedance is strictly applicable only at the place of impressed force,
therefore. But to avoid coining a new word, I shall extend its use, and
term p anywhere the " equivalent impedance." It is with the equivalent
impedance at the far end of the circuit, say z = I, that we are principally
concerned. Call it /, this being the ratio of the amplitude of the
impressed force at z = to that of the current at z = I. Let
LSv*=l, X/Ln=f, . K/Sn = g, ................. (2)
where nftir is the frequency. Also let
P or e = "(l) t {(l+/ 2 ) i (l +f}(fg- 1)}* ............. (3)
On these understandings, the value of / is
provided the line be short-circuited at both ends. Terminal apparatus
will be considered later.
If S 0, L = 0, K= 0, then I=Rl t the steady resistance of the circuit.
If only S=Q, K= 0, then /= l(E* + L 2 n 2 )*, the magnetic impedance. If
= 0, K=0, then
I=i(A\ {" + e- 8 "- 2 cos 2^}*, .................. (5)
in which Pl = (\nR3lrf ............................... (6)
342 ELECTRICAL PAPERS.
Now the significance of (4) depends materially upon the values of the
ratios /, g, and on the frequency. First as regards g. A leakage-
resistance of 1 megohm per kiloni. makes K= 10~ 20 , and a permittance
of 1 microf. per kilom. makes S= 10~ 20 also. Therefore on a land-line
of 1 megohm per kilom. insulation-resistance and '01 microf. per kilom.
permittance, we have g = 100//1. Thus g is important at low frequencies,
and becomes a small fraction at high frequencies, even with this rela-
tively low insulation. Thus, ?i=1000 makes #='1, and n = 20,000
makes g='OQ5. These correspond to frequencies of about 160 and
3200. We see that in telephony, even with poor insulation, g is always
small. By bettering the insulation it is made smaller still. Therefore
we may practically take g = Q in telephony through a fairly well-in-
sulated line. Notice here that the effect of g in attenuating the current
may be considerable when the frequency is low, and yet be small when
the frequency is high.
Now the frequency is low on long submarine cables. Consequently
g, if there is sensible leakage, has an important attenuating effect. But
the above formula does not inform us what other effects leakage has,
except by examination through a large range of frequencies. It has a
remarkable effect in removing the distortion of the signals, by neutralis-
ing the effect of electrostatic retardation. This is marked when the
frequency is low, and becomes less marked when it is high. But in the
latter case, if the frequency be only high enough, there is little distor-
tion even when the insulation is perfect, or g = 0, provided the resistance
be small. Thus g has a large attenuating and also a large rectifying
effect when the frequency is low; when it is high, then it does not
attenuate so much and does not rectify so much, nor is so much rectifi-
cation wanted. But the full nature of this rectifying action will be seen
later in the distortionless circuit.
Now consider /. This depends on the resistance, inductance, and
frequency. Now 1 ohm per kilom. makes -ff=10 4 ; consequently, if
r be the resistance in ohms per kilom.,
f=W*r/Ln ................................. (7)
In a long submarine cable r is small, but n is also small, and L is small,
or certainly not great ; therefore / is big. So we may take its reciprocal
to be zero ; or, what will come to the same thing, take L = 0. We have
then the formula (5) for the equivalent impedance (unless leakage is
important) ; and since we can work up to such frequencies that e 2 ^ is
big, we may then write
(8)
or p = I/El = Pl (8Pl)-\ ...................... (9)
where PI is as in (6). This PI may be as big as 10 on an Atlantic
cable. Equation (8) shows the extent to which the line's resistance
appears to be multiplied, and is according to Sir W. Thomson's theory.
Now consider buried wires of 45 ohms per mile, such as are used in
telephony by the Post Office. Being twin wires, L is small ; so, when
n is even as high as 10 1 , / is made rather large. Consequently we may
ON TELEGRAPH AND TELEPHONE CIRCUITS. 343
still apply the electrostatic theory, even in telephony, so far as the
buried wires mentioned are concerned, although it will somewhat fail
at the higher frequencies : and we see that it is by reason of their
high resistance and low inductance that we can ignore the influence
of inertia in them. But this does not apply to the suspended wires
which are in circuit with the buried wires, as we shall see pre-
sently.
Consider a pair of open or suspended wires. Take 20 ohms per
kilom. as the resistance, or 10 ohms each wire. This will, by (7), make
/=2 if Z=10 and n= 10,000; and /=-2 if =100. Now the last
value of L is extreme. It could only be got with an iron wire, and its
inductivity would need to be large even then ; besides that, the fre-
quency would need to be low in order to allow the large L to operate,
on account of the increased resistance due to the tendency to skin-
conduction at high frequencies. Such a large value of L may usually
be put on one side, so far as practical work is concerned; but = 50
would be more reasonable, remembering that in L is included the part
due to the dielectric surrounding the wire. The data regarding the
inductivity of iron telegraph-wires are not copious ; from my own
observations, I believe that, with the weak magnetic forces concerned in
telephony, /* = 200 is high, and it may be as low as 100. The point is,
however, that /, from being large, may be made small by increasing the
inductance without other changes. Still, however, with the assumed
steady resistance of 20 ohms per kilom., we could not treat /as a small
fraction, especially as the increased resistance due to the imperfect
penetration of the magnetic induction into the wires will increase /, as
will also the reduced inductance due to the same cause. Thus / must
be kept in the formula for the equivalent impedance, though not to
be treated as either very large or very small in general. That is,
we have the form of theory of class (3) mentioned above. Similar
remarks apply to long suspended copper wires if the resistance be
several ohms per kilom., and they be at the usual distance apart ; for
although with high frequencies / will be small, yet it will not be
small enough at the low frequencies to allow of its treatment as a
small quantity. We should therefore use equation (4) with only g =
in general.
But now come to a copper wire of only 1 ohm per kilom., in
loop with a similar wire, making R= 20 4 or r = 2. Now %=10 4
/.2/i; (10)
from which we see that / may be so small a fraction as to lead to a
simplified form of theory. We now have the fourth class of circuits ;
well-insulated, of low resistance, and of fairly high inductance, making
RjLn small, and a tolerably close approach to distortionless trans-
mission.
To estimate the value of L, go back to equation (2) defining v. Here
v is a speed, always less than that of light, but of the same order of
magnitude. If the wires are of iron, it is considerably less ; but if of
copper it is so little less that we may neglect the difference. Now
344 ELECTRICAL PAPERS.
and 1 microf. = 10~ 15 , so that if S Q is the permittance in microf.
per kilom.,
L=($s )-\ ................................ (11)
which is useful in giving an immediate notion of the size of L in
terms of the permittance, when that is known. Thus '01 microf.
per kilom. makes L = 11, so that /= T 2 T when n= 10,000, when the
resistance per kilom. is 2 ohms ; and / is only T X T at the higher
frequency 20,000/27r.
But this estimate (11) will always be too small a one, and sometimes
much too small, if S Q be the measured permittance per kilom. It was
found by Professor Jenkin that the measured permittance was twice as
great as that calculated on the assumption that the wire was solitary.
The explanation (or a part of it) which I have before given [Art. xn.,
vol. L, p. 42, and XXXVIL, vol. IL, p. 159] is that the neighbouring
wires themselves largely increase the permittance. Therefore, if s be
the measured permittance in presence of earthed wires, the real L must
be considerably greater than by equation (11). On the other hand,
there is a set-off by reason of L being reduced by the induction of
currents in the neighbouring wires, though not so greatly as to
counteract the preceding effect. Again, the magnetic field pene-
trates the earth, which increases L. But, to avoid these complexities,
which require us to consider the various mutual effects of circuits,
let our circuit be quite solitary. Then, if r = radius of each wire, and
s = distance apart,
L= 1-1-4 log (s/r) ............................ (12)
when yu, = 1, as with copper wires, the 1 standing for J/AJ + J/J 2 , if /^ and
H 2 are the inductivities of the two wires. These terms are important
in the case of iron wires ; but riot with copper, unless the wires are very
close, when they become relatively important on account of the small-
ness of the total inductance. The other part of L is the inductance of
the dielectric, and it is this which, when multiplied by S, gives the
reciprocal of the square of the speed of light, subject to the proper
limitations. Now L = 20 requires s/r = 148 ; or if r be inch (which is
about what is wanted to make the resistance 1 ohm per mile), s must
be 18 J inches. We therefore see that L = 20 is quite a reasonable value
with copper loop-circuits. It gives /= 1 when n= 1000, and -^ when
n = 10,000. Thus /is less than unity throughout the whole range of
telephonic frequencies, and becomes a small fraction even at practical
frequencies.
Take, then, g = and / small in (3) and (4). We get
1 n R R n
and the equivalent impedance formula (4) reduces to
(14)
in the fourth class of circuits.
ON TELEGRAPH AND TELEPHONE CIRCUITS. 345
The further significance of this formula will depend materially upon
the value of the ratio ltt/2Lv (that is, the value of PI), the ratio of the
resistance of the circuit to 2Lv, which is, in the present case, 1 200 ohms.
If the length of the circuit be a small fraction of 600 kiloms., the
impedance depends upon the frequency in a fluctuating manner, going
down nearly to Uil and then running up nearly to Lv, as the circular
function goes from - 1 to + 1, on raising the frequency. Thus the
least possible equivalent impedance at z = l is one half the steady
resistance of the line, and the greatest is Lv.
According to (14) this would go on indefinitely, as the frequency was
raised continuously. But another effect would come into play, viz., the
increased resistance due to skin-conduction, with a corresponding small
change in L. As the result of this increased resistance the value of
Il/2Lv will rise, and the range in the fluctuations of / decrease ; and if
the frequency be pushed high enough the fluctuations will tend to
disappear. But this could not happen in telephony at any reasonable
frequency, say n = 20,000.
The physical cause of the low value \El at certain frequencies is the
timing together of the impressed force at the beginning of the circuit
and the reflected waves. It is akin to resonance. Thus, if the line
had no resistance at all we should have
I=Lv$in(nl/v), ............................ (15)
with the circular function taken always positive. When nl/v = 7r,
1=0. Then 27r/n = 2l/v, or the period of the impressed force coincides
with the time of a double transit (to the end of the circuit and
back again).
In connection with (15) I may mention that an approximate formula
for the impedance, when nl/v is in the first quadrant, and especially in
its early part, is
which shows the beginning of the action of the permittance in reducing
the impedance from its magnetic value as the frequency is raised.
But to use wires of such low resistance for comparatively short lines
would be wastefully extravagant. Such wires admit of very long
circuits being worked. Therefore increase the length of the line in
equation (14) ; as we do this the range in the oscillation in / falls,
until, when fil = 2Lv, I does not depend much upon the circular
function. We may then, and at all higher frequencies, write simply
............................. (17)
--
Compare with (8), the corresponding cable-formula, and note the differ-
ences. The impedance is now nearly independent of the frequency,
and there is nearly distortionless transmission of signals, provided H/Ln
be small, and Bl/Lc = 2 or 3 or more.
346
ELECTRICAL PAPERS.
The following table gives the values of p calculated by (14), which
only assumes that RjLn is small, for a series of values of PdjLv = y.
y-
Min. p.
Mean p.
Max. p.
y>
/>
y>
P-
\
505
1-500
2-063
6
1-678
12
16-81
521
878
1-128
7
2-65
14
39-3
2
587
686
771
8
3-378
16
93-2
2-0653
594
685
766
9
5-000
18
225
3
710
748
784
10
7-420
20
550
4
907
924
940
5
1-210
1-218
1-226
Here the "mean," "maximum," and " minimum " values of p mean the
values when the cosine is 0, + 1, and - 1. The fluctuations are very
large when y is small, going from \Rl to Lv ; but they are insensible
when y is bigger. Kemember that the line is short-circuited. The
receiving apparatus, by absorbing energy, reduces the fluctuations, and
we shall see later that they can be nearly abolished.
When RljLv y is variable, the value of IjRl is made a minimum by
taking 7^ = 2-06 Lv, say 2Lv. This is a little over 1200 ohms in our
example of Z = 20; and makes the length of circuit be 600 kilom.,
when the resistance is 2 ohms per kilom. After y = 3 we may disregard
the fluctuations.
Now this length of only 600 kilom. is still far too short to make it
necessary to employ so expensive a wire. One of much higher
resistance would answer quite well enough for practical telephony, in
which a considerable amount of distortion is permissible, because
transmission would be nearly perfect over 600 kilom. according to the
above data. The question arises, upon what principles can we compare
one circuit with another, and is it possible to lay down the law from
theory as to the limiting distance of telephony ? The answer is plainly
that it is not possible, because the types of telephonic circuits differ.
A cable or other circuit with inertia ignored is radically different from
one in which there is a marked approach to elastic wave-propagation.
Even if we fix the type, and take, say, the above example of low
resistance, 2 ohms per kilom. and L = 20 per centim., and the question
be asked, How far can you telephone ? the answer is that there is no
fixed limit, as it depends upon so many circumstances, some of which
are unstated, and are hardly susceptible of measurement when stated.
Consider, first, the circuit without terminal influences. We may
distinguish two connected, but yet entirely different, things in opera-
tion. We set up electromagnetic vibrations at A somehow, not regular
vibrations of one frequency, but irregular, and of almost any type.
Now, during transmission along the circuit, the vibrations are attenuated
for one thing, and distorted, or changed in type, for another. With
perfect transmission there would be neither attenuation nor distortion.
This would require perfect conductors, which would not permit the
ON TELEGRAPH AND TELEPHONE CIRCUITS. 347
waves to enter them from the dielectric and be dissipated, but would
let them slip along like greased lightning. Then there is a kind of
circuit which is distortionless, but in which there is considerable attenu-
ation. Here, plainly, any distance can be worked through, provided
the attenuation is not too great. Trial alone could settle how far it
would be practicable with a given type. Coming to more practical
cases, there is the approximately distortionless circuit above described.
Here the attenuation is not nearly so great as in the distortionless
circuit of the same type (that is, only differing in the leakage needed
to remove the remaining distortion), so that the distance to be worked
through is much greater with similarly sensitive instruments, or with
instruments graduated to make the currents received and sounds
produced be about equal in the different cases compared. Here, again,
trial alone can settle how far we may work safely. Supposing, for
instance, we had reached a practical limit with nearly distortionless
transmission, it is clear that we could increase that limit by the simple
expedient of increasing the current sent out or the sensitiveness of the
receiver. So we cannot fix a limit at all on theoretical principles.
But undoubtedly the distortion will increase as the circuit is lengthened
(except in the ideal distortionless circuit) ; this will tend to fix a limit,
though we cannot precisely define it, independently of the attenuation.
Nor should interferences be forgotten, and their distorting effects.
When thousands of miles are in question, many other things may
come in to interfere, all tending to fix a limit. Independently of the
line, too, there are the terminal arrangements to be considered. A
practical limit in a given case might be fixed merely by the inadequate
intensity of the received currents to work the receiver suitably. But
apart from intensity of action, both the transmitter and the receiving
telephone distort the proper "signals" themselves. The distortion
due to the electrical part of the receiver may, however, be minimized
by a suitable choice of its impedance, and especially by making its
inductance the smallest possible consistent with the possession of the
other necessary qualifications. The conditions as regards perfect
silence in reception are also of importance. Finally, there is " personal
equation." It is clear, then, that in such a mixed-up problem as this
is, we cannot safely estimate what amount of distortion is permissible
in transit along the circuit, and how much attenuated and distorted
we may allow the vibrations to become before human speech ceases to
be recognisable as such, and to be intelligibly guessable.
It is, however, surprising what a large amount of distortion is
permissible, not merely on long lines, but on short ones. It is, indeed,
customary, or certainly was on the first introduction of the telephone,
and for long after, for people to enlarge upon the wonderful manner
in which a receiving telephone exactly reproduces, in all details, the
sounds that are communicated to the transmitter, and to be astonished
at the power the disc possesses of doing it, and to explain it by
harmonic analysis, and so forth. Well, the disc does not do it. If it
did, as it would be in quite mechanical obedience to the forces acting
upon it, there would be nothing to wonder at ; or the reason for wonder
348 ELECTRICAL PAPERS.
would be shifted elsewhere. It would be really wonderful if we could
get perfect reproduction of speech. The best telephony is bad to the
critical ear, if a high standard be selected, and not one based upon
mere intelligibility. (As a commentary upon the reports of " perfect
articulation," etc., I may mention that we sometimes see the amusingly
innocent remarks added that even whistling could be heard, and one
voice distinguished from another.) Consider the difficulties in the
way. We cannot even make the diaphragm of the transmitter precisely
follow the vibrations set up by the vocal organs (which vibrations are, by
the way, distorted between the larynx and the diaphragm, though this
is not an important matter), because it is not a dead-beat arrangement,
and responds differently to different tones. Here is one cause of
distortion. A second occurs in trying to make the primary current
variations copy the motion of the diaphragm. A third is in the
transformation to the secondary circuit, though perhaps this and the
last transformation may be taken together with advantage. So to
begin with, we have considerably distorted our signals before getting
them on to the telephone line. Then, there is the distortion in transit,
which may be very little or very great, according to the nature of the line.
Next, the received-current variations ought to be exactly copied by the
magnetic stress between the disc and magnet of the receiver. But the
inductance of the receiver prevents that, even if the resistance be
suitably chosen to nearly stop the reaction of the instrument on the
line. Then we should get the disc of the telephone to exactly copy
the magnetic- force variations, which it cannot do at all well, on account
of the want of dead-beatness, and the augmentation of certain tones
and weakening of others. The remaining transformations, from the
brain to the vocal organs at one end, and from the disc to the brain
via the air and ear at the other end of the circuit, we need not consider.
And yet, after all these transformations and distortions, practical
telephony is possible. The real explanation is, I think, to be found
in the human mind, which has been continuously trained during a
lifetime (assisted by inherited capacity) to interpret the indistinct
indications impressed upon the human ear ; of which some remarkable
examples may be found amongst partially deaf persons, who seem to
hear very well even when all they have to go by (which practice makes
sufficient) is as like articulate speech as a man's shadow is like the man.
In connection with these transformations, I may mention that one
of them, viz., in the telephone receiver itself, was until recently un-
explained. Writers have before now remarked upon the necessity of a
permanent magnetic field, and speculated as to its cause, and recently
Prof. Silvanus Thompson recalled attention to the matter, and candidly
confessed his ignorance of the explanation, beyond what was furnished
by M. Giltay, who had also considered the matter, and found that the
permanent field was needed to eliminate the vibrations of doubled
frequency that would result were there no permanent field. This is
true in a sense ; but it is not the really important part of what is,
I think, the true explanation, because the vibrations of doubled
frequency would be very feeble. What the permanent field does is
ON TELEGRAPH AND TELEPHONE CIRCUITS. 349
to vastly magnify the effect of the weak telephonic currents, and make
them workable. The disc is attracted by the magnet, and the stress
between them varies as the square of the intensity of magnetic force in
the intermediate space. We want the disc to vibrate sensibly by very
weak variations of magnetic force. If the permanent magnet were not
there, we should have insensible vibrations of doubled frequency. But
the permanent field makes the stress-variations vary as the product of
the intensity of the permanent field and that of the weak variation due
to the current-variations ; they are therefore proportional to the received
current-variations, and are also greatly magnified, so that the telephone
becomes efficient. [See Art. xxxvi., vol. n., p. 155.]
Returning to the telephone-circuit itself, the following would appear
to be what should be aimed at (apart from improvements in terminal
transmission and reception) in efficient long-distance telephony. Setting
up an arbitrary train of disturbances at one end, causing the despatch
of a continuously varying train of waves into the circuit, the waves
should travel to the distant end of the line as little distorted as possible,
and with as nearly equal attenuation as possible, which attenuation
should not be too great ; and, finally, on reaching the terminal
telephone, the waves should be absorbed by it, as nearly as possible,
without reflex action. This ideal may be illustrated by a long cord,
along which we can, by forcibly agitating one end, despatch a train
of waves, which travel along it only slightly distorted, and which
should then be absorbed by some mechanical arrangement at the
further end. Theoretically this only needs the further end to have its
motion resisted by a force proportional to its velocity, the coefficient
of resistance depending upon the mass and tension of the cord.
At any intermediate point we may correctly register the disturbances
passing it. It is evident that the reflected wave from the distant end
should be done away with, in order that the disturbances passing (and
reaching the distant end) may be a correct copy of those originally
despatched. This ideal state of things is fairly-well reached in the
fourth class of circuits above mentioned, and perfectly in the fifth
class, whilst the low-resistance long-distance circuits introduced in
America are somewhere between the third and the fourth classes.
In passing from the fourth class to the third, by increasing the
resistance of the line from very low to more common values, the effect
is to introduce a considerable amount of distortion which may be
(somewhat imperfectly) ascribed to electrostatic retardation. The
limiting distance of telephony will therefore now depend more upon
the circuit itself (apart from terminal arrangements) than before. Still
we cannot fix it. Only by passing to the extreme case of such high
resistance of the line acting in conjunction with the permittance that
the effect of inertia is really insensible, do we so magnify the effect of
the distortion in transit as to make the limiting distance be determined
approximately by the value of the electrostatic time-constant JtSl 2 .
We now come to the first class we began with, and Sir W. Thomson's
law of the squares may be applied in making comparisons. The dis-
tortion in transit is very great, if the line be long, and we therefore to
350
ELECTRICAL PAPERS.
some extent swamp the terminal apparatus as regards the total dis-
tortion.
But there is only a tendency to the electrostatic theory, not a com-
plete fulfilment. In the case of a cable of the Atlantic type, used as a
telephone-circuit (of course not across the Atlantic) the resistance is
rather low, and this is quite sufficient, in conjunction with the induct-
ance, to greatly improve matters from the electrostatic theory, in spite
of the large permittance. In fact, a small amount of inductance is
sufficient to render telephony possible under circumstances which would
preclude possibility were it non-existent. To show this, consider the
following table :
n.
L=0.
L = 2'5.
L=5.
= 10.
1250
1-723
1-567
1-437
1-235
2500
3-431
2-649
2-251
1-510
5000
10-49
5-587
3-176
1-729
10,000
58-87
10-496
4-169
1-825
20,000
778
16-707
4-670
1-854
In the first column we have the frequency-constant n ITT x frequency,
so that the frequency ranges through four octaves. It is supposed that
the resistance is 4 ohms and the permittance J microf. per kilom., being
somewhat like what obtains in an Atlantic cable. The remaining
columns show the values of the equivalent impedance p at the distant
end according to the already-given formula (4), with the values of L
given at the tops of the columns. (Take # = in (4).)
Thus in the second column we have the figures given by the electro-
static theory, showing such an extremely rapid increase of attenuation
with the frequency that telephony would I think be quite impossible.
But the third column shows that the small inductance of 2-5 per
centim. immensely improves matters, especially with the great fre-
quencies.
The fourth column, with L = 5, shows a far greater improvement,
and I should think good telephony would be possible.
The fifth column, with L = W, is very remarkable, as it shows an
approach to distortionless transmission.
This remarkable result is wholly due to the inductance, in presence
of the rather low resistance. Whereabouts the effective inductance
really lies it is hard to say, but it must surely be greater than 2-5,
though it may not be much more, as the iron sheathing does not make
the effective L run up in the way that might be supposed at first sight.
With Z = 0, n = 10,000 makes /> = 58, or the received current 1/58 of
the steady current. To have the same result in our low-resistance
circuit, we see by the first table that Pd = \5Lv about does it, giving
HI =15 x 600 = 9000 ohms, and Z = 4500 kilom. Now is it possible to
work a telephone fairly well through a mere resistance of 58 x 9000 or
say 50,000 ohms (ignoring complications due to the telephone not
being a mere resistance), remembering that our currents will be fairly
ON TELEGRAPH AND TELEPHONE CIRCUITS. 351
uniformly attenuated ? If so, then this circuit of 4500 kilom. will work
with good articulation, under favourable conditions freedom from
interferences, etc. But I do not fix this limit, nor any, for reasons
before given.
This difference should be noted. In the case of the cable of no
inductance, the reduction to 1/58 part applies only to %=10,000. If
?i=1250, at the lower limit, the reduction is only to 10/17 of the
steady current ; thus there is plenty of sound, but very inarticulate.
This is the reverse of what occurs in our other case, in which there is
little sound, but with good articulation, and therefore usefully admitting
of magnification.
If, on the other hand, we take the electrostatic time-constant as *02
second, the attenuation at n = 10,000 is, by the second table, to 1/778 of
the steady current ; and this value, by the first table, gives PdjLv = say
20, and HI 12,000 and = 6000 kilom., and the equivalent impedance
= 778 x 12,000 ohms. Of course this is excessively large. If com-
ponent vibrations on a cable really suffer attenuation to 1/778 part,
such vibrations might as well be altogether omitted, leaving only the
lower tones. On the other hand, a sufficient magnification in the
6000 kilom. case would render telephony possible. But the probable
fact is that '01 second with L Q is not possible, far less '02 second.
When it is said to be done, the reason is that L is not zero. In the
north of England examples there are usually buried wires and overhead
wires in sequence, so that it is still more true that self-induction comes
in to help, although the theory of such composite circuits cannot be
easily brought down to numerical calculation.
But, returning to the 4500 kilom. example, it appears reasonable that
the circuit might be worked under favourable circumstances. Let us
see what its electrostatic time-constant is. We get, by (11), SQ
microf. per kilom. Hence
which is no less than 22 times the supposed maximum of -01 second.
Even if we make a large allowance, and suppose that an attenuation to
^ part only of the steady current, instead of -% part, is the utmost
allowable, we shall see by the table that this makes Pd = \\Lv (instead
of the previous 15), so that the electrostatic time-constant is still a
large multiple of the value -01 obtained by observation of wires of high
resistance.
Again, to contrast the two theories, let us inquire what length of line
makes '01 sec. the electrostatic time-constant. The result is 300^10
or say 900 kilom., of resistance 1800 ohms, which is only three times Lv ;
so that there is nearly perfect transmission on the line of low resistance,
whilst there is extreme distortion on the circuit having the same electro-
static time-constant if destitute of inductance.
Since there is a minimum value of the attenuation-ratio I/ HI when
the ratio El/Lv is variable, let it be merely L that is variable, without
change of length or resistance. This may be done by simply varying
352 ELECTRICAL PAPERS.
the distance between the two wires in the circuit. The minimum
attenuation at the distant end comes about (by first table) when
Rl El JRlohms
When / = 600 kilom. we have L = 20, as we saw before. If / = 300 kilom.,
then L = 1 0, which change is easily made by bringing the wires closer.
But if I = 1200 kilom., we require L = 40, and a wide separation is neces-
sary, according to equation (12). But there is another thing to be
remembered. The distance between the wires should continue to be a
small fraction of the height above the ground, in order that the property
LSv 2 = 1 should remain fairly true. Although the permittance does not
appear explicitly in formula (14), it is implicitly present in v, and in
such a way that a doubling of S and halving of L are equivalent. (But
this does not apply to the table, where L and S may vary independently.)
Now, if we separate wires very widely without raising them any higher,
S tends to become simply the reciprocal of the sum of the elastances
from the first wire to earth and from the earth to the second wire ;
that is, half the permittance of either. It therefore tends to constancy
instead of varying inversely as L, which goes on increasing slowly as
the wires are further separated. Hence the necessity of raising the
wires, as well as of separating them, if the full advantage of L is to be
secured when it is large.
In passing, I may add that if the earth were perfectly conducting, so
as to shut out the magnetic field from itself, the product LSv' 2 , where L
is the inductance of the dielectric and S its permittance, calculated so
as to suit the propagation of plane-waves, would remain unity always,
however the wires were shifted, provided parallelism were maintained.
It seems at first sight anomalous that when the permittance is so
small that we might expect the common magnetic formula to apply, we
should increase the amplitude of current of any (not too low) frequency
by increasing the inductance. It seems to show how careful we should
be not to extend too widely the application of professedly approximate
formulae. Equation (4) has quite different significations under varied
circumstances ; and, general as it is, it is yet not general enough to
meet extreme cases, even when, as in my original statement of it (The
Electrician, July 23, 1886) [vol. II., p. 61], the increased resistance
and reduced inductance due to the tendency towards skin-conduction
are allowed for. Besides the propagation of disturbances through the
dielectric following the wires, after the manner of plane- waves, there is
an outward propagation from the source of energy, which seems to me,
however, to be quite a secondary matter, and insignificant, especially
when the circuit is a metallic loop, which concentrates the electro-
magnetic field considerably. But when there is an earth-return, there
is a wide extension of the magnetic field, and distances from the line
should be compared with its length, in making estimates of the range of
disturbances of appreciable magnitude, appreciable by cumulative action
on a distant wire. There are also the modifications due to the presence
of neighbouring wires, which may be calculated by the equations of a
ON TELEGEAPH AND TELEPHONE CIRCUITS. 353
system of parallel wires. But perhaps the most important modifying
influence of all is that of the terminal apparatus.
I have considered the effect of any terminal apparatus in my paper,
" On the Self-induction of Wires," Part V., [vol. IL, p. 247]. It is very
complex in general. But so far as relates to a long circuit of low resist-
ance, we do not want the full formulae. Take (17) as the formula
when the wires are short-circuited at the sending and receiving ends.
Then, when we put on terminal apparatus containing no impressed
force except the one sinusoidally varying force at the beginning of the
circuit, (which may be in any part of the main circuit of the terminal
apparatus there), the result is to alter the attenuation-ratio from the
former /> to p v given by
ft-pxejxflf, ............................ (19)
where G$ and G$ are the terminal factors for the sending and receiving
ends, to be calculated in the following manner. Let R^ and L^ be the
" effective " resistance and inductance of the apparatus at the receiving
end, then
(20)
without assumptions regarding the size of / and g. Now take g = 0,
and / a small fraction, and we reduce (20), when the fraction fLfl/Lv
is small, to
G^l+EJLv ............................ (21)
Therefore (19) becomes
P^bLv.e^l+EJLv^l+BJLv) ................ (22)
Note that the full expression for G is obtainable from (20) by changing
A\ and L^ to R Q and L Q . But if we only assume / to be small and g
zero, then, instead of (21), we have
1 = (1 + E l /Lv^ + (L l n/Lv)(L 1 n/Lv-f)+f 2 (R l /Lv) ......... (23)
Now let it be merely a telephone that is the receiving apparatus, of
resistance and inductance R 1 and L lt or something equivalent to a mere
coil. If it be a mere coil, and also, though less easily, if a telephone,
we may vary L^ independently by changing the form of the coil or by
inserting non-conducting iron. We see, then, that the terminal factor
is made a minimum, with L^ alone variable, when
which, with 72 = 20 4 and w=10 4 makes 2Z 1 = 60 6 , quite a reasonable
value for a small telephone. But if w = 20 3 , the result is 150 7 , twenty-
five times as large.
Next let it be, not the current, but the magnetic force of the coil
that is a maximum, on the assumption that L-JRy the time-constant of
the coil, is fixed. This is nearly true when the size of the wire is
varied, if it be a mere coil that is concerned, and is an approach to the
H.E.P. -VOL. II. Z
354 ELECTRICAL PAPERS.
truth when there is iron. It is now G l /R l that has to be a minimum,
subject to RiJL-L = constant. This happens when
)* = Li>, ........................... (24)
or when the impedance of the coil equals the critical Lv.
I showed in my paper " On Electromagnets," etc. [Art. xvn., vol. I.,
p. 99], that in the magnetic theory the condition of maximum magnetic
force of the coil is that its impedance should equal that of the rest of
the circuit, which contains the impressed force. We may easily verify
that Lv is the impedance in the present case (with / small). Now
Lv = 600 ohms when L = 20 ; this is the extreme value of the resistance
of the coil, which should really be less on account of the term L^n.
For instance, if the time-constant be '0002 second, and ?i=10 4 , we
require 2'24 : B l = Lv. We see further that this does make fL-^n/Lv
small, because / is small, and L l njLv< 1. Therefore, using (23), we
have
nW)-}, ........... (25)
which, with % = 10 4 and the time-constant a = -0002, becomes G% = 1*7.
This is, of course, a far larger value of the terminal factor than need
be. In fact, the conditions of maximum magnetic force of the coil and
of maximum received current are not usually identical, and may be
quite antagonistic. For instance, if we should make the terminal
factor nearly unity, we should have the biggest current, but with the
least power.
But a remarkable property should be mentioned, which may be
proved by the general formula from which (19) is derived. It is that
if the receiver be a mere resistance, the choice of its resistance to equal
Lv will, when RjLn is small, nearly annihilate the reflected wave, and
so do away with the fluctuations and the distortion due to them,
whether the circuit be a long or a short one. Under these circum-
stances we have practically perfect reception of signals.
The general condition making G^R a minimum on a long circuit,
subject to constancy of a, is by (19) and (20),
The right member expresses the square of the impedance of the circuit to
a S.H. impressed force at its end. When/ and g are small we obtain the
former result. The property of equal impedances is, however, a general
one, so that all we do in verifying it is to see that no glaring error has
crept in. If a coil connect two points of any arrangement in which a
S.H. state is kept up by impressed force, and we vary the size of wire
without varying the size and shape of the coil, we bring the magnetic
force of the coil to a maximum by making its impedance equal to that
external to it, if the thickness of covering vary similarly to that of the
wire.
RESISTANCE AND CONDUCTANCE OPERATORS. 355
XLIL ON RESISTANCE AND CONDUCTANCE OPERATORS,
AND THEIR DERIVATIVES, INDUCTANCE AND PER-
MITTANCE, ESPECIALLY IN CONNECTION WITH
ELECTRIC AND MAGNETIC ENERGY.
[Phil. Mag., December, 1887, p. 479.]
General Nature of the Operators.
1. IF we regard for a moment Ohm's law merely from a mathematical
point of view, we see that the quantity E, which expresses the resist-
ance, in the equation V=RC, when the current is steady, is the
operator that turns the current C into the voltage V. It seems, there-
fore, appropriate that the operator which takes the place of R when
the current varies should be termed the resistance-operator. To
formally define it, let any self-contained electrostatic and magnetic
combination be imagined to be cut anywhere, producing two electrodes
or terminals. Let the current entering at one and leaving at the other
terminal be C, and let the voltage be P] this being the fall of potential
from where the current enters to where it leaves. Then, if V= ZC be
the differential equation (ordinary, linear) connecting V and C, the
resistance-operator is Z.
All that is required to constitute a self-contained system is the
absence of impressed force within it, so that no energy can enter or
leave it (except in the latter case by the irreversible dissipation con-
cerned in Joule's law) until we introduce an impressed force; for
instance, one producing the above voltage J^at a certain place, when
the product VQ expresses the energy-current, or flux of energy into the
system per second.
The resistance-operator Z is a function of the electrical constants of
the combination and of d/dt, the operator of time-differentiation, which
will in the following be denoted by p simply. As I have made ex-
tensive use of resistance-operators and connected quantities in previous
papers,* it will be sufficient here, as regards their origin and manipu-
lation, to say that resistance-operators combine in the same way as if
they represented mere resistances. It is this fact that makes them of
so much importance, especially to practical men, by whom they will be
much employed in the future. I do not refer to practical men in the
very limited sense of anti- or extra-theoretical, but to theoretical men
who desire to make theory practically workable by the ' simplification
and systematisation of methods which the employment of resistance-
operators and their derivatives allows, and the substitution of simple
for more complex ideas. In this paper I propose to give a connected
account of most of their important properties, including some new ones,
especially in connection with energy, and some illustrations of extreme
cases, which are found, on examination, to " prove the rule."
2. If we put p = in the resistance-operator of any system as above
defined, we obtain the steady resistance, which we may write Z Q . If
all the operations concerned in Z involve only differentiations, it is
* Especially Part III. , and after, "On the Self-induction of Wires," [vol. n.,
pp. 201 to 361 generally. Also vol. I., p. 415].
356 ELECTRICAL PAPERS.
clear that when C is given completely, V is known completely. But if
inverse operations (integrations) have to be performed, we cannot find
V immediately from C completely ; but this does not interfere with the
use of the resistance-operator for other purposes.
It is sometimes more convenient to make use of the converse method.
Thus, let Y be the reciprocal of Z, so that C = YV. If we make p
vanish in Y, the result, say Y Q) is the conductance of the combination.
Therefore F is the conductance -operator.
The fundamental forms of Y and Z are
................................. (1)
(2)
In the first case, it is a coil of resistance R and inductance L that is in
question, with the momentum LC and magnetic energy ^LC 2 . In the
second case, it is a condenser of conductance K and permittance S, with
the charge iSFand electric energy ^SF 2 ; or its equivalent, a perfectly
nonconducting condenser having a shunt of conductance K.
In a number of magnetic problems (no electric energy) the resistance-
operator of a combination, even a complex one, reduces to the simple
form (1). The system then behaves precisely like a simple coil, so far
as externally impressed force is concerned, and is indistinguishable
from a coil, provided we do not inquire into the internal details. I
have previously given some examples.* Substituting condensers for
coils, permittances for inductances, we see that corresponding reductions
to the simple form (2) occur in electrostatic combinations (no magnetic
energy).
But such cases are exceptional; and, should a combination store
both electric and magnetic energy, it is not possible to effect the above
simplifications except in some very extreme circumstances. There are,
however, two classes of problems which are important practically, in
which we can produce simplicity by a certain sacrifice of generality.
In the first class the state of the whole combination is a sinusoidal or
simple-harmonic function of the time. In the second class we ignore
altogether the manner of variation of the current, and consider only
the integral effects in passing from one steady state to another, which
are due to the storage of electric and magnetic energy.
S.H. Fixations, and the effective K', I/, K', and S'.
3. If the voltage at the terminals be made sinusoidal, the current
will eventually become sinusoidal in every part of the system, unless it
be infinitely extended, when consequences of a singular nature result.
At present we are concerned with a finite combination. Then, if nftir
be the periodic frequency, we have the well-known property p 2 = - n 2 ;
which substitution, made in Z and F, reduces them to the forms
................................... (3)
.................................. (4)
* "On the Self-induction of Wires," Parts VI. and VII. [vol. n., pp. 268 and
292.]
RESISTANCE AND CONDUCTANCE OPERATORS. 357
where R f , Z/, K f , S f are functions of the electrical constants and of w 2 ,
and are therefore constants at a given frequency.
In the first case we compare the combination to a coil whose resist-
ance is R' and inductance L f , so that R f and L' are the effective resist-
ance and inductance of the combination, originally introduced by Lord
Rayleigh* for magnetic combinations. In my papers, however, there is
no limitation to cases of magnetic energy only,f and it would be highly
inconvenient to make a distinction.
In a similar way, in the second case we compare the combination to
a condenser, and we may then call K f the effective conductance and S f
the effective permittance at the given frequency. R' reduces to Z^
and K f to Y Q at zero frequency. But it is important to remember that
the two comparisons are of widely different natures : and that the
effective resistance [in the coil-comparison] is not the reciprocal of the
effective conductance [in the condenser-comparison].
Fand Z in (3) and (4) are reciprocal, or YZ= 1, just as the general
Y and Z of (1) and (2) are reciprocal.
If (V) and (C) denote the amplitudes of Fand (7, we have, by (3)
and (4),
= I, say, ..................... (5)
, say ...................... (6)
/ and / are also reciprocal. The former, /, being the ratio of the force
to the flux (amplitudes), is the impedance of the combination. It is
naturally suggested to call / the " admittance " of the combination.
But it is not to be anticipated that this will meet with so favourable a
reception as impedance, which term is now considerably used, because
the methods of representation (1), (3), and (5) are more useful in
practice than (2), (4), and (6) ; although theoretically the two sets are
of equal importance. }
To obtain the relations between R' and K r , and L f and /S", we have
.................... (7)
', .................... (8)
from which we derive
i
\ ............
, }
(9)
R'\K' = P = - L'l&,
all of which are useful relations.
* Phil. Mag., May, 1886.
t In Part V. of " On the Self -Induction of Wires " I have given a few examples
of mixed cases of an elementary nature, in connexion with the problem of finding
the effect of an impressed force in a telegraph circuit.
% The necessity of the term impedance (or some equivalent) to take the place of
the various utterly misleading expressions that have been used, has come about
through the wonderful popularisation of electromagnetic knowledge due to the
dynamo, and its adoption to Sir W. Thomson's approval of it and of one or two
other terms.
358 ELECTRICAL PAPERS.
4. By (3) and (4) we have the equations of activity
(10)
....................... (11)
in general. Now, if we take the mean values, the differentiated terms
go out, leaving
VC = R'~& = K'V\ ............................ (12)
the bars denoting mean values. The three expressions in (12) each
represent the mean dissipativity, or heat per second. E' and K' are
therefore necessarily positive. It should be noted that R f C* or K'V^
do not represent the dissipativity at any moment. The dissipativity
fluctuates, of course, because the square of the current fluctuates ; but
besides that, there is usually a fluctuation in the resistance, because the
distribution of current varies, and it is only by taking mean values that
we can have a definite resistance at a given frequency.
If the combination be magnetic, and T denote the magnetic energy,
its mean value is given by
T=$I/C* t ............................... (13)
so that L f is necessarily positive and S f negative. But ^I/C Z is not
usually the magnetic energy at any moment.
If the combination be electrostatic, and U denote the electric energy,
its mean value is
Z7=i'F 2 , ............................. (14)
so that S' is positive and U negative. The electric energy at any
moment is not usually S'V**.
But, in the general case of both energies being stored, we have
T- U=$UC*=-\S'V*. ....................... (15)
If the mean magnetic energy preponderates, the effective inductance
is positive, and the permittance negative ; and conversely if the electric
energy preponderates. If there be no condensers, the comparison
with a coil is obviously most suitable, and if there be no magnetic
energy we should naturally use the comparison with a condenser ; but
when both energies coexist, which method of representation to adopt is
purely a matter of convenience in the special application concerned.
If the mean energies, electric and magnetic, be equal, then
I/ = = 8', R'K'=\, I=R', J = K' ............ (16)
That is, by equalising the mean energies we bring the current and
voltage into the same phase, annihilate the effective inductance (and
also permittance), and make the effective conductance the reciprocal of
the effective resistance, which now equals the impedance itself. It
should be noted that the vanishing of the energy-difference only refers
to the mean value. The two energies are not equal and do not vanish
simultaneously. Sometimes, however, their sum is constant at every
moment, but this is exceptional. (Example, a coil and a condenser in
sequence.)
RESISTANCE AND CONDUCTANCE OPERATORS. 359
Impulsive Inductance and Permittance. General Theorem relating to the
Electric and Magnetic Energies.
5. Passing now to the second class referred to in 2, imagine, first,
the combination to be magnetic, and that V is steady, producing a
steady (7, dividing in the system in a manner solely settled by the dis-
tribution of conductivity. Although we cannot treat the combination
as a coil as regards the way the current varies when the impressed force
is put on, we may do so as regards the integral effect at the terminals
produced by the magnetic energy. The last is the well-known
quadratic function of the currents in different parts of the system,
T=\Ll + MC& + \L\ + (17)
Now put every one of these C"s in terms of the C, the total current at
the terminals, which may be done by Ohm's law. This reduces T to
T=$L C*, (18)
where L Q is a function of the real inductances, self and mutual, of the
parts of the system, and of their resistances. This L may be called the
impulsive inductance of the system. For although it is, in a sense, the
effective steady inductance, taking the current C at the terminals as a
basis, being, in fact, the value of the sinusoidal inductance L r at zero
frequency; yet, as it is only true for impulses that the combination
behaves as a coil of inductance Z , it is better to signify this fact in the
name, to avoid confusion. This will be specially useful in the more
general case in which both energies are concerned.
Secondly, let the system be electrostatic. Then, in a similar way, we
may write the electric energy in the form
U=iS^, (19)
in terms of the T^at the terminals, where S is a function of the real
permittances and of the resistances. $ is the impulsive permittance of
the combination. It is also the sinusoidal S r at zero frequency.
In (18) L Q is positive, arid in (19) $ is positive. The momentum or
electromotive impulse [or the voltaic impulse, if we use the modern
"voltage" to signify the old "electromotive force"] at the terminals in
the former case is L C, and in the latter case is - S^RV, where R is the
steady resistance. The true analogue of momentum, however, is charge,
or time-integral of current, and this, at the terminals, is - $ V, corre-
sponding to L Q C.
6. Passing to the general case, and connecting with the resistance-
operator, let F be the current at the terminals at time t when varying,
so that
F-&m(fi+'pBi+&*F+..;.W (20)
where the accents denote differentiations to p, and the zero suffixes
indicate that the values when p = are taken. The coefficients of the
powers of p are therefore constants. Integrating to the time,
+ $zi'[t] + (21)
360 ELECTRICAL PAPERS.
If the current be steady at beginning and at end,
^(F-z Q r)dt=z>[ri .......................... (22)
and if the initial current be zero, and the final value be C,
^C; ........................... (23)
so that ZQ is the voltaic impulse employed in setting up the magnetic
and the electric energy of the steady state due to steady V at the
terminals. Thus
L = Z' ................................. (24)
finds the impulsive inductance from the resistance-operator. Or,
L Q = (Z-Z Q )p~' i with p = Q .................... (25)
In a similar manner, we may show that
S = F{= -Z?Z[ ............................ (26)
finds the impulsive permittance from the conductance-operator. L Q C
and -iS ^ Fare equivalent expressions for the voltaic impulse.
If Z Q should be infinite, then use Y. For instance, the insertion of a
nonconducting condenser of permittance S l in the main circuit of the
current makes Z infinite, since the resistance-operator of the condenser
is (Stf)' 1 . There is no final steady current, and L Q is infinite. We
should then use (26) instead of (24), especially as the energy is wholly
electric in the steady state.
7. To connect with the energy, multiply (23) by (7, the final current,
and, for simplicity, let V be steady ; giving
((7-RT}Cdt=Z^=(F(C-T)dt ................ (27)
It may be anticipated from the preceding that these equated quantities
express twice the excess of the magnetic over the electric energy.
In connexion with this I may quote from Maxwell, vol. ii., art. 580.
A purely electromagnetic system is in question. "If the currents are
maintained constant by a battery during a displacement in which a
quantity of work, W, is done by electromotive force, the electrokinetic
energy of the system will be at the same time increased by W. Hence
the battery will be drawn upon for a double quantity of energy, or 2/F,
in addition to that which is spent in generating neat in the circuit.
This was first pointed out by Sir W. Thomson. Compare this result
with the electrostatic property in art. 93." The electrostatic property
referred to relates to conductors charged by batteries. If " their poten-
tials are maintained constant, they tend to move so that the energy of
the system is increased, and the work done by the electrical forces
during the displacement is equal to the increment of the energy of the
system. The energy spent by the batteries is equal to double of either
of these quantities, and is spent half in mechanical, half in electrical
work."
Although of a somewhat similar nature, these properties are not
RESISTANCE AND CONDUCTANCE OPERATORS. 361
what is at present required, which is contained in the following general
theorem given by me* : Let any steady impressed electric forces be
suddenly started and continued in a medium permitting linear relations
between the two forces, electric and magnetic, and the three fluxes
conduction current, electric displacement, and magnetic induction (but
with no rotational property allowed, even for conduction current) ; the
whole work done by the impressed forces during the establishment of
the steady state exceeds what would have been done had this state been
instantly established (but then without any electric or magnetic energy)
by twice the excess of the electric over the magnetic energy. That is,
(28)
where e stands for an element of impressed force, F the current-density
at time t, F the final value, and 2 the space-integration to include all
the impressed forces. (Black letters for vectors.) The theorem (28)
seems the most explicit and general representation of what has been
long recognised in a general way, that permitting electric displacement
increases the activity of a battery, whilst permitting magnetisation
decreases it. The one process is equivalent to allowing elastic yielding,
and the other to putting on a load (not to increasing the resistance, as
is sometimes supposed).
Applying (28) to our present case of one impressed voltage V, pro-
ducing the final current C, we obtain
T), ,...(29)
comparing which with (27), we see that
T- U-WP-tLjP- - JS F* (30)
confirming the generality of our results.
General Theorem of Dependence of Disturbances solely on the Curl of the
Impressed Forcive.
8. It is scarcely necessary to remark that the properties of Z and Z f
previously discussed do not apply merely to combinations consisting of
coils of fine wire and condensers ; the currents may be free to flow in
conducting masses or dielectric masses. Solid cores, for example, may
be inserted in coils within the combination. The only effect is to make
the resultant resistance-operator at a given place more complex.
But a further very remarkable property we do not recognise by
regarding only common combinations of coils and condensers. If we,
in the complex medium above defined, select any unclosed surface, or
surface bounded by a closed line, and make it a shell of impressed
voltage (analogous to a simple magnetic shell), thereby producing a
potential-difference V between its two faces, and C be the current
through the shell in the direction of the impressed voltage, there must
be a definite resistance-operator Z connecting them, depending upon
* Electrician, April 25, 1885, p. 490, [vol. i., p. 464.]
362 ELECTRICAL PAPERS.
the distribution of conductivity, permittivity, and inductivity through
all space, and determinable by a sufficiently exhaustive analysis. The
remarkable property is that the resistance-operator is the same for any
surfaces having the same bounding-edge. For a closed shell of im-
pressed voltage of uniform strength can produce no flux whatever.
This is instructively shown by the equation of activity,
........................... (31)
indicating that the sum of the activities of the impressed forces, or the
energy added to the system per second, equals the total dissipativity Q,
plus the rate of increase of the stored energies, electric and magnetic,
throughout the system. Now here F is circuital; if, therefore, the
distribution of e be polar, or e be the vector space-variation of a single-
valued scalar potential, of which a simple closed shell of impressed
force is an example, the left member of (31) vanishes, so that the dis-
sipation, if any, is derived entirely from the stored energy. Start,
then, with no electric or magnetic energy in the system ; then the
positivity of Q, U, and T ensures that there never can be any, under
the influence of polar impressed force. Hence two shells of impressed
force of equal uniform strength produce the same fluxes if their edges
be the same ; not merely the steady fluxes possible, but the variable
fluxes anywhere at corresponding moments after commencing action.
The only difference made when one shell is substituted for the other is
in the manner of the transfer of energy at the places of impressed force;
for we have to remember that the effective force producing a flux, or
the "force of the flux," equals the sum of the impressed force and the
" force of the field " ; whereas the transfer of energy is determined by
the vector product of the two forces of the field, electric and magnetic
respectively. In (31) no count is taken of energy transferred from one
seat of impressed force to another, reversibly, all such actions being
eliminated by the summation.
It is well to bear in mind, when considering the consequences of this
transferability of impressed force, especially in cases of electrolysis or
the Volta-force, not only that the three physical properties of con-
ductivity, permittivity, and inductivity, though sufficient for the state-
ment of the main facts of electromagnetism, are yet not comprehensive,
but also that they have no reference to molecules and molecular actions;
for the equations of the electromagnetic field are constructed on the
hypothesis of the ultimate homogeneity of matter, or, in another form,
only relate to elements of volume large enough to allow us to get rid of
the heterogeneity.
As the three fluxes are determined solely by the vorticity (to borrow
from liquid motion) of the vector impressed force, we cannot know the
distribution of the latter from that of the former, but have to find
where energy transformations are going on ; for the denial of the law
that eF not only measures the activity of an impressed electric force e
on the current F, but represents energy received by the electromagnetic
system at the very same place, lands us in great difficulties.
Again, as regards the " electric force of induction." We cannot find
RESISTANCE AND CONDUCTANCE OPERATORS. 363
the distribution through space of this vector from the Faraday-law
that its line-integral in a closed circuit equals the rate of decrease of
induction through the circuit. We may add to any distribution
satisfying this law any polar distribution without altering matters,
except that a different potential function arises. In this case we do
not even alter the transfer of energy. The electric force of the field is
always definite but when we divide it into two distinct distributions,
and call one of them the electric force of induction, and the other the
force derived from electric potential, it is then quite an indeterminate
problem how to effect the division, unless we choose to make the quite
arbitrary assumption that the electric force of induction has nothing of
the polar character about it (or has no divergence anywhere), when of
course it is the other part that possesses the whole of the divergence.
This fact renders a large part of some mathematical work on the
electromagnetic field that I have seen redundant, as we may write
down the final results at the beginning. In the course of some in-
vestigations concerning normal electromagnetic distributions in space
I have been forcibly struck with the utter inutility of dividing the
electric field into two fields, and by the simplicity that arises by not
doing so, but confining oneself to the actual forces and fluxes, which
describe the real state of the medium and have the least amount of
artificiality about them. Similar remarks apply to Maxwell's vector-
potential A. Has it divergence or not ? It does not matter in the
least, on account of the auxiliary polar force. When the electric force
itself is made the subject of investigation, the question of divergence of
the vector-potential does not present itself at all.
The lines of vorticity, or vortex-lines of the vector impressed force,
are of the utmost importance, because they are the originating places
of all disturbances. This is totally at variance with preconceived
notions founded upon the fluid analogy, which is, though so useful in
the investigation of steady states, utterly misleading when variable
states are in question, owing to the momentum and energy belonging
to the magnetic field, not to the electric current. Every solution
involving impressed forces consists of waves emanating from the vortex-
lines of impressed force (electric or magnetic as the case may be, but
only the electric are here considered), together with the various
reflected waves produced by change of media and other causes. At
the first moment of starting an impressed force the only disturbance
is at the vortex-lines, which are the first lines of magnetic induction.
Examples of the Forced Vibrations of Electromagnetic Systems.
(a). Thus a uniform field of impressed force suddenly started over all
space can produce no effect. For, either there are no vortex-lines at
all, or they are at an infinite distance, so that an infinite time must
elapse to produce any effect at a finite distance from the origin.
(b). Copper and zinc put in contact. Whether the Volta-lbrce be at
the contact or over the air-surfaces away from and terminating at the
contact (if perfectly metallic), the vortex-line is the common meeting-
364 ELECTRICAL PAPERS.
place of air, zinc, and copper ; the first line of magnetic force is there,
and from it the disturbance proceeds into the metals and out into the
air, which ends in the steady electric field.*
Since the vortex-lines or tubes are closed, we need only consider one
at present say, that due to a simple shell of impressed force. If it be
wholly within a conductor, the initial wave emanating from it is so
rapidly attenuated by the conductivity (the process being akin to
repeated internal reflexions, say reflexion of 9 parts and transmission
of 1 part, repeated at short intervals) that the transmission to a distance
through the conductor (if good) becomes a very slow process, that of
diffusion. Consequently, when the impressed force is rapidly alternated,
there is no sensible disturbance except at and near the vortex-line.
But if there be a dielectric outside the conductor, the moment dis-
turbances reach it, and therefore instantly if the vortex-line be on the
boundary, waves travel through the dielectric at the speed of light
unimpeded, and without the attenuating process within the conductor,
which therefore becomes exposed to electric force all over its boundary
in a very short time ; hence diffusion inward from the boundary. The
electric telegraph would be impossible without the dielectric. It would
take ages if the wire itself had to be the seat of transfer of energy.
(c). In the magnetic theory of the rise of current in a wire we have,
at first sight, an exception to the law that at the first moment there
is no disturbance except at the vortex-lines of impressed force. But it
is that theory which is incorrect, in assuming that there is no displace-
ment. This is equivalent to making the speed of propagation through
the dielectric infinitely great ; so that we have results mathematically
equivalent to distributing the impressed force throughout the whole
circuit, and therefore its vortex-lines over the whole boundary.! In
reality, with finite speed, the disturbances come from the real vortex-
lines in time.
There is still a limitation of the disturbances to the neighbourhood
of the vortex-lines when they are on the boundary of the conductor,
and the periodic frequency is sufficiently great, the impressed force being
within the conductor. [The attenuation by resistance is referred to.]
But in a nonconducting dielectric this effect does not occur, at least
in any case I have examined. On the contrary, as the frequency is
raised, there is a tendency to constancy of amplitude of the waves sent
out from the edge of a simple sheet of impressed force, or from a shell
of vortex-lines of the same, in a dielectric. Very remarkable results
follow from the coexistence of the primary and reflected waves. Thus :
(d). If a spherical portion of an infinitely extended dielectric have a
uniform field of alternating impressed force within it, and the radius a,
the wave-frequency n/2ir, and the speed v be so related that
na na
tan s= .
v v
*"Some Remarks on the Volta Force," Journal 8. T. E. d; E., 1885 [vol. I.,
p. 425].
t The Electrician, June 25, 1886, p. 129 [vol. n., p. 60],
RESISTANCE AND CONDUCTANCE OPERATORS. 365
there is no disturbance outside the sphere. There are numerous
similar cases ; but this is a striking one, because, from the distribution
of the impressed force, it looks as if there must be external displacement
produced by it. There is not, because the above relation makes the
primary wave outward from the surface of the sphere, which is a shell
of vorticity, be exactly neutralised by the reflexion, from the centre,
of the primary wave inward from the surface.
(e). If, instead of alternating, the uniform field of impressed force in
(d) be steady, the final steady electric field due to it takes the time
(/ + a)/v to be established at distance r from the centre. The moment
the primary wave inward reaches the centre, the steady state is set up
there; and as the reflected wave travels out, its front marks the
boundary between the steady field (final) and a spherical shell of
depth 2a, within which is the uncancelled first portion of the primary
wave outward from the surface; which carries out to an infinite
distance an amount of energy equal to that of the final steady electric
field. This is the loss by radiation. (The magnetic energy in this
shell equals half the final electric energy on the whole journey ; the
electric energy in the shell is greater, but ultimately becomes the
same.) In practical cases this energy would be mostly, perhaps wholly
dissipated in conductors.
(/). If a uniformly distributed impressed force act alternatingly
longitudinally within an infinitely long circular cylindrical portion of a
dielectric, the axis is the place of reflexion of the primary wave inward,
and the reflected wave cancels the outward primary wave when
so that there is no external disturbance, except at first. Here a = radius
of cylinder.
(g). There is a similar result when the vorticity of impressed force
takes the place of impressed force in (/).
(h). If the alternating impressed force act uniformly and longi-
tudinally in a thin conducting-tube of radius a, with air within and
without, then
destroys the external field and makes the conduction-current depend
upon the impressed force only. And if we put a barrier at distance x
to serve as a perfect reflector, that is, a tube of infinite conductivity,
J Q (nx/v) = Q
makes the electric force of the field in the inner tube be the exact
negative of the impressed force ; so that there is no conduction-current.
The electromagnetic field is in stationary vibration. If the inner tube
be situated at one of the nodal surfaces of electric force, the vibrations
mount up infinitely.
(i). If, in case (h), the impressed force act circularly about the axis
of the inner tube (which may be replaced by a solenoid of small depth),
/ 1 (wa/v) =
destroys the external field, and
J^nx/v) =
366 ELECTRICAL PAPERS.
makes the electric force of the field the negative of the impressed force,
and so destroys the conduction -current.
(j). We can also destroy the longitudinal force of the field in a con-
ductor without destroying the external field. Let it be a wire of
steady resistance in a dielectric, and the impressed force in it be
e =: e Q cos nix cos nt
per unit length. Then m = n/v makes e be the force of the flux, in the
wire ; so that the current is Ke, if K be the conductance of unit length.
These examples are mostly selected from a paper I am now writing
on the subject of electromagnetic waves, which I hope to be permitted
to publish in this Journal.
If the electric and magnetic energies, and the dissipation of energy,
in a given system be bounded in their distribution, it is clear that the
resistance operator is a rational function of p. But should the field be
boundless, as when conductors are contained in an infinitely extended
dielectric, then just as complete solutions in infinite series of normal
solutions may become definite integrals by the infinite extension, so
may the resistance-operator become irrational. We may also have to
modify the meaning of the sinusoidal R' from representing mean
resistance only, on account of the never-ceasing outward transfer of
energy so long as the impressed force continues.
Induction-Balances General, Sinusoidal, and Impulsive.
9. Returning to a finite combination represented by V=ZG, there
are at least three kinds of induction-balances possible. First, true
balances of similar systems, where we balance one combination against
another which either copies it identically or upon a reduced scale,
without any reference to the manner of variation of the impressed
force. Along with these we may naturally include all cases in which
the Z of a combination, in virtue of peculiar internal relations, reduces
to a simpler form representing another combination, equivalent so far
as V and are concerned. The telephone may be employed with
great advantage, and is, in fact, the only proper thing to use, especially
for the observation of phenomena.
There are, next, the sinusoidal-current balances. These are also
true, in being independent of the time, so that the telephone may be
used; but are of course of a very special character otherwise. Here
any combination is made equivalent to a mere coil if L' be positive, or
to a condenser if S' be positive ( 3 and 4), and so may be balanced by
one or the other. But intermittences of current cannot be safely taken
to represent sinusoidality, and large errors may result from an assumed
equivalence.
In the third kind of balances it is the impulsive inductance that is
balanced against some other impulsive inductance, positive or negative
as the case may be; or perhaps the impulsive inductance of a com-
bination is made to vanish, by equating the electric and magnetic
energies in it when its state is steady. The rule that the impulsive
balance in a Christie arrangement without mutual induction between
RESISTANCE AND CONDUCTANCE OPERATORS. 367
the four sides is given by equating to zero the coefficient of p in
the expansion of Z-^Z^ - Z^Z.^ in powers of p, where Z v etc. are the
resistance operators of the four sides,* is in agreement with the rule
derived from (24) or (25) above, to make the impulsive inductance of
one combination vanish. Impulsive, or "kick" balances, naturally
require a galvanometer. Even then, however, the method is sometimes
unsatisfactory, when the opposing influences which make up the
impulse are not sufficiently simultaneous, as has been pointed out by
Lord Rayleigh.f
There is also the striking method of cumulation of impulses employed
by Ayrton and Perry, J employing false resistance-balances. It seems
complex, and of rather difficult theory ; but, just as a watch is a
complex piece of mechanism, and is yet thoroughly practical, so
perhaps the secohmmeter may have a brilliant career before it.
Several interesting papers relating to the comparison of inductances
and permittances have appeared lately. It is usually impulsive balances
that are in question, probably because it is not the observation of
phenomena that is required, but a direct, even if rough, measurement
of the inductance or permittance concerned, often under circumstances
that do not well admit of the use of the telephone. Only one of these
papers, however, contains anything really novel, scientifically, viz., that
of Mr. W. H. Preece, F.R.S., who concludes, from his latest researches,
that the "coefficient of self-induction" of copper telegraph-circuits is
nearly zero, the results he gives being several hundred times smaller
than the formula derived from electromagnetic principles asserts it to
be. Here is work for the physicist.
10. To equate the expressions for the electric and magnetic energies
of a combination is, I find, in simple cases, the easiest and most direct
way of furnishing the condition that the impulsive inductance shall
vanish. Thus, if there be but one condenser and one coil, SF' 2 = LC 2 is
the condition, S and L being the permittance and the inductance
respectively, F the voltage of the condenser, and C the current in the
coil. The relation between V and C will be, of course, dependent upon
the resistances concerned. || But in complex cases, and to obtain the
value of the impulsive inductance when it is not zero, equation (24) is
most useful.
The Resistance Operator of a Telegraph Circuit.
The following illustration of the properties of Z and Z$ is a complex
one, but I choose it because of its comprehensive character, and because
it leads to some singular extreme cases, interesting both mathematically
* "On the Self-induction of Wires," Part VI., Phil. Mag., Feb. 1887 [vol. n.,
p. 263].
t Electrical Measurements, p. 65.
Journ. Soc. Tel. Engineers and Electricians, 1887.
B.A. Meeting, 1887: "On the Coefficient of Self-induction of Iron and
Copper Wires."
|| If the condenser shunts the coil, making V=RC, we get the case brought
before the S.T.E. & E. by Mr. Sumpner, with developments.
368 ELECTRICAL PAPERS.
and in the physical interpretation of the apparent anomalies. Let the
combination be a telegraph-circuit, say a pair of parallel copper wires,
of length / ; resistance /, permittance S, inductance L, and leakage-
conductance K, all per unit length, and here to be considered strictly
constants, or independent of p. Let the two wires be joined through
an arrangement whose resistance-operator is Z at the distant end B ;
then the resistance-operator at the beginning A of the circuit is given
by*
z _ (R + Lp)l{(ta,n ml) /ml] + Z l /o. 2 \
1 + K+ fi/tan mlml
if -m? = (R + Lp)(K+Sp) ....................... (33)
Take Z 1 = Q for the present, or short-circuit at B. This makes
Z=(E + Lp)l(tenml)lml, ..................... (34)
and the steady resistance at A is therefore
, .......................... (35)
if 77^ = RK. Also, differentiating (34) to p, and then making p = 0,
we find
, T 17 tanmJ/ r RS\ 17 ,/,- RS\ /QA x
Z{ = L = $1 ml Q (L - -g) + |l sec% ^Z + _ j ....... (36)
represents the impulsive inductance.
If we put $ = in (36) we make the arrangement magnetic, and then
L is positive. If we put L = 0, we make it electrostatic, and L Q is nega-
tive, or S , the impulsive permittance, is positive. It is to be noticed that
there is no confusion when both energies are present; that is, there
are no terms in Z' Q containing products of real permittances and induct-
ances, which is clearly a general property of resistance-operators,
otherwise the two energies would not be independent.
We may make L Q vanish by special relations. Thus, if there be no
leakage, or JT=0, (36) is
L = U-kffl.RSP; ..... .................... (37)
so that the magnetic must be one third of the electrostatic time-
constant to make the "extra-current" and the static charge balance.
(The length of the circuit required for this result may be roughly stated
as about 60 kilometres if it be a single copper wire of 6 ohms per
kilometre, 4 metres high, with return through the ground; but it
varies considerably, of course.)
But if leakage be now added, it will increase the relative importance
of the magnetic energy, so that the length of the circuit requires to be
increased to produce a balance. This goes on until K reaches the
value JKS/L, when, as an examination of (36) will show, the length of
the circuit needs to be infinitely great. The same formula also shows
that if K be still greater, L cannot be made to vanish at all, being then
always positive.
*"On the Self-induction of Wires," Part IV., Phil. Mag., Nov., 1886
[vol. TI., p. 232 ; also p. 247 and p. 105.]
RESISTANCE AND CONDUCTANCE OPERATORS. 369
11. Now let the circuit be infinitely long. Equation (35) reduces to
the irrational form
..................... (38)
with ambiguity of sign. Of course the positive sign must be taken.
The negative appears to refer to disturbances coming from an infinite
distance, which are out of the question in our problem, as there can be
no reflexion from an infinite distance. But equation (38) may be
obtained directly in a way which is very instructive as regards the
structure of resistance-operators. Since the circuit is infinitely long, Z
cannot be altered by cutting-off from the beginning, or joining on, any
length. Now first add a coil of resistance 7^ and inductance L^ in
sequence, and a condenser of conductance K l and permittance S 19 in
bridge, at A, the beginning of the circuit. The effect is to increase Z
to Z 2 , where
Z^{K l + S lP + (R l + L l p + Z)-^ ] .............. (39)
i.e., the reciprocal of the new Z 2 , or the new conductance-operator,
equals the sum of the conductance-operators of the two branches in
parallel, one the conducting condenser, the other the coil and circuit in
sequence. (39) gives the quadratic
S 1 p)^ ............ (40)
Now choose R V L V K V S lt in exact proportion to fi,L,K, and S, and then
make the former set infinitely small. The result is that we have added
to the original circuit a small piece of the same type, so that Z 2 and Z
are identical, and that the coefficient of the first power of ^ 2 in (40)
vanishes. Therefore (40) becomes
This fully serves to find the sinusoidal solution. Differentiating it, we
find
corroborating the previous result as to the vanishing of L Q when the
circuit is infinitely long by equality of RS and KL, and the positivity
of L when KL>RS.
The Distortionless Telegraph Circuit.
1 2. Now, in the singular case of R/L = K/S, we have, by (41) and (42),
Z=Lv, Z = 0, ........................... (43)
if v = (LS)-*, the speed of transmission of disturbances along the circuit.
The resistance-operator has reduced to an absolute constant, and the
current and transverse voltage are in the same phase, altogether
independent of the frequency of wave-period, or indeed of the manner
of variation. The quantity Lv, or L x 30 ohms, approximately, if the
dielectric be air, is strictly, and without any reservation, the impedance
of the circuit at A, but it is only exceptionally the resistance,
IJ.E.P. VOL. ii. 2 A
370 ELECTRICAL PAPERS.
Make Vf(t) t at A, an arbitrary function of the time ; then, if V x
and C x are the transverse voltage and the current at distance x from A
at time t, we shall have
F x =f(t-xlv)e-w, C x -=F x /Lv, (44)
or all disturbances originating at A are transmitted undistorted along
the circuit at the speed v, attenuating at a rate indicated by the
exponential function. (I have elsewhere* full} developed the properties
of this distortionless circuit, and only mention such as are necessary to
understand the peculiarities connected with the present subject-matter.)
The electric and magnetic energies are always equal, not only on the
whole, but in any part of the circuit ; this accounts for the disappear-
ance of L Q , and the bringing of V x and C x to the same phase, as we
should expect from 4. But in the present case Z^ or Lv, or E', for
they are all equal, is only the resistance when the steady state due to
the steady V at A is arrived at (asymptotically), or the effective
resistance at a given frequency when Fis sinusoidal, and sufficient time
has elapsed to have allowed V x and C x to become sinusoidal to such a
distance from A that we can neglect the remainder of the circuit into
which greatly attenuated disturbances are still being transmitted.
13. Now, since the impedance is unaltered by joining on at A any
length of circuit of the same type, and is a constant, it follows that the
impedance at A of a distortionless circuit as above described, but of
finite length, stopping at B, where x = l, with a resistance of amount Lv
inserted at B, is also a constant, viz. the same Lv. To corroborate, take
RS = KL and Z^ = Lv in the full formula (32). The result is Z = Lv.
The interpretation in this case is that all disturbances sent from A are
absorbed completely by the resistance at B immediately on arrival,
so that the finite circuit behaves as if it were infinitely long. The
permanent state due to a steady V at A is arrived at in the time l/v.
The impedance and the resistance then become identical.
14. If, in the case of 12, we further specialize by taking R = Q,
K=Q, producing a perfectly insulated circuit of no resistance, the
impedance is, as before, Lv but no part of it is resistance, or ever can
be, in spite of the identity of phase of V and C. However long we
may keep on a steady Fat A, we keep the impressed force working at
the same rate, the energy being entirely employed in increasing the
electric and magnetic energies at the front of the wave, which is
unattenuated, and cannot return.
But if we cut the circuit at B, at a finite distance /, and there insert
a resistance Lv, the effect is that, as soon as the front of the wave
reaches B, the inserted resistance immediately becomes the resistance
of the whole combination ; or the impedance instantly becomes the
resistance, without change of value.
15. As a last example of singularity, substitute a short-circuit for the
terminal resistance Lv just mentioned. Since there is now no resistance
in any part of the system, if we make the state sinusoidal everywhere,
* "Electromagnetic Induction and its Propagation," Sections XL. to L., Electri-
cian, 1887 [vol. ii., pp. 119 to 155].
RESISTANCE AND CONDUCTANCE OPERATORS. 371
by V sinusoidal at A, R f musfc vanish, or V and C be in perpendicular
phases, due to the infinite series of to-and-fro reflexions. We now
have, by (32),
Z' = L^^ = Dp^*M, (45)
ph/v nl/v
if n/2ir = frequency, and R f has disappeared.
If, on the other hand, V be steady at A, the current increases without
limit, every reflexion increasing it by the amount VjLv at A or at B
(according to which end the reflexion takes place at), which increase
then extends itself to B or A at speed v. The magnetic energy mounts
up infinitely. On the other hand, the electric energy does not, fluctu-
ating perpetually between when the circuit is uncharged, and %SIF 2
when fully charged. The impedance of the circuit to the impressed
force at A is Lv for the time 2l/v after starting it; then \Lo for a
second period 21 /v ; then \Lv for a third period, and so on.
It will have been observed that I have, in the last four paragraphs,
used the term impedance in a wider sense than in 3, where it is the
ratio of the amplitude of the impressed force to the amplitude of the flux
produced at the place of impressed force when sufficient time has elapsed
to allow the sinusoidal state to be reached, when that is possible. The
justification for the extension of meaning is that, since in the distortion-
less circuit of infinite length, or of finite length with a terminal resistance
to take the place of the infinite extension, we have nothing to do with
the periodic frequency, or with waiting to allow a special state to be
established, it is quite superfluous to adhere to the definition of the
last sentence ; and we may enlarge it by saying that the impedance of
a combination is simply the ratio of the force to the flux, when it
happens to be a constant, which is very exceptional indeed. I may
add that R, L, K, and S need not be constants, as in the above, to pro-
duce the propagation of waves without tailing. All that is required is
R/L = K/S, and Li) = constant ; so that R and L may be functions of x.
The speed of the current, and the rate of attenuation, now vary from
one part of the circuit to another.
The Use of the Resistance-Operator in Normal Solutions.
16. In conclusion, consider the application of the resistance-operator
to normal solutions. If we leave a combination to itself without
impressed force, it will subside to equilibrium (when there is resistance)
in a manner determined by the normal distributions of electric and
magnetic force, or of charges of condensers and currents in coils ; a
normal system being, in the most extended sense, a system that, in
subsiding, remains similar to itself, the subsidence being represented
by the time-factor e x , where p is a root of the equation Z=0. It is
true that each part of the combination will usually have a distinct
resistance-operator ; but the resistance-operators of all parts involve,
and are contained in, the same characteristic function, which is merely
the Z of any part cleared of fractions. It is sometimes useful to
remember that we should clear of fractions, for the omission to do so
372 ELECTRICAL PAPERS.
may lead to the neglect of a whole series of roots ; but such cases
are exceptional and may be foreseen; whilst the employment of a
resistance-operator rather than the characteristic function is of far
greater general utility, both for ease of manipulation and for physical
interpretation.
Given a combination containing energy and left to itself, it is upon the
distribution of the energy that the manner of subsidence depends, or
upon the distribution of the electric and magnetic forces in those parts
of the system where the permittivity and the inductivity are finite, or
are reckoned finite for the purpose of calculation. Thus conductors,
if they be not also dielectrics, have only to be considered as regards the
magnetic force, whilst in a dielectric we must consider both the electric
and the magnetic force. (The failure of Maxwell's general equations of
propagation arises from the impossibility of expressing the electric
energy in terms of his potential function. " The variables should always
be capable of expressing the energy.) Now the internal connexions of
a system determine what ratios the variables chosen should bear to one
another in passing from place to place in order that the resultant system
should be normal; and a constant multiplier will fix the size of the
normal system. Thus, supposing u and w are the normal functions of
voltage and current, which are in most problems the most practical
variables, the state of the whole system at time t will be represented by
.................. (46)
V being the real voltage at a place where the corresponding normal
voltage is u, and C the real current where the normal current is w, the
summation extending over all the ^>-roots of the characteristic equation.
The size of the systems, settled by the A's (one for each p) are to be
found by the conjugate property of the vanishing of the mutual energy-
difference of any pair of ^-systems, applied to the initial distributions
of Fand C.
17. To find the effect of impressed force is a frequently recurring
problem in practical applications; and here the resistance-operator is
specially useful, giving a general solution of great simplicity. Thus,
suppose we insert a steady impressed force e at a place where the
resistance-operator is Z, producing e = ZC thereafter. Find C in terms
of e and Z. The following demonstration appears quite comprehensive.
Convert the problem into a case of subsidence first, by substituting a
condenser of permittance S, and initial charge Se, for the impressed
force. By making S infinite later we arrive at the effect of the steady e.
In getting the subsidence solution we have only to deal with the energy
of the condenser, so that a knowledge of the internal connexions of the
system is quite superfluous.
The resistance-operator of the condenser being (Sp)~ l , that of the
combination, when we use the condenser, is Z lt where
Z 1 = (Sp)-^ + Z. ........................... (47)
Let V and C be the voltage and the current respectively, at time t after
insertion of the condenser, and due entirely to its initial charge.
RESISTANCE AND CONDUCTANCE OPERATORS. 373
Equations (46) above express them, if u and w have the special ratio
proper at the condenser, given by
w= -Spu, ................................. (48)
because the current equals the rate of decrease of its charge. Initially,
we have e = 2Au and 2,Aw = Q. So, making use of the conjugate
property,* we have
Seu=2(U p -T p )A, .......................... (49)
if U p be the electric and T p the magnetic energy in the normal system.
But the following property of the resistance-operator is also true,*
2(r,-0,) = ; ........................... (50)
that is, dZJdp is the impulsive inductance in the p system at a place
where the resistance-operator is Z lt p being a root of Z l = Q; just as
dZJdp with p = is the impulsive inductance (complete) at the same
place. Using (50) in (49) gives
(5!)
Now use (48) in (51) and insert the resulting A in the second of (46),
and there results
o- .......................... ...... < 52 >
where the accent means differentiation to p. This is the complete
subsidence solution. Now increase S infinitely, keeping e constant
Z l ultimately becomes Z ; but, in doing so, one root of Z l = becomes
zero. We have, by (47), and remembering that Z l = 0,
pZ{= -(Sp)-i+pZ' = Z+pZ f ' ) ..................... (53)
so, when $=oo and Z = Q, we have pZ{=pZ r for all roots except the
one just mentioned, in which case p tends to zero and Z f is finite,
making in the limit pZ{ = Z^ by (53), where Z Q is the^? = value of Z,
or the steady resistance. Therefore, finally,
where the summation extends over the roots of Z--=0, shows the
manner of establishment of the current by the impressed force e. The
use of this equation (54), even in comparatively elementary problems,
leads to a considerable saving of labour, whilst in cases involving partial
differential equations it is invaluable.! To extend it to show the rise
of the current at any other part of the system than where the impressed
* "On the Self-induction of Wires," Phil. Mag., Oct. 1886 [vol. n., pp. 202
to 206].
t In Part III. of " On the Self -Induction of Wires," I employed the Condenser
Method, with application to a special kind of combination ; but, as we have seen
from the above proof, (54) is true for any electrostatic and electromagnetic com-
bination provided it be finite.
374 ELECTRICAL PAPERS.
force is, it is necessary to know the connections, so that we may know
the ratio of the current in a normal system at the new place to that at
the old; inserting this ratio in the summation, and modifying the
external Z Q to suit the new place, furnishes the complete solution there.
Or, use the more general resistance-operator Z xy , such that e x = Z xy C y ,
connecting the. impressed force at any place x with the current at
another place y.
18. When the initial current is zero, as happens when there is self-
induction without permittance at the place of e t and in other cases, (54)
gives
showing that the normal systems may be imagined to be arranged in
parallel, the resistance of any one being ( -pZ f ).
To express the impulsive inductance Z' Q in terms of the normal ^s,
multiply (54) by e and take the complete time-integral. We obtain
Uc-}dt = 2(U-T)= --, (56)
J \ ZJQ/ p /j
remembering (29). Or, using (26),
(57)
In electrostatic problems the roots of Z=0 are real and negative, as
is also the case in magnetic problems. There are never any oscillatory
results in either case, and the vanishing of Z 1 is then accompanied by
vanishing of the corresponding normal functions, to prevent the oscilla-
tions which seem on the verge of occurring by the repetition of a root
which Z' = Q implies.* When both energies are present, the real parts
of the imaginary roots are always compelled to be negative by the
positivity of 7", T, and of Q the dissipativity.
When Z is irrational, it is probable that the complete solution
corresponding to (54) might be immediately derived from Z. In the
case of (41),f however, the application is not obvious, although there is
no difficulty in passing from the (54) solution to the corresponding
definite integrals which arise when the length of the circuit is infinitely
increased.
"* [See p. 529, vol. i. Also Thomson and Tait, Part I. , 343c and after, relating
to Routh's Theorem, given in his Adam's Prize Essay, "Stability of Motion."]
t [Done in "El. Mag. Waves," 1888. Arts. XLIII. and XLIV. later.]
ON ELECTROMAGNETIC WAVES. PART I. 375
XLIIL ON ELECTROMAGNETIC WAVES, ESPECIALLY IN
RELATION TO THE VORTICITY OF THE IMPRESSED
FORCES ; AND THE FORCED VIBRATIONS OF ELECTRO-
MAGNETIC SYSTEMS.
[Phil. Mag., 1888; Parti., February, p. 130; Part II., March, p. 202; Part III.,
May, p. 379 ; Part IV., October, p. 360; Part V., November, p. 434; Part VL,
December, 1888, p. 488.]
PART I.
Summary of Electromagnetic Connections.
1 . To avoid indistinctness, I start with a short summary of Maxwell's
scheme, so far as its essentials are concerned, in the form given by me
in January, 1885.*
Two forces, electric and magnetic, E and H, connected linearly with
the three fluxes, electric displacement D, conduction-current C, and
magnetic induction B ; thus
B = /zH, C = &E, D = (c/47r)E (1)
Two currents, electric and magnetic, T and G, each of which is
proportional to the curl or vorticity of the other force, not counting
impressed ; thus,
curl (H - h) = 47TF, (2)
curl(e-E) = 47rG; (3)
where e and h are the impressed parts of E and H. These currents
are also directly connected with the corresponding forces through
r = C + D, G = B/47r (4)
An auxiliary equation to exclude unipolar magnets, viz.
divB = 0, (5)
expressing that B has no divergence. The most important feature of
this scheme is the equation (3), as a fundamental equation, the natural
companion to (2).
The derived energy-relations are not necessary, but are infinitely too
useful to be ignored. The electric energy 7, the magnetic energy T 7 ,
and the dissipativity Q, all per unit volume, are given by
Z7=iED, T=JHB/47r, (J = EC (6.)
The transfer of energy W per unit area is expressed by a vector product,
W = V(E-e)(H-h)/47r, (7)
and the equation of activity per unit volume is
er + hG = +?7+r+divW, (8)
from which W disappears by integration over all space.
The equations of propagation are obtained by eliminating either E or
* See the opening sections of " Electromagnetic Induction and its Propagation,"
Electrician, Jan. 3, 188o, and after [Art. xxx., vol. i., p. 429].
376 ELECTRICAL PAPERS.
H between (2) and (3), and of course take different forms according to
the geometrical coordinates selected.
In a recent paper I gave some examples* illustrating the extreme
importance of the lines of vorticity of the impressed forces, as the
sources of electromagnetic disturbances. Those examples were mostly
selected from the extended developments which follow. Although,
being special investigations, involving special coordinates, vector
methods will not be used, it will still be convenient occasionally to use
the black letters when referring to the actual forces or fluxes, and to
refer to the above equations. The German or Gothic letters employed
by Maxwell I could never tolerate, from inability to distinguish one
from another in certain cases without looking very hard. As regards
the notation EC for the scalar product of E and C (instead of the
quaternionic - SEC) it is the obvious practical extension of EC, the
product of the tensors, what EC reduces to when E and C are parallel, f
Plane Sheets of Impressed Force in a Nonconducting Dielectric.
2. We need only refer to impressed electric force e, as solutions relat-
ing to h are quite similar. Let an infinitely extended nonconducting
dielectric be divided into two regions by an infinitely extended plane
(x, y), on one side of which, say the left, or that of - z, is a field of e of
uniform intensity e, but varying with the time. If it be perpendicular
to the boundary, it produces no flux. Only the tangential component
can be operative. Hence we may suppose that e is parallel to the
plane, and choose it parallel to x. Then E, the force of the flux, is
parallel to x, of intensity E say, and the magnetic force, of intensity
H, is parallel to y. Let e =f(t) ; the complete solutions due to the
impressed force are then
E-iwH- -i/(*-*/i>) ......................... (9)
on the right side of the plane, where z is + , and
-E = nvH= -$f(t + z/v) ...................... (10)
on the left side of the plane, where z is - . In the latter case we must
deduct the impressed force from E to obtain the force of the field, say
F, which is therefore
* Phil. Mag. Dec. 1887, " On Resistance and Conductance Operators," 8, p.
487 [Art. XLII., vol. II., p. 363].
t In the early part of my paper " On the Electromagnetic Wave- Surf ace," Phil.
Mag., June, 1885 [Art. xxxi., vol. n., p. 1] I have given a short introduction to
the Algebra of vectors (not quaternions) in a practical manner, i.e., without
metaphysics. The result is a thoroughly practical working system. The matter
is not an insignificant one, because the extensive use of vectors in mathematical
physics is bound to come (the sooner the better), and my method furnishes a way
of bringing them in without any study of Quaternions (which are scarcely wanted
in Electromagnetism, though they may be added on), and allows us to work
without change of notation, especially when the vectors are in special type, as
they should be, being entities of widely different nature from scalars. I denote a
vector by (say) E, its tensor by E, and its x, y, z components, when wanted, by
E I} E%, E y The perpetually occurring scalar product of two vectors requires no
prefix. The prefix V of a vector product should be a special symbol.
ON ELECTROMAGNETIC WAVES. PART I. 377
The results are most easily followed thus : At the plane itself, where
the vortex-lines of e are situated, we, by varying e, produce simultaneous
changes in H, thus,
-H=e/2pv, ................................ (12)
at the plane. This disturbance is then propagated both ways undis-
torted at the speed v = (/AC)~*.
On the other hand, the corresponding electric displacements are
oppositely directed on the two sides of the plane.
Since the line-integral of H is electric current, and the line-integral
of e is electromotive force, the ratio of e to H is the resistance-operator
of an infinitely long tube of unit area ; a constant, measurable in ohms,
being 60 ohms in vacuum, or 30 ohms on each side. Why it is a con-
stant is simply because the waves cannot return, as there is no reflecting
barrier in the infinite dielectric.
3. If the impressed force be confined to the region between two
parallel planes distant 2a from one another, there are now two sources
of disturbances, which are of opposite natures, because the vorticity of
e is oppositely directed on the two planes, so that the left plane sends
out both ways disturbances which are the negatives of those simultane-
ously emitted by the right plane. Thus, if the origin of z be midway
between the planes, we shall have
............ (13)
on the right side of the stratum of e, and
............ (H)
on the left side. If therefore e vary periodically in such a way that
f(t)=f(t + 2a/v), ............................ (15)
there is no disturbance outside the stratum, after the initial waves have
gone off, the disturbance being then confined to the stratum of impressed
force.
Decreasing the thickness of the stratum indefinitely leads to the
result that the effect due to e =f(t) in a layer of thickness dz&tz-Q is,
on the right side,
since ^v' 2 = 1 ; on the left side the + sign is required.
We can now, by integration, express the effect due to ef(z t t\ viz.,
In these, however, a certain assumption is involved, viz. that e vanishes
at ~s^ both ways, because we base the formulae upon (16), which concerns
378 ELECTRICAL PAPERS
a layer of e on both sides of which e is zero. Now the disturbances
really depend upon de/dz, for there can be none if this be zero. By
(12) the elementary de/dz through distance dz instantly produces
U=%d* ............................ (19)
2pv dz
at the place. If, therefore, e =f(z, t), the If-solution at any point con-
sists of the positive waves coming from planes of de/dz on the left, pro-
ducing say, H lt and of H 2 , due to the negative waves from the planes
of de/dz on the right side, making the complete solution
H=H l + H 2 , =K#i-tf 2 ); ............... (20)
where
This is the most rational form of solution, and includes the case of
e =f(t) only. The former may be derived from it by effecting the
integrations in (21) and (22) ; remembering in doing so that the
differential coefficient under the sign of integration is not the complete
one with respect to z f y as it occurs twice, but only to the second z r , and
further assuming that e = at infinity.
Waves in a Conducting Dielectric. How to remove the Distortion
due to the Conductivity.
4. Let us introduce a new physical property into the conducting
medium, namely that it cannot support magnetic force without dissipa-
tion of energy at a rate proportional to the square of the force, a
property which is the magnetic analogue of electric conductivity. We
make the equations (2) and (3) become, ifp-d/dt,
......................... (23)
........................ (24)
if there be no impressed force at the spot, where g is the new coefficient
of magnetic conductivity, analogous to k.
Let
47T&/2C = q v ft + q 2 = q, E = '*
, 9 ~
q 2 = s, = l ."
Substitution in (23), (24) leads to
(26)
(27)
If s = 0, these are the equations of electric and magnetic force in a non-
conducting dielectric. If therefore the new g be of such magnitude as
to make s = 0, we cause disturbances to be propagated in the conducting
dielectric in identically the same manner as if it were nonconducting,
ON ELECTROMAGNETIC WAVES. PART I. 379
but with a uniform attenuation at a rate indicated by the time-
factor ~ qt .
Undistorted Plane Waves in a Conducting Dielectric.
5. Taking z perpendicular to the plane of the waves, we now have,
as special forms of (23), (24),
(28)
(29)
E being the tensor of E, parallel to x, and H the tensor of H, parallel
to y, and both being functions of z and t.
Given E E and H=H Q &t time t = 0, functions of z only, decompose
them thus,
(30)
(31)
Here f l makes the positive and / 2 the negative wave, and at time t the
solutions are, due to the initial state, when s = 0,
................... (32)
(33)
The only difference from plane waves in a nonconducting dielectric is
in the uniform attenuation that goes on, due to the dissipation of
energy, which is so balanced on the electric and magnetic sides as to
annihilate the distortion the waves would undergo were s finite, whether
positive or negative.
Practical Application. Imitation of this Effect.
6. When I introduced * the new property of matter symbolized by
the coefficient g, it was merely to complete the analogy between the
electric and magnetic sides of electromagnetism. The property is non-
existent, so far as I know. But I have more recently found how to
precisely imitate its effect in another electromagnetic problem, also
relating to plane waves, making use of electric conductivity to effect
the functions of both k and g in 4 and 5. In the case of 5, first
remove both conductivities, so that we have plane waves unattenuated
and undistorted. Next put a pair of parallel wires of no resistance in
the dielectric, parallel to z, and let the lines of electric force terminate
upon them, whilst those of magnetic force go round the wires. We
shall still have these plane electromagnetic waves with curved lines of
force propagated undistorted and unattenuated, at the same speed v.
If Fbe the line-integral of E across the dielectric from one wire to the
other, and kirC be the line-integral of H round either wire, we shall
have
(34)
(35)
*See first footnote [p. 375].
380 ELECTRICAL PAPERS.
(34) taking the place of (29), and (35) of (28), with k and g both zero.
Here L and S are the inductance and permittance of unit length of the
circuit of the parallel wires, and v = (LS) ~ *.
Next let the wires have constant resistance R per unit length to
current in them, and let the medium between them be conducting (to a
very low degree), making K the conductance per unit length across
from one wire to the other. We then turn the last equations into
(36)
(37)
and have a complete imitation of the previous unreal problem. The
two dissipations of energy are now due to R in the wires, and to K in
the dielectric, it being that in the wires which takes the place of the
unreal magnetic dissipation. The relation RjL = K/S, which does not
require excessive leakage when the wires are of copper of low resist-
ance, removes the distortion otherwise suffered by the waves. I have,
however, found that when the alternations of current are very rapid, as
in telephony, there is very little distortion produced by copper wires,
even without the leakage required to wholly remove it, owing to RjLn
becoming small, n/2ir being the frequency ; an effect which is greatly
assisted by increasing the inductance (see Note A, [p. 392]). Of course
there is little resemblance between this problem and that of the long
and slowly-worked submarine cable, whether looked at from the
physical side or merely from the numerical point of view, the results
being then of different orders of magnitude. A remarkable misconcep-
tion on this point seems to be somewhat generally held. It seems to be
imagined that self-induction is harmful* to long-distance telephony.
The precise contrary is the case. It is the very life and soul of it, as is
proved both by practical experience in America and on the Continent
on very long copper circuits, and by examining the theory of the
matter. I have proved this in considerable detail ; f but they will not
believe it. So far does the misconception extend that it has perhaps
contributed to leading Mr. W. H. Preece to conclude that the coefficient
of self-induction in copper circuits is negligible (several hundred times
smaller than it can possibly be), on the basis of his recent remarkable
experimental researches.
The following formula, derived from my general formulae |, will show
the rdle played by self-induction Let R and L be the resistance and
inductance per unit length of a perfectly insulated circuit of length /,
short-circuited at both ends. Let a rapidly sinusoidal impressed force
of amplitude e Q act at one end, and let C be the amplitude of the
*W. H. Preece, F.R.S., "On the Coefficient of Self -Induction of Copper
Wires," B. A. Meeting, 1887.
t"El. Mag. Ind. and its Propagation," Electrician, Sections XL. to L. (1887)
[vol. ii., pp. 119 to 155].
J See the sinusoidal solutions in Part II. and Part. V. of " On the Self -Induction
of Wires," Phil. Mag., Sept. 1886 and Jan. 1887 [vol. n., pp. 194 and 247.
Also p. 62].
ON ELECTROMAGNETIC WAVES. PART I. 381
current at the distant end. Then, if the circuit be very long,
(7 =^o -w (38)
Li'
where v is the speed (LS)~~ = (/^c)~^, provided E/Ln be small, say J.
It may be considerably greater, and yet allow (38) to be nearly true.
We can include nearly the whole range of telephonic frequencies by
using suspended copper wires of low resistance. *
It is resistance that is so harmful, not self-induction ; as, in combina-
tion with the electrostatic permittance, it causes immense distortion of
waves, unless counteracted by increasing the inductance, which is not
often practicable (see Note B, [p. 393]).
Distorted Plane Waves in a Conducting Dielectric.
7. Owing to the fact that, as above shown, we can fully utilize solu-
tions involving the unreal g, by changing the meaning of the symbols,
whilst still keeping to plane electromagnetic waves, we may preserve g
in our equations (28) and (29), remembering that H has to become G',
E become T 7 , kirk become K, c become S, kirg become E, and p, become
L, when making the application to the possible problem ; whilst, when
dealing with a real conducting dielectric, g has to be zero.
Required the solutions of (28) and (29) due to any initial states E Q
and H , when s is not zero. Using the notation and transformations of
(25), (or direct from (26), (27)), we produce
(39)
(40)
from which
^ffiHJd^-tf-VHv (41)
with the same equation for E v
The complete solution may be thus described. Let, at time = 0,
there be H=H Q through the small distance a at the origin. This
immediately splits into two plane waves of half the amplitude, which
travel to right and left respectively at speed v, attenuating as they
progress, so that at time t later, when they are at distances vt from
the origin, their amplitudes equal
i#o<-", (42)
with corresponding E's, viz.,
frvHtf-* and -favH c-, (43)
on the right and left sides respectively. These extend through the
* The explanation of the \Lv dividing e in (38), instead of the Lv we might
expect from the /JLV resistance-operator of a tube of unit section infinitely long one
way only, is that, on arrival at the distant end of the line, the current is immedi-
ately doubled in amplitude by the reflected wave. The second and following
reflected waves are negligible, on account of the length of the line.
382 ELECTRICAL PAPERS.
distance a. Between them is a diffused disturbance, given by
(45)
in which v 2 t 2 > z 2 .
In a similar manner, suppose initially E = E Q through distance a at
the origin. Then, at time t later, we have two plane strata of depth a
at distance vt to right and left respectively, in which
E = \Ef-= fjivH, ......................... (46)
the + sign to be used in the right-hand stratum, the - in the left.
And, between them, the diffused disturbance given by
............... (47)
Knowing thus the effects due to initial elements of E Q and H Q , we
have only to integrate with respect to z to find the solutions due to
any arbitrary initial distributions. I forbear from giving a detailed
demonstration, leaving the satisfaction of the proper conditions to be
the proof of (42) to (48) ; since, although they were very laboriously
worked out by myself, yet, as mathematical solutions, are more likely
to have been given before in some other physical problem than to
be new.
Another way of viewing the matter is to start with s = 0, and then
examine the effect of introducing s, either + or - . Let an isolated
plane disturbance of small depth be travelling along in the positive
direction undistorted at speed v. We have E = pvH in it. Now
suddenly increase k, making s positive. The disturbance still keeps
moving on at the same speed, but is attenuated with greater rapidity.
At the same time it leaves a tail behind it, the tip of which travels out
the other way at speed v t so that at time t after commencement of the
tailing, the whole disturbance extends through the distance 2vt. In
this tail H is of the same sign as in the head, and its integral amount is
such that it exactly accounts for the extra-attenuation suffered by H in
the head. On the other hand, E in the tail is of the opposite sign to
E in the head ; so that the integral amount of E in head and tail
decreases faster. As a special case, let, in the first place, there be no
conductivity, k = and = 0. Then, keeping g still zero, the effect of
introducing k is to cause the above-described effect, except that as there
was no attenuation at first, the attenuation later is entirely due to k,
whilst the line-integral of H along the tail, or
including H in the head, remains constant. This is the persistence of
momentum.
ON ELECTROMAGNETIC WAVES. PART I. 383
If, on the other hand, we introduce g, the statements made regarding
// are now true as regards E, and conversely. The tail is of a different
nature, E being of same sign in the tail as in the head, and H of the
opposite sign. Hence, of course, when we have both k and g of the
right amounts, there is no tailing. This subject is, however, far better
studied in the telegraphic application, owing to the physical reality
then existent, than in the present problem, and also then by elementary
methods.*
8. Owing to the presence of d/dz in (45) and (47) we are enabled to
give some integral solutions in a finite form. Thus, let H= H Q (constant)
and E = initially on the whole of the negative side of the origin, with
no E or H on the positive side. The E at time t later is got by
integrating (45), giving
which holds between the limits z= vt, there being no disturbance
beyond, except the H on the left side. When g=0 and z/vt is small,
it reduces to
This is the pure-diffusion solution, suitable for good conductors.
If initially E = E , constant, on the left side of the origin, and zero on
the right side, then at time t the H due to it is, by (48),
The result of taking c = 0, g = 0, in this formula is zero, as we may
see by observing that c in (49) becomes /* in (51). It is of course
obvious that as the given initial electric field has no energy if c = 0, it
can produce no effect later.
The ^-solution corresponding to (49) cannot be finitely expressed.
which, integrated, gives
H.
where all the J'a operate on stJ - 1 ; thus, e.g. (Bessel's),
* "Electromagnetic Induction and its Propagation, " Electrician, Sections XLIIJ.
to L. (1887) [vol. ii., pp. 132 to 155].
384 ELECTRICAL PAPERS.
But a much better form than (52), suitable for calculating the shape
of the wave speedily, especially at its start, may be got by arranging in
powers of z - vt t thus
true when z < vt, where / lf / 2 , etc., are functions of t only, of which the
first five are given by
st
At the origin, II is given by
H-lHjf**, .............................. (54)
and is therefore permanently \H Q when g = Q. At the front of the
wave, where z = vt,
#=pr o -*< ................................. (55)
Now, to represent the J^-solution corresponding to (51), we have only
to turn HtoE and H Q to E in (53), and change the sign of s throughout,
i.e. explicit, and in the/'s. Similarly in (52). Thus, at the origin,
=p: - 2 <><, ............................... (56)
and at the front of the wave
E = \E^ ................................. (57)
9. Again, let H=H on the left side, and H= -%H Q on the right
side of the origin, initially. The E that results from each of them is
the same, and is half that of (49); so that (49) still expresses the
^-solution. This case corresponds to an initial electric current of
surface-density HQ/^TT on the z = Q plane, with the full magnetic field
to correspond, and from it immediately follows the ^-solution due to
any initial distribution of electric current in plane layers.
Owing to H being permanently JJT at the origin in the case (49),
(54), when # = 0, we may state the problem thus: An infinite con-
ducting dielectric with a plane boundary is initially free from magnetic
induction, and its boundary suddenly receives the magnetic force %H =
constant. At time t later (49) and (52) or (53) give the state of the
conductor at distance zz. But on arrival of the wave C l at B, V becomes zero,
and C doubled by the reflected wave that then commences to travel
from B to A. This wave may be imagined to start when t = from a
ON ELECTROMAGNETIC WAVES. PART I. 389
point distant I beyond B, and be the precise negative of the first wave
as regards V but the same as regards ft Thus
expresses the second wave, starting from B when t = l/v, and reaching
A when t = 2l/v. The sum of C l and C 2 now expresses (71) where the
waves coexist, and C l alone expresses (71) in the remainder of the
circuit.
The reflected wave arising when this second wave reaches A may be
imagined to start when t = from a point distant 21 from A on its
negative side, and be a precise copy of the first wave. Thus
expresses the third wave; and now (71) means C 1 + C 2 + C B in those
parts of the circuit reached by (7 3 , and C l + C 2 in the remainder.
The fourth wave is, similarly,
starting from B when t = 3l/v, and reaching A when t - il/v. And so
on, ad inf.*
If we take L = in this problem, we make v = QO , and bring the
whole of the waves into operation immediately. (70) becomes
and similarly for (7 2 , <7 3 , etc. In this simplified form the identity is
that obtained by Sir W. Thomson f in connexion with his theory of
the submarine cable; also discussed by A. Cayley J and J. W. L.
Glaisher. [See also vol. I., p. 88.]
In order to similarly represent the history of the establishment of
F , we require to use the series for E due to E Q , corresponding to
(53), or some equivalent. In other respects there is no difference.
Whilst it is impossible not to admire the capacity possessed by solu-
tions in Fourier series to compactly sum up the effect of an infinite
series of successive solutions, it is greatly to be regretted that the
Fourier solutions themselves should be of such difficult interpretation.
* It is not to be expected that in a real telegraph -circuit the successive waves
have abrupt fronts, as in the text. There are causes in operation to prevent this,
and round off the abruptness. The equations connecting V and C express the
first approximation to a complete theory. Thus the wires are assumed to be
instantaneously penetrated by the magnetic induction as a wave passes over their
surfaces, as if the conductors were infinitely thin sheets of the same resistance.
It is only a, very partial remedy to divide a wire into several thinner wires, unless
we at the same time widely separate them. If kept quite close it would, with
copper, be no remedy at all.
t Math, and Physical Papers, vol. ii., art. Ixxii. ; with Note by A. Cayley.
SPhil. Mag., June 1874.
390 ELECTRICAL PAPERS.
Perhaps there will be discovered some practical way of analysing them
into easily interpretable forms.
Some special cases of (66), (67) are worthy of notice. Thus V is
established in the same way when fi = Q as when K=0, provided the
value of K/S in the first case be the same as that of E/L in the second.
Calling this value 2q, we have in both cases
F= Jl - f) - S&.-'SELE'fcoB u + q sin A A ......... (76)
\ I / I Tfl \ A /
But the current is established in quite different manners. When it
is K that is zero, (71) is the solution; but if R vanish instead, then
(67) gives
. (77)
C now mounts up infinitely. But the leakage-current, which is KV,
becomes steady, as (76) shows.
In connexion with this subject I should remark that the distortionless
circuit produced by taking RjL = K/S is of immense assistance, as its
properties can be investigated in full detail by elementary methods, and
are most instructive in respect to the distortional circuits in question
above.*
Modifications made by Terminal Apparatus. Certain Cases easily
brought to Full Realization.
13. Suppose that the terminal conditions in the preceding are
V= Zjb and VZ-f^ Z Q and Z l being the "resistance-operators " of
terminal apparatus at A and B respectively. In a certain class of cases
the determinantal equation so simplifies as to render full realization
possible in an elementary manner. Thus, the resistance-operator of the
circuit, reckoned at A, ist
pJlZ^ten ml)/ml y '
where m 2 = - (R + Lp)(K + Sp) ........................ (79)
That is, e = $C is the linear differential equation of the current at A.
Now, to illustrate the reductions obviously possible, let Z Q = 0, and
Z^nJtf + Lp) ............................ (80)
This makes the apparatus at B a coil whose time-constant is LfR, and
reduces to
j. /r> T \ 7 /tanwz , \f, 2 yatan???^" 1
=*(R + Lp)U -- +n l \l 1 -msnjl 2 -- , ....... (81)
so that the roots of $ = are given by
(82)
tan m/ + 7/1^ = 0; .................... '. ...... (83)
*" Electromagnetic Induction and its Propagation," Arts. XL. to L. [vol. II.,
p. 119].
t "On the Self-Induction of Wires," Part IV. [vol. IL, p. 232].
ON ELECTROMAGNETIC WAVES. PART I. 391
i.e., a solitary root p = -E/L, and the roots of (83), which is an
elementary well-known form of determinantal equation.
The complete solution due to the insertion of the steady impressed
force e (} at A will be given by*
...................... (85)
where the summations range over all the p roots of > = 0, subject to
(79) ; whilst u and w are the V and C functions in a normal system,
expressed by
w = cos mz, u = m sin mz -f (K + Sp) ; ............ (86)
and F , C are the final steady V and C. In the case of the solitary
root (82) we shall find
^), ......................... (87)
but for all the rest
I dm 2 ,..
+V> ................ (88)
Realizing (84), (85) by pairing terms belonging to the two j?'s associ-
ated with one m 2 through (79), we shall find that (66), (67) express the
solutions, provided we make these simple changes : Divide the general
term in both the summations by (1 +n 1 cos%/), and the term following
C outside the summation in (67) by (1 4-?^). Of course the m's have
now different values, as per (83), and F" , 6 y are different.
14. There are several other cases in which similar reductions are
possible. Thus, we may have
Z l = n^R + Lp) + n((K + Sp)~\
simultaneously, n , n' , n v n{ being any lengths. That is, apparatus at
either end consisting of a coil and a condenser in sequence, the time-
constant of the coil being L/R and that of the condenser S/K. Or, the
condenser may be in parallel with the coil. In general we have, as an
alternative form of > = 0, equation (78),
ml 1 - mWZ^ { (R + Lp)l} ~ 2 '
from which we see that when
and
. .
(R + Lp}l
are functions of ml, equation (89) finds the value of m 2 immediately, i.e.
not indirectly as functions of p. In all such cases, therefore, we may
*lb. Parts III. and IV. Phil. Mag., Oct. and Nov. 1886; or "On Resistance
and Conductance Operators," Phil. Mag., Dec. 1887, 17, p. 500 [vol. TI., p. 373].
392 ELECTRICAL PAPERS.
advantageously have the general solutions (84), (85) put into the realized
form. They are
mz + tan 6 cos mz)m~
-
d(ml)
tanw
I same denominator
where q, A, s , s 2 are as in (64), (65). The differentiation shown in the
denominator is to be performed upon the function of ml to which tan ml
is equated in (89), after reduction to the form of such a function in the
way explained ; and 6 depends upon Z Q thus,
tan 6 = - mr\K+ Sp)Z , sec 2 6> = 1 + m~ 9 Zf(K+ Sp)*, (92)
which are also functions of ml. It should be remarked that the terms
depending upon solitary roots, occurring in the case m 2 = 0, are not
represented in (90), (91). They must be carefully attended to when
they occur.
NOTE A. The Electromagnetic Theory of Light.
An electromagnetic theory of light becomes a necessity, the moment one
realizes that it is the same medium that transmits electromagnetic dis-
turbances and those concerned in common radiation. Hence the electro-
magnetic theory of Maxwell, the essential part of which is that the vibra-
tions of light are really electromagnetic vibrations (whatever they may be),
and which is an undulatory theory, seems to possess far greater intrinsic
probability than the undulatory theory, because that is not an electro-
magnetic theory. Adopting, then, Maxwell's notion, we see that the only
difference between the waves in telephony (apart from the distortion and
dissipation due to resistance) and light-waves is in the wave-length ; and the
fact that the speed, as calculated by electromagnetic data, is the same as that
of light, furnishes a powerful argument in favour of the extreme relative
simplicity of constitution of the ether, as compared with common matter in
bulk. There is observational reason to believe that the sun sometimes causes
magnetic disturbances here of the ordinary kind. It is impossible to
attribute this to any amount of increased activity of emission of the
sun so long as we only think of common radiation. But, bearing in mind the
long waves of electromagnetism, and the constant speed, we see that
disturbances from the sun may be hundreds or thousands of miles long
of one kind (i.e. without alternation), and such waves, in passing the earth,
would cause magnetic " storms," by inducing currents in the earth's
crust and in telegraph-wires. Since common radiation is ascribed to
molecules, we must ascribe the great disturbances to movements of large
masses of matter.
There is nothing in the abstract electromagnetic theory to indicate whether
the electric or the magnetic force is in the plane of polarization, or rather,
surface of polarization. But by taking a concrete example, as the reflexion
of light at the boundary of transparent dielectrics, we get Fresnel's formula
for the ratio of reflected to incident wave, on the assumption that his " dis-
placement" coincides with the electric displacement ; and so prove that it is
the magnetic flux that is in the plane of polarization.
ON ELECTROMAGNETIC WAVES. PART I. 393
NOTE B. The Beneficial Effect of Self-Induction.
I give these numerical examples :
Take a circuit 100 kilom. long, of 4 ohms and | microf. per kilom. and no
inductance in the first place, and also no leakage in any case. Short-circuit
at beginning A and end B. Introduce at A a sinusoidal impressed force,
and calculate the amplitude of the current at B by the electrostatic theory.
Let the ratio of the full steady current to the amplitude of the sinusoidal
current be />, and let the frequency range through 4 octaves, from ft = 1250 to
n = 20,000 ; the frequency being H/ZTT. The values of p are
1-723, 3-431, 10-49, 58'87, 778.
It is barely credible that any kind of speaking would be possible, owing
to the extraordinarily rapid increase of attenuation with the frequency.
Little more than murmuring would be the result.
Now let Z = 2^ (very low indeed), L being inductance per centim.
Calculate by the combined electrostatic and magnetic formula. The
corresponding figures are
1-567, 2-649, 5 '587, 10'496, 16'607.
The change is marvellous. It is only by the preservation of the currents
of great frequency that good articulation is possible, and we see that
even a very little self-induction immensely improves matters. There
is no "dominant" frequency in telephony. What should be aimed at
is to get currents of any frequency reproduced at B in their proper pro-
portions, attenuated to the same extent.
Change L to 5. Results :
1-437, 2-251, 3-176, 4'169, 4'670.
Good telephony is now possible, though much distortion remains.
Increase L to 10. Results :
1-235, 1-510, 1-729, 1'825, 1'854.
This is first class, showing approximation towards a distortionless circuit.
Now this is all done by the self-induction carrying forward the waves
undistorted (relatively) and also with much less attenuation.
I should add that I attach no importance to the above figures in point
of exactness. The theory is only a first approximation. In order to
emphasize the part played by self-induction, I have stated that by sufficiently
increasing it (without other change, if this could be possible) we could
make the amplitude of current at the end of an Atlantic cable greater
than the steady current (by the g'wem'-resonance).
NOTE C. The Velocity of Electricity.
In Sir W. Thomson's article on the " Velocity of Electricity " (Nichols's
Cyclopaedia, 2nd edition, 1860, and Art. Ixxxi. of 'Mathematical and Physical
Papers,' vol. ii.) is an account of the chief results published up to that
date relating to the "velocity" of transmission of electricity, and a very
explicit statement, except in some respects as regards inertia, of the
theoretical meaning to be attached to this velocity under different circum-
stances. This article is also strikingly illustrative of the remarkable
contrast between Sir W. Thomson's way of looking at things electrical
(at least at that time) and Maxwell's views ; or perhaps I should say
Maxwell's plainly evident views combined with the views which his followers
have extracted from that mine of wealth ' Maxwell,' but which do not lie on
the surface. (As charity begins at home, I may perhaps illustrate by a
personal example the difference between the patent and the latent, in
394 ELECTRICAL PAPERS.
Maxwell. If I should claim (which I do) to have discovered the true
method of establishment of current in a wire that is, the current starting
on its boundary, as the result of the initial dielectric wave outside it,
followed by diffusion inwards, I might be told that it was all "in Max-
well." So it is ; but entirely latent. And there are many more things
in Maxwell which are not yet discovered.) This difference has been the
subject of a most moving appeal from Prof. G. F. Fitzgerald, in Nature,
about three years since. There really seemed to be substance in that
appeal. For it is only a master-mind that can adequately attack the
great constructional problem of the ether, and its true relation to matter ;
and should there be reason to believe that the master is on the wrong track,
the result must be, as Prof. Fitzgerald observed (in effect) disastrous to
progress. Now Maxwell's theory and methods have stood the test of
time, and shown themselves to be eminently rational and developable.
It is not, however, with the general question that we are here concerned,
but with the different kinds of "velocity of electricity." As Sir W.
Thomson points out, his electrostatic theory, by ignoring magnetic in-
duction, leads to infinite speed of electricity through the wire. Inter-
preted in terms of Maxwell's theory, this speed is not that of electricity
through the wire at all, but of the waves through the dielectric, guided by
the wire. It results, then, from the assumption /z = 0, destroying inertia
(not of the electric current, but of the magnetic field), and leaving only
forces of elasticity and resistance.
But he also points out another way of getting an infinite speed, when we,
in the case of a suspended wire, not of great length, ignore the static charge.
This is illustrated by the pushing of incompressible water through an
unyielding pipe, constraining the current to be the same in all parts of the
circuit. This, in Maxwell's theory, amounts to stopping the elastic dis-
placement in the dielectric, and so making the speed of the wave through it
infinite. As, however, the physical actions must be the same, whether
a wire be long or short, the assumption being only warrantable for purposes
of calculation, I have explained the matter thus. The electromagnetic
waves are sent to and fro with such great frequency (owing to the shortness
of the line) that only the mean value of the oscillatory V at any part can be
perceived, and this is the final value ; at the same time, by reason of current
in the negative waves being of the same sign as in the positive, the current
C mounts up by little jumps, which are, however, packed so closely together
as to make a practically continuous rise of current in a smooth curve,
which is that given by the magnetic theory. This curve is of course
practically the same all over the circuit, because of the little jumps being
imperceptible.
But in any case this speed is not the speed of electricity through the wire,
but through the dielectric outside it. Maxwell remarked that we know
nothing of the speed of electricity in a wire supporting current ; it may be
an inch in an hour, or immensely great. This is on the assumption,
apparently, that the electric current in a wire really consists in the transfer
of electricity through the wire. I have been forced, to make Maxwell's
scheme intelligible to myself, to go further, and add that the electricity may
be standing still, which is as much as to say that there is no current, in
a literal sense, inside a conductor. (The slipping of electrification over the
surface of a wire is quite another thing. That is merely the movement
of the wave through the dielectric, guided by the wire. It occurs in a
distortionless circuit, owing to the absence of tailing, in the most plainly
evident manner.) In other words, take Maxwell's definition of electric
current in terms of magnetic force as a basis, and ignore the imaginary
fluid behind it as being a positive hindrance to progress, as soon as one
ON ELECTROMAGNETIC WAVES. PART I. 395
leaves the elementary field of stead)/ currents and has to deal with variable
states.
The remarks in the text on the subject of the speed of waves in conductors
relates to a speed that is not considered in Sir W. Thomson's article, It is
the speed of transmission of magnetic disturbances into the wire, in
cylindrical waves, which begins at any part of a wire as soon as the primary
wave through the dielectric reaches that part. It would be no use trying to
make signals through a wire if we had not the outer dielectric to carry the
magnetizing and electrizing force to its boundary. The slowness of diffusion
in large masses is surprising. Thus a sheet of copper covering the earth,
only 1 centim, in thickness, supporting a current whose external field imitates
that of the earth, has a time-constant of about a fortnight. If the copper
extended to the centre of the earth, the time-constant of the most slowly sub-
siding normal system would be millions of years.
In the article referred to. Sir W. Thomson mentions that Kirch-
hoff's investigation, introducing magnetic induction, led to a velocity
of electricity considerably greater than* that of light, which is so far in
accordance with Wheatstone's observation. Now it seems to me that
we have here a suggestion of a probable explanation of why Sir W.
Thomson did not introduce self-induction into his theory. There were
presumably more ways than one of doing it, as regards the measure of
the electric force of induction. When we follow Maxwell's equations, there
is but one way of doing it, which is quite definite, and leads to a speed which
cannot possibly exceed that of light, since it is the speed (/xc)~ through
the dielectric, and cannot be sensibly greater than 3 x 10 10 centim., though
it may be less. Kirchhoff's result is therefore in conflict with Maxwell's
statement that the German methods lead to the same results as his.
Besides that, Wheatstone's classical result has not been supported by any
later results, which are always less than the speed of light, as is to
be expected (even in a distortionless circuit). But a reference to Wheat-
*(Note by SIR WILLIAM THOMSON.) In this statement I inadvertently did injustice to
Kirchhoff. In the unpublished investigation referred to in the article Electricity,
Velocity of [Nichols's Cyclopaedia, second edition, 1860; or my 'Collected Papers,' vol.
ii. page 135 (3)], I had found that the ultimate velocity of propagation of electricity in a
long insulated wire in air is equal to the number of electrostatic units in the electro-
magnetic unit ; and I had correctly assumed that Kirchhoff's investigation led to
the same result. But, owing to the misunderstanding of two electricities or one,
referred to in 317 of my ' Electrostatics and Magnetism,' I imagined Weber's measure-
ment of the number of electrostatic units in the electromagnetic to be 2x3'lxl0 10
centimetres per second, which would give for the ultimate velocity of electricity through
a long wire in air twice the velocity of light. In my own investigation, for the sub-
marine cable, I had found the ultimate velocity of electricity to be equal to the number
of electrostatic units in the electromagnetic unit divided by Vk ; k denoting the specific
inductive capacity of the gutta-percha. But at that time no one in Germany (scarcely
any one out of England) believed in Faraday's "specific inductive capacity of a
dielectric."
Kirchhoff himself was perfectly clear on the velocity of electricity in a long insulated
wire in air. In his original paper, "Ueber die Bewegung der Electricitat in Drahten"
(Pogg. Ann. Bd. c. 1857; see pages 146 and 147 of Kirchhoff's Volume of Collected
Papers, Leipzig, 1882), he gives it as c/\/2, which is what I then called the number of
electrostatic units in the electromagnetic unit ; and immediately after this he says,
" ihr Werth ist der von 41950 Meilen in einer Sekunde, also sehr nahe gleich der
Geschwindigkeit des Lichtes im leeren Raume."
Thus clearly to Kirchhoff belongs the priority of the discovery that the velocity of
electricity in a wire insulated in air is very approximately equal to the velocity of light.
[Note by THE AUTHOR. In Maxwell's theory, however, as I understand it, we are not
at all concerned with the velocity of electricity in a wire (except the transverse velocity
of lateral propagation). The velocity is that of the waves in the dielectric outside
the wire.]
396 ELECTRICAL PAPERS.
.stone's paper on the subject will show, first, that there was confessedly
a good deal of guesswork ; and, next, that the repeated doubling of the
wire on itself made the experiment, from a modern point of view, of
too complex a theory to be examined in detail, and unsuitable as a test.
PART II.
NOTE ON PART I. The Function of Self-Induction in the Propagation of
Waves along Wires*
An editorial query, the purport of which I did not at first understand,
has directed my attention to Prof. J. J. Thomson's paper " On Electrical
Oscillations in Cylindrical Conductors" (Proc. Math. Soc., vol. xvii.,
Nos. 272, 273), a copy of which the author has been so good as to send
me. His results, for example, that an iron wire of \ centim. radius,
of inductivity 500, carries a wave of frequency 100 per second about
100,000 miles before attenuating it from 1 to c" 1 , and similar results,
summed up in his conclusion that the carrying-power of an iron-wire
cable is very much greater than that of a copper one of similar dimen-
sions, are so surprisingly different from my own, deduced from my
developed sinusoidal solutions, in the accuracy of which I have perfect
confidence (having had occasion last winter to make numerous practical
applications of them in connexion with a paper which was to have been
read at the S. T. E. and E.) [see Art. XLL, vol. IL, p. 323], that I felt
sure there must be some serious error of a fundamental nature running
through his investigations. On examination I find this is the case,
being the use of an erroneous boundary condition in the beginning,
which wholly vitiates the subsequent results [relating to the effect of
magnetisation]. It is equivalent to assuming that the tangential com-
ponent of the flux magnetic induction is continuous at the surface of
separation of the wire and dielectric, where the inductivity changes
value, from a large value to unity, when the wire is of iron. The true
conditions are continuity of tangential force and of normal flux.
As regards my own results, and how increasing the inductance is
favourable, the matter really lies almost in a nutshell ; thus. In order to
reduce the full expression of Maxwell's connexions to a practical
working form I make two assumptions. First, that the longitudinal
component of current (parallel to the wires) in the dielectric is negli-
gible, in comparison with the total current in the conductors, which
makes C one of the variables, C being the current in either conductor ;
and next, what is equivalent to supposing that the wave-length of
disturbances transmitted along the wires is a large multiple of their
distance apart. The result is that the equations connecting Fand C
become
- dr/dz = R"C, - dCjdz = KV+ SV\
S being the permittance and K the conductance of the dielectric per
unit length of circuit, whilst R" is a " resistance-operator," depending
* This note may be regarded as a continuation of Note B [p. 393, vol. n.].
ON ELECTROMAGNETIC WAVES. PART II. 397
upon the conductors, and their mutual position, which, in the sinusoidal
state of variation, reduces to
where R f and L' are the effective resistance and inductance of the
circuit respectively, per unit length, to be calculated entirely upon
magnetic principles. It follows that the fully developed sinusoidal
solution is of precisely the same form as if the resistance and induct-
ance were constants. Disregarding the effect of reflexions, we have
r=r o - fz sin (nt-Qz),
due to VQ sin nt impressed at z = ; where P and Q are functions of
R', L', S, K, and n.
Now if R'lL'n is large, and leakage is negligible (a well-insulated
slowly-worked submarine cable, and other cases), we have
as in the electrostatic theory of Sir W. Thomson. There is at once
great attenuation in transit, and also great distortion of arbitrary
waves, owing to P and Q varying with n.
But in telephony, n being large, P and Q may have widely different
values, because R'jUn may be quite small, even a fraction. In such
case we have no resemblance to the former results. If R'jL'n is small,
P and Q approximate to
P = R'l^L'vf + K/2S^ Q = /t/,
where v' = (L f S)-^. This also requires KjSn to be small. But it is
always very small in telephony.
Now take the case of copper wires of low resistance. L f is practically
L ot the inductance of the dielectric, and v f is practically v, the speed of
undissipated waves, or of all elementary disturbances, through the
dielectric, whilst R' may be taken to be R, the steady resistance, except
in extreme cases. Hence, with perfect insulation,
P = /2L Q v, Q = n/v,
or the speed of the waves is v, and the attenuating coefficient P is practi-
cally independent of the frequency, and is made smaller by reducing
the resistance, and by increasing the inductance of the dielectric.
The corresponding current is
very nearly, or V and C are nearly in the same phase, like undissipated
plane waves. There is very little distortion in transit.
How to increase L Q is to separate the conductors, if twin wires, or
raise the wire higher from the ground, if 'a single wire with earth-
return. It is not, however, to be concluded that L could be increased
indefinitely with advantage. If / is the length of the circuit,
Iil = 2L Q v
shows the value of L Q which makes the received current greatest. It
is then far greater than is practically wanted, so that the difficulty of
increasing L sufficiently is counterbalanced by the non-necessity. The
best value of L is, in the case of a long line, out of reach ; so that we
may say, generally, that increasing the inductance is always of advant-
age to reduce the attenuation and the distortion.
398 ELECTRICAL PAPERS.
Now if we introduce leakage, such that E/L = K/S, we entirely
remove the distortion, not merely when EjL^n is small, but of any sort
of waves. It is, however, at the expense of increased attenuation. The
condition of greatest received current, L being variable, is now
W = L Q v.
We have thus two ways of securing good transmission of electromag-
netic waves : one very perfect, for any kind of signals ; the other less
perfect, and limited to the case of fi/L n small, but quite practical.
The next step is to secure that the receiving-instrument shall not intro-
duce further distortion by the quasi-resonance that occurs. In the truly
distortionless circuit this can be done by making the resistance of the
receiver be L^v (whatever the length of the line) ; this causes complete
absorption of the arriving waves. In the other case, ofE/L Q n small, with
good insulation, we require the resistance of the receiver to be also L e
to secure this result approximately. I have also found that this value
of the receiver's resistance is exactly the one that (when size of wire in
receiver is variable) makes the magnetic force, and therefore the
strength of signal, a maximum. Some correction is required on
account of the self-induction of the receiver ; but in really good tele-
phones of the best kind, with very small time-constants, it is not great.
We see therefore that telephony, so far as the electrical part of the
matter is concerned, can be made as nearly perfect as possible on lines
of thousands of miles in length. But the distortion that is left, due to
imperfect translation of sound waves into electromagnetic waves at the
sending-end, and the reproduction of sound-waves at the receiving-end,
is still very great ; though, practically, any fairly good telephonic
speech is a sufficiently good imitation of the human voice.
There is one other way of increasing the inductance which I have
described, viz., in the case of covered wires to use a dielectric impreg-
nated with iron dust. I have proved experimentally that L Q can be
multiplied several times in this way without any increase in resistance ;
and the figures I have given above (in Note B) prove what a wonderful
difference the self-induction makes, even in a cable, if the frequency is
great. Hence, if this method could be made practical, it would greatly
increase the distance of telephony through cables.
Now, passing to iron wires, the case is entirely different, on account
of the great increase in resistance that the substitution of iron for
copper of the same size causes, which increases P and the attenuation.
Taking for simplicity the very extreme case of such an excessive
frequency as to make the formula
nearly true, R being the steady and E r the actual resistance, we see
that increasing either R or /x, increases R' and therefore P, because ZV
tends to the value L v. Thus the carrying power of iron is not greatly
above, but greatly below that of copper of the same size.
I have, however, pointed out a possible way of utilizing iron (other
than that above mentioned), viz., to cover a bundle of fine iron wires
with a copper sheath. The sheath is to secure plenty of conductance ;
ON ELECTROMAGNETIC WAVES. PART II.
399
the division of the iron to facilitate the penetration of current, and so
lower the resistance still more, to the greatest extent, whilst at the same
time increasing the inductance. But the theory is difficult, and it is
doubtful whether this method is even theoretically legitimate. First
class results were obtained by Van Rysselberghe on a 1000-mile circuit
in America (2000 miles of wire), using copper-covered steel wire. Here
the resistance was very low, on account of the copper, and the induct-
ance considerable, on account of the dielectric alone ; so that there is no
certain evidence that the iron did any good except by lowering the
resistance. But about the advantage of increasing the inductance of
the dielectric there can, I think, be no question. It imparts momentum
to the waves, and that carries them on.
In Note B to the first part of this paper [p. 393 ante], I gave four sets
of numerical results showing the influence of increasing the inductance,
selecting a cable of large permittance (constant) in order to render the
illustrations more forcible. The formula used was equation (82), Part II.
of my paper "On the Self-induction of Wires" [p. 195 ante], which is
-2 cos 2$)"*;
where
P or =
Here C is the amplitude of current at z = l due to impressed force
"
F" sin nt at z = 0, with terminal short-circuits.
enough to make t~ ri small, we obtain
When the circuit is long
as the expression for the ratio /> of the steady current to the amplitude
of the sinusoidal current.
The following table is constructed to show the fluctuating manner of
variation of the amplitude with the frequency. Drop the accents, and
let R/Ln be small. Then, approximately,
where y = Itl/Lv,
under no restriction as regards the length of the circuit. Now give y a
succession of values, and calculate p with the cosine taken as -1,0, and
+ 1. Call the results the maximum, mean, and minimum values of />.
y-
Min. p.
Mean p.
Max. p.
y-
P-
y-
P-
i
2
505
1-500
2-063
6
1-678
12
16-81
1
521
878
1-128
7
2-365
14
39-3
2
587
686
771
8
3-378
16
93-2
2-065
594
685 1 -766
9
5-000
18
225
3
710
748
784
10
7-420
20
550
4
907
924
940
5
1-210
1-218
1-226
400 ELECTRICAL PAPERS.
It will be seen that when the resistance of the circuit varies from a
small fraction to about the same magnitude as Lv (which may be
from 300 to 600 ohms in the case of a suspended copper wire), the
variation in the value of p as the frequency changes through a
sufficiently wide range, is great, merely by reason of the reflexions
causing reinforcement or reduction of the strength of the received
current. The theoretical least value of p is J, when RjLn is vanishingly
small, indicating a doubling of the amplitude of current. But, as y
increases, the range of p gets smaller and smaller. After y = 5 it is
negligible.
It is, however, the mean p that is of most importance, because the
influence of terminal resistances is to lower the range in />, and to a
variable extent. The value y= 2*065, or, practically, El = 2Lv } makes
the mean p a minimum. As I pointed out in the paper before referred
to, these fluctuations can only be prejudicial to telephony. In the
present Note I have described how to almost entirely destroy them.
The principle may be understood thus. Let the circuit be infinitely long
first. Then its impedance to an intermediate impressed force alternat-
ing with sufficient frequency to make R/Ln small will be 2Lv, viz., Lv
each way. The current and transverse voltage produced will be in the
same phase, and in moving away from the source of energy they will be
similarly attenuated according to the time-factor e -^/ 2 ^. In order that
the circuit, when of finite length, shall still behave as if of infinite
length, the constancy of the impedance suggests to us that we should
make the terminal apparatus a mere resistance, of amount Lv, by which
the waves will be absorbed without reflexion.
That this is correct we may prove by my formula for the amplitude
of received current when there is terminal apparatus, equation (195),
Part V. "On the Self-Induction of Wires" (Phil. Mag., Jan. 1887). It
is
Here C Q is the amplitude of received current at z = I due to V Q sin nt
impressed force at = 0, R' and L' the effective resistance and induct-
ance per unit length of circuit ; K and S the leakage-conductance and
permittance per unit length,
P or Q = (I
6r , H Q , are terminal functions depending upon the apparatus at z = ;
G v H v upon that at z = I ; the apparatus being of any kind, specified by
resistance-operators, making R f Q , L f the effective resistance and induct-
ance of apparatus at z = 0, and R{, L{, at z = I. G is given by
from which H is derived by changing the signs of P and Q ; whilst
ON ELECTROMAGNETIC WAVES. PART II. 401
#j and ZTj are the same functions of R{, L( as G and H are of
RQ, LQ.
Now drop the accents, since we have only copper wires of low resist-
ance (but not very thick) in question, and the terminal apparatus are
to be of the simplest character. K/Sn will be vanishingly small prac-
tically, so take K=0. Next let R/Ln be small, and let the apparatus
at z = l be a mere coil, R v of negligible inductance first. We shall
now have
P = Jt/2Lv, Q = n/v,
and these make G? = ( 1 + RJLv), H$ = (\- RJLv).
Thus R 1 = Lv makes H^ vanish, whatever the length of lin(
terms due to reflexions disappear.
We now have
where 6r~i expresses the effect of the apparatus at z = in reducing the
potential-difference there, F" being the impressed force, and the value
of GQ being unity when there is a short-circuit.
Now, to show that R l = Lv makes the magnetic force of the receiver
the greatest, go back to the general formula, let ~ pl be small, and let
the size of the wire vary, whilst the size of the receiving-coil is fixed.
It will be easily found, from the expression for G lt that the magnetic
force of the coil is a maximum when
\l
)'
where we keep in L v the inductance of the receiver. Or, when R/Ln
and KjSn are both small,
or, as described, R l = Lv when the receiver has a sufficiently small time-
constant. The rule is, equality of impedances.
We may operate in a similar manner upon the terminal function at
the sending end. Suppose the apparatus to be representable as a
resistance containing an electromotive force, and that by varying the
resistance we cause the electromotive force to vary as its square root.
Then, according to a well-known law, the arrangement producing the
maximum external current is given by R Q = Lv, equality of impedances
again. This brings us to
as if the circuit were infinitely long both ways, with maximum efficiency
secured at both ends.
Lastly, the choice of L such that Rl - 2Lv makes the circuit, of given
resistance, most efficient.
In long-distance telephony using wires of low resistance, the waves
are sent along the circuit in a manner closely resembling the trans-
mission of waves along a stretched elastic cord, subject to a small
amount of friction. In order to similarly imitate the electrostatic
H.E.P. VOL. ii. 2c
402 ELECTRICAL PAPERS.
theory, we must so reduce the mass of the cord, or else so exaggerate
the friction, that there cannot be free vibrations. We may suppose
that the displacement of the cord represents the transverse voltage in
both cases. But the current will be in the same phase as the transverse
voltage in one case, and proportional to its variation along the circuit
in the other.
We may conveniently divide circuits, so far as their signalling
peculiarities are concerned, into five classes. (1). Circuits of such
short length, or so operated upon, that any effects due to electric
displacement are insensible. The theory is then entirely magnetic, at
least so far as numerical results are concerned. (2). Circuits of such
great length that they can only be worked so slowly as to render
electromagnetic inertia numerically insignificant in its effects. Also
some telephonic circuits in which fi/Ln is large. Then, at least so far
as the reception of signals is concerned, we may apply the electrostatic
theory. (3). The exceedingly large intermediate class in which both
the electrostatic and magnetic sides have to be considered, not separ-
ately, but conjointly. (4). The simplified form of the last to which we
are led when the signals are very rapid and the wires of low resistance.
(5). The distortionless circuit, in which, by a proper amount of uniform
leakage, distortion of signals is abolished, whether fast or slow.
Regarded from the point of view of practical application, this class lies
on one side. But from the theoretical point of view, the distortionless
circuit lies in the very focus of the general theory, reducing it to simple
algebra. I was led to it by an examination of the effect of telephones
bridged across a common circuit (the proper place for intermediate
apparatus, removing their impedance) on waves transmitted along the
circuit. The current is reflected positively, the charge negatively, at a
bridge. This is opposite to what occurs when a resistance is put in
the main circuit, which causes positive reflexion of the charge, and
negative of the current. Unite the two effects and the reflexion of the
wave is destroyed, approximately when the resistance in the main
circuit and the bridge-conductance are finite, perfectly when they are
infinitely small, as in a uniform distortionless circuit.
PART III.
SPHERICAL ELECTROMAGNETIC WAVES.
15. Leaving the subject of plane waves, those next in order of
simplicity are the spherical. Here, at the very beginning, the question
presents itself whether there can be anything resembling condensational
waves ?
Sir W. Thomson (" Baltimore Lectures", as reported by Forbes in
Nature, 1884) suggested that a conductor charged rapidly alternately +
and - would cause condensational waves in the ether. But there is no
other way of charging it than by a current from somewhere else, so he
suggested two conducting spheres to be connected with the poles of an
ON ELECTROMAGNETIC WAVES. PART III. 403
alternating dynamo. The idea seems to be here that electricity would
be forced out of one sphere and into the other to and fro with great
rapidity, and that between the spheres there might be condensational
waves.
But in this case, according to the Faraday law of induction, the
result would be the setting up of alternating electromagnetic disturb-
ances in the dielectric, exposing the bounding surfaces of the two
spheres to rapidly alternating magnetizing and electrizing force, causing .
waves, approximately spherical at least, to be transmitted into the
spheres, in the diffusion manner, greatly attenuating as they progressed
inward.
Perhaps, however, there can be condensational waves if we admit
that a certain quite hypothetical something called electricity is com-
pressible, instead of being incompressible, as it must be if we in
Maxwell's scheme make the unnecessary assumption that an electric
current is the motion through space of the something. In fact, Prof.
J. J. Thomson has calculated* the speed of condensational waves
supposed to arise by allowing the electric current to have convergence.
But a careful examination of his equations will show that the con-
densational waves there investigated do not exist, i.e., the function
determining them has the value zero.f
16. To construct a perfectly general spherical wave we may proceed
thus. The characteristic equation of H, the magnetic force, in a homo-
geneous medium free from impressed force is, by (2) and (3),
V2H = (4w/i^ + /tfjp*)H (93)
Now, let r be the vector distance from the origin, and Q any scalar
function satisfying this equation. Let
H = curl(rQ) (94)
Then this derived vector will satisfy (93), and have no convergence,
and have no radial component, or will be arranged in spherical sheets
From it derive the other electromagnetic quantities. Change H to E
to obtain spherical sheets of electric force.
This method leads to the spherical sheets depending upon any kind
of spherical harmonic. They are, however, too general to be really
useful except as mathematical exercises. For the examination of the
manner of origin and propagation of waves, zonal harmonics are more
useful, besides leading to the solution of more practical problems. It
is then not difficult to generalize results to suit any kind of spherical
harmonics.
The Simplest Spherical Waves.
17. Let the lines of H be circles, centred upon the axis from which
is measured, and let r be the distance from the origin. We have no
concern with < (longitude) as regards H, so that the simple specification
* B.A. Report on Electrical Theories.
1 1 ought to qualify this by adding that the investigation seems very obscure,
so that, although I cannot make the system work, yet others may.
404 ELECTRICAL PAPERS.
of its intensity H fully defines it. Under these circumstances the
equation (93) becomes
(95)
= q*rH, say,
where the acute accent denotes differentiation to r, and the grave accent
to cos 6 or n, whilst v stands for sin 9. The inductivity will be now /* ,
to avoid confusing with the p of zonal harmonics. Equation (95) also
defines q in the three forms it can assume in a conductor, dielectric, and
conducting dielectric.
Now try to make rH be an undistorted spherical wave, i.e. H varying
inversely as the distance, and travelling inward or outward at speed v.
Let
rH=Af(r-vt), ............................. (96)
where A is independent of r and t. Of course we must have & = 0,
making q =p/v. Now (96) makes
v\rH)" = rfH', .................................. (97)
which, substituted in (95), gives
v(v#)^ = 0; ...................................... (98)
therefore Av = A l ^ + S l .............................. (99)
From these we find the required solutions to be
(100)
(101)
where F is any function, A l and B l constants, E and F the two com-
ponents of the electric force, F being the radial component out, and E
the other component coinciding with a line of longitude, the positive
direction being that of increasing 0, or from the pole. Similarly, if the
lines of E be circular about the axis, we have the solutions
S- -ftpfl.- -^4^FI,(r-vt), ............. (102)
H r -^FJf-^, ........................ (103)
where H r and H e are the radial and tangential components of H.
But both these systems involve infinite values at the axis. We must
therefore exclude the axis somehow to make use of them. Here is one
way. Describe a conical surface of any angle 6 V and outside it another
of angle # 2 , and let the dielectric lie between them. Make the tan-
gential component of E at the conical surfaces vanish, requiring infinite
conductivity there, and we make F vanish in (101), and produce the
solution
(104)
ON ELECTROMAGNETIC WAVES. PART III. 405
exactly resembling plane waves as regards rvE. Here B is the same as
/V-#i and/ the same as F f , in equation (100).*
18. Now bring in zonal harmonics. Split equation (95) into the two
(rHY' = { q * + ?+VyH, (105)
^ / rr\\> 7/l\Til T 1 ) TT /irk\
-^(vJti) = i - '-it V*^"/
The equation (106) has for solution
where A is independent of (9, and is to be found from (105).
The most practical way of getting the r functions is that followed by
Professor Rowland in his paper f wherein he treats of the waves
emitted when the state is sinusoidal with respect to the time. We shall
come across the same waves in some problems.
Let H=P m -vQ^ ............................. (107)
Then the equation of P m is, by insertion of (107) in (105),
(108)
* In order to render this arrangement (104) intelligible in terms of more every-
day quantities, let the angles 6 l and 6% be small, for simplicity of representation ;
then we have two infinitely conducting tubes of gradually increasing diameter
enclosing between them a non-conducting dielectric. Now change the variables.
Let V be the line-integral of E across the dielectric, following the direction of the
force ; it is the transverse voltage of the conductors. Let 4cirC be the line-integral
of H round the inner tube ; it is the same for a given value of r, independent of 6 ;
C is therefore what is commonly called the current in the conductor. We shall
have
V= LvC, C=SvV, LSv 2 = 1 ;
where L is the inductance and 8 the permittance, per unit length of the circuit.
The value of L is
L = 2/t log [(tan |0 2 ) (tan \OJ\ ;
so that the circuit has uniform inductance and permittance. The value of G in
terms of (104) is
When the tubes have constant radii c^ and a 2 , the value of L reduces to the well
known
of concentric cylinders. The wave may go either way, though only the positive
wave is mentioned.
-\-PhiL Mag., June 1884, " On the Propagation of an Arbitrary Electromagnetic
Disturbance, Spherical Waves of Light, and the Dynamical Theory of Refraction."
Prof. J. J. Thomson has also considered spherical waves in a dielectric in his paper
" On Electrical Oscillations and Effects produced by the Motion of an Electrified
Sphere," Proc. London Math. Soc. vol. xv., April 3, 1884. [See also Stoke's
Mathematical and Physical Papers, and Rayleigh's Sound on the subject of
these functions.!
406 ELECTRICAL PAPERS.
and the solution, for practical purposes with complete harmonics, is
m(m 2 -l 2 )(m+2) m(m 2 -l)(m 2 -2 2 )(m+3)
2qr
We shall find the first few useful, thus :
P 1= =l- (qr)-\ }
, .............. (110)
Now let U= eT, so that U is the r function in Hr. If we change
the sign of q in U, producing, say, W, it is the required second solution
of (105). Thus
_
in the very important case of Q lt when m= 1.
The conjugate property of Z7 and W is
U'tr=-2q, ........................ (112)
which is continually useful.
We have next to combine U and W so as to produce functions suitable
for use inside spheres, right up to the centre, and finite there. Let
u = \(U+W), w = i(U-W), .............. (113)
It will be found that when m is even, w/r is zero and u/r infinite at the
origin ; but that when m is odd, it is u/r that is zero at the origin and
w infinite.
The conjugate property of u and w is
uw' - u'w = q, ............................ (114)
corresponding to (112).
Construction of the Differential Equations connected with a Spherical Sheet of
Vorticity of Impressed Force.
19. Now let there be two media one extending from r = to r = a, in
which we must therefore use the w-function or ^-function, according as
m is odd or even, and an outer medium, or at least one in which q has a
different form in general. Then, within the sphere of radius a, we have
H=Ar-iu, ............................... (115)
-^ = ^r-V, .............................. (116)
where ^ = 4=7rk + cp, and we suppose m odd. It follows that
E 1 u' (H7)
r=-^
In the outer medium use W, if the medium extends to infinity, or both
U and W if there be barriers or change of medium. First, let it be an
infinitely extended medium. Then, in it,
H=Br~\u-w\ ............................ (118)
r-\u f -w r ], .......................... (119)
ON ELECTROMAGNETIC WAVES. PART III. 407
where k 2 = kirk + cp in the outer medium. From these
B\*- f ............................ (120)
H k 2 u-w
(117) and (120) show the forms of the resistance-operators on the two
sides.*
Now, at the surface of separation, r = a, H is continuous (unless we
choose to make it a sheet of electric current, which we do not) ; so that
the H in (117) and (120) are the same. We only require a relation
between the E's to complete the differential equation.
Let there be vorticity of impressed force on the surface r = a, and
nowhere else (the latter being already assumed). Then
curle = curlE .......................... (121)
is the surface-condition which follows ; or, if / be the measure of the
curl of e,
/-,-* ............................. (122)
E 2 meaning the outer and E l the inner E. Therefore
(123)
H a denoting the surface H. So, by (117) and (120), used in (123),
/i 4 1 "L^SW (r=a)) ............. (124)
\& U, k z U z -wJ
the required differential equation. Observe that u^ only differs from
u 2 and w 1 from w 2 in the different values of q inside and outside (when
different), and that r = a in all.
* Some rather important considerations are presented here. On what principles
should we settle which functions to use internally and externally, seeing that these
functions U and W are not quantities, but differential operators ? First, as regards
the space outside the surface of origin of disturbances. The operator e r turns
J\t) iutojlt + r/v), and can therefore only be possible with a negative wave, coming
to the origin. But there cannot be such a wave without a barrier or change of
medium to produce it. Hence the operator e~9 r alone can be involved in the exter-
nal solution when the medium is unbounded, and we must use W. Next, go inside
the sphere r = a. It is clear that both U and W are now needed, because disturb-
ances come to any point from the further as well as from the nearer side of the
surface, thus coming from and going to the centre. Two questions remain : Why
take U and W in equal ratio ; and why their sum or their difference, according as
m is odd or even ? The first is answered by stating the facts that, although it is
convenient to assume the origin to be a place of reflection, yet it is really only a
place where disturbances cross, and that the H produced at any point of the sur-
face is (initially) equal on both sides of it. The second question is answered by
stating the property of the Q} n function, that it is an even function of /j. when m is
odd, and conversely ; so that when m is odd the H disturbances arriving at any
point on a diameter from its two ends are of the same sign, requiring U+ W ', and
when m is even, of opposite sign, requiring U - W.
Similar reasoning applies to the operators concerned in other than spherical
waves. Cases of simple diffusion are brought under the same rules by generalizing
the problem so as to produce wave-propagation with finite speed. On the other
hand, when there are barriers, or changes of media, there is no difficulty, because
the boundary conditions tell us in what ratio U and W must be taken.
408 ELECTRICAL PAPERS.
Equation (124) applies to any odd m. When m is even, exchange u
and w, also u' and w'. In the m th system we may write
the form of < being given in (124). The vorticity of the impressed
force is of course restricted to be of the proper kind to suit the m th zonal
harmonic. Thus, any distribution of vorticity whose lines are the lines
of latitude on the spherical surface may be expanded in the form
2/ m vCi, .............................. (126)
and it is the m th of these distributions which is involved in the preceding.
20. Both media being supposed to be identical, < reduces to
= ! _ 2 _ , ...(127)
*!.(.->.)
by using (114) in (124). This is with m odd ; if even, we shall get
< = !- jJL -- .......................... (128)
^ *.(. -w.)
In a non-dielectric conductor, & 1 = 47r&, and cf = ^Tr^p; so that,
keeping to m odd,
__ ..................... (129)
U a (U a -W a )
In a non-conducting dielectric, ^ = cp, and q =p/v ; so
= ,*>* . .............................. (130)
.(.- O
In this case the complete differential equation is
- Mo
when there is any distribution of impressed force in space whose vor-
ticity is represented by (126).
Outside the sphere, consequently,
a"" r >(132)
(out)
.(133)
understanding that when no letter is affixed to u or w, the value at
distance r is meant. We see at once that u a = makes the external
field vanish, i.e., the field of the particular / concerned. This happens
when / is a sinusoidal function of the time, at definite frequencies.
Also, inside the sphere,
(135)
ON ELECTROMAGNETIC WAVES. PART III. 409
As for the radial component F t it is not often wanted. It is got thus
from H:
-cpF=r-\vH)\ .......................... (136)
where for cp write 4?r^ + cp in the general case. Thus, the internal F
corresponding to (135) is
(in) tp*-2fil)K-OWU .............. (137)
Practical Problem. Uniform Impressed Force, in the Sphere.
21. If there be a uniform field of impressed force in the sphere,
parallel to the axis, of intensity f v its vorticity is represented by / x sin 6
on the surface of the sphere. It is therefore the case m = 1 in the above.
Let this impressed force be suddenly started. Find the effect produced.
We have, by (132),*
(out) H=u a (u-w)^- } ........................... (138)
or, in full, referring to the forms of u and w, equations (110) to (113),
->A - JLYl + IW -*, and makes the last method fail completely, all trace of the roots having
disappeared. But if we pass continuously from one case to the other, then the last
formula becomes a definite integral. On the other hand, we can immediately
integrate /= (f>H in its simplified form, and obtain an interpretable equivalent for
the definite integral, the latter being more ornamental than useful. In the simpli-
fied form, may be either rational or irrational. The integration of the irrational
forms will be given in some later problems.
410 ELECTRICAL PAPERS.
It is particularly to be noticed that the ^ part of (141) only comes into
operation when ^ reaches zero, and similarly as regards the t 2 part.
Thus, the first part expresses the primary wave out from the surface ;
the second, arriving at any point 2a/v later than the first, is the reflected
wave from the centre, arising from the primary wave inward from the
surface.
The primary wave outward may be written
where vt>(r-a), and the second wave by its exact negative, with
vt>(r + a). Now, by comparing (132) with (134), we see that the
internal solution is got from the external by exchanging a and r in the
{}'s in (139) and (141), including also in ^ and t 2 . The result is that
(142) represents the internal H in the primary inward wave, vt having
to be >(a-r)-, whilst its negative represents the reflected wave,
provided vt>(a + r).
The whole may be summed up thus. First, vt is a, H is given by the same formula between the limits r = vt-a and
vt + a. In both cases jETis zero outside the limits named.
The reflected wave, superimposed on the primary, annuls the H
disturbance, which is therefore, after the reflexion, confined to a
spherical shell of depth 2a containing the uncancelled part of the
primary wave outward.
The amplitude of H at the front of the two primary waves, in and
out, before the former reaches the centre, is
After the inward wave has reached the centre, however, the amplitude
of H on the front of the reflected wave is the negative of that of the
primary wave at the same distance, which is itself negative.
The process of reflexion is a very remarkable one, and difficult to
fully understand. At the moment t = a/v that the disturbance reaches
the centre, we have H = (f l v) + (4/* 0), constant, all the way from r =
to 2&, which is just half the initial value of H on leaving the surface of
the sphere. But just before reaching the centre, H runs up infinitely
for an infinitely short time, infinitely near the centre ; and just after
the centre is reached we have H = GO infinitely near the centre, where
the ^-disturbance is always zero, except in this singular case when it
is seemingly finite for an infinitely short time, though, of course, v is
indeterminate.
With respect to this running-up of the value of H in the inward
primary wave, it is to be observed that whilst H is increasing so fast at
and near its front, it is falling elsewhere, viz., between near the front
and the surface of the sphere ; so that just before the centre is reached
H has only half the initial value, except close to the centre, where it is
enormously great.
After reflexion has commenced, the ^-disturbance is negative in the
hinder part of the shell of depth 2a which goes out to infinity, positive
ON ELECTROMAGNETIC WAVES. PART III. 41 1
of course still in the forward part. At a great distance these portions
become of equal depth a ; at the front of the shell H=(f l va)(2p Q vr)~ l ,
at its back H= - ditto ; using of course a different value of r.
22. As regards the electric field, we have, by (133),
(out) E= --L-H. \(u' -*)/i; .............. (143)
which, expanded, is
comparing which with (139), we see that
qa
We have, therefore, only to develop the second part, which is not in
the same phase with H. It is, in the same manner as before,
, U6)
only operating when vt l = vt r + a, and vt z = vt r-a are positive. Or,
1 and 2 referring to the two waves. So, when vt> (r + a), and the two
are coincident, we have the sum
which is the tangential component of the steady electric field left
behind.
The radial component F is, by (137),
(out)
where the unwritten term . . . may be obtained from the preceding by
changing the sign of a. Or
...... " 49)
where vt 1 = vt + a- r. Or,
-*--*-''#+~* < 150 >
so that, when both waves coincide, we have their sum,
F 2 /i a3cos ^
'"
which is the radial component of the steady field left behind by the
part of the primary wave whose magnetic field is wholly cancelled.
412 ELECTRICAL PAPERS.
To verify ; the uniform field of impressed force of intensity f lt by
elementary principles, produces the external electric potential
whose derivatives, radial and tangential, taken negatively, are (151)
and (147). The corresponding internal potential is
ft = J// cos 6.
But its slope does not give the force E left behind within the sphere,
because this E is the force of the flux. Any other distribution of
impressed force, with the same vorticity, will lead to the same E. Our
equation (135) and its companion for F, derived from (134) by using
(136), lead to the steady field (residual)
E=-$f l m6, ^f/icosfl, ............... (152)
the components of the true force of the flux. Add e to the slope of ft
to produce E.*
F is always zero at the front of the primary wave outward, and
E = fJ'QvH. At the front of the primary wave inward F is also zero,
and E = - p^H. After reflection, F at the front of the reflected wave
is still zero, but now E = ^vH.
The electric energy U l set up is the volume-integral of the scalar
product ^eD. That is,
Di-ttx^x***^ ................... (153)
But the total work done by e is 2 U v by the general law that the
whole work done by impressed forces suddenly started exceeds the
amount representing the waste by Joule-heating at the final rate (when
there is any), supposed to start at once, by twice the excess of the
electric over the magnetic energy of the steady field set up. It is
clear, then, that when the travelling shell has gone a good way out, and
it has become nearly equivalent to a plane wave, its electric and mag-
netic energies are nearly equal, and each nearly J U^ in value. I did
not, however, anticipate that the magnetic energy in the travelling
shell would turn out to be constant, viz., %U l during the whole journey,
from t = a/v to t = oo , so that it is the electric energy in the shell which
gradually decreases to J U r Integrate the square of H according to
(142) to verify.
23. The most convenient way of reckoning the work done, and also
the most appropriate in this class of problems, is by the integral of the
* Sometimes the flux is apparently wrongly directed. For example, a uniform
field of impressed force from left to right in all space except a spherical portion
produces a flux from right to left in that portion. This is matte intelligible by
the above. Let the impressed force act in the space between r=a and r=b, a
being small and b great. In the inner sphere the first effects are those due to the
r=a vorticity, and the flux left behind is against the force. But after a time
comes the wave from the r=b vorticity, which sets matters right. The same
applies in the case of conductors, when, in fact, a long time might have to elapse
before the second and real permanent state conquered the first one.
ON ELECTROMAGNETIC WAVES. PART III. 413
scalar product of the curl of the impressed force and the magnetic force.
Thus, in our problem
> ...... (154)
where dS is an element of the surface r = a. So we have to calculate
the time-integral of the magnetic force at the place of vorticity of e, the
limits being and 2a/v. This can be easily done without solving the
full problem, not only in the case of m= 1, but m = any integer. The
result is, if U m be the electric energy of the steady field due to f m ,
and, therefore, by surface-integration according to (154),
(156)
J U m is the magnetic energy in the m th travelling shell. I have entered
into detail in the case of m = 1, because of its relative importance, and
to avoid repetition. In every case the magnetic field of the primary
wave outward is cancelled by that of the reflection of the primary
wave inward, producing a travelling shell of depth 2a, within which is
the final steady field. There, are, however, some differences in other
respects, according as m is even or odd.
Thus, in the case m=- 2, we have, by (110) to (113),
1 + 3 \ _ ^H-a,/! + 1 + 3 \ I x A 3 3 \
qa q 2 a?J \ go, qWJ) \ qr fl*J
Making this operate upon / 2 , zero before and constant after t = 0, we
obtain, by (132), (140), and Taylor's theorem,
In the wave represented, vt>(r-a), it being the primary wave out.
The unrepresented part, to be obtained by changing the sign of a
within the {}, is the reflected wave, in which vt>(r + a).
To obtain the internal H exchange a and r within the {} in (158).
The result is that
............ < 159 >
expresses the IT-solution always, provided that when vt < a the limits
for r are a - vt and a + vt; but when vt > a, they are vt - a and vt + a.
At the surface of the sphere,
from = to 2a/v. It vanishes twice, instead of only once, inter-
mediately, finishing at the same value that it commenced at, instead of
at the opposite, as in the m = 1 case.
414 ELECTRICAL PAPERS.
The radial component F of E is always zero at the front of either of
the primary waves or of the reflected wave, and E = ^vH, according
as the wave is going out or in. In the travelling shell H changes sign
m times, thus making m + 1 smaller shells of oppositely directed
magnetic force. At its outer boundary
....................... (161)
and at the inner boundary the same formula holds, with prefixed
according as m is even or odd.
In case m = 3, the magnetic force at the spherical surface is
-f* v Sift* 15**** 5
" ~
from t = to 2fl/i> ; after which, zero.
Spherical Sheet of Radial Impressed Force.
24. If the surface r = a be a sheet of radial impressed force, it is clear
that the vorticity is wholly on the surface. Let the intensity be inde-
pendent of <, so that
e = ?e m Q m (163)
The steady potential produced is
(in) r 1 --v^a^Lt4 r /ry' ) ( i64)
............. (165)
because, at r = a, these make
Fs-r^e, and dFJdr^dFJdr; ............. (166)
i.e., potential-difference e, and continuity of displacement. The normal
component of displacement is
therefore, integrating over the sphere, the total work done by e is
(168)
which agrees with the estimate (156), because
/._* r*
add a dp
finds the vorticity, /, from the radial impressed force e ; or, taking
e = e m Q m , * ^ m vQlfl,- l = vorticity,
so that the old f m = e m /a.
Single Circular Vortex Line.
25. There are some advantages connected with transferring the
impressed force to the surface of the sphere, as it makes the force of the
ON ELECTROMAGNETIC WAVES. PART III. 415
flux and the force of the field identical both outside and inside. At
the boundary F is continuous, E discontinuous.
Let the impressed force be a simple circular shell of radius a, and
strength e. Let it be the equatorial plane, so that the equator is the
one line of vorticity. Substitute for this shell a spherical shell of
strength \e on the positive hemisphere, - \e on the negative, the
impressed force acting radially. Expand this distribution in zonal
harmonics. The result is
15 - L3 - 5 mm
..... (170)
so that we are only concerned with the odd ra's. This equation settling
the value of e m , the vorticity is
?e m a-i v Ql = ?f m vQl ........................ (171)
We know therefore, by the preceding, the complete solution due to
sudden starting of the single vortex-line. That is, we know the
individual waves in detail produced by e lt e z , etc. The resultant
travelling disturbance is therefore confined between two spherical
surfaces of radii vt-a and vt + a, after the centre has been reached,
or of radii a vt and a + vt before the centre is reached. But it
cannot occupy the whole of either of the regions mentioned.
The actual shape of the boundaries, however, may be easily found.
It is sufficient to consider a plane section through the axis of the
sphere. Let A and B be the points on this plane cut by the vortex-
line. Describe circles of radius vt with A and B as centres. If vt < a,
the circles do not intersect ; the disturbance is therefore wholly within
them. But when vt is > a, the intersecting part contains no H, and
only the E of the steady field due to the vortex-line, which we know
by 24.
That within the part common to both circles there is no H we may
prove thus. The vortex-line in question may be imagined to be a line
of latitude on any spherical surface passing through A and B, and
centred upon the axis. Let a x be the radius of any sphere of this kind.
Then, at a time making vt>a, the disturbance must lie between the
surfaces of spheres of radii vt - a x and vt + a v whose centre is that of
the sphere a^. Now this excludes a portion of the space between the
vt - a and vt + a circles, referring to the plane section ; and by varying
the radius a^ we can find the whole space excluded. Thus, find the
locus of intersections of circles of radius
with centre at distance z from the origin, upon the axis. The equation
of the circle is
or x* + y*-2xz = vW + a z -2vt(a? + z*) ............... (172)
Differentiate with respect to z t giving
x*) =ax, .......................... (173)
416 ELECTRICAL PAPERS.
and eliminate z between (173) and (172). After reductions, the result is
x* + (ya)* = vW, ........................... (174)
indicating two circles, both of radius vt, whose centres are at A and B.
Within the common space, therefore, the steady electric field has been
established.
If this case be taken literally, then, since it involves an infinite
concentration in a geometrical line of a finite amount of vorticity of e,
the result for the steady field is infinite close up to that line, and the
energy is infinite. But imagine, instead, the vorticity to be spread
over a zone at the equator of the sphere r = a, half on each side of
it, and its surface-density to be /jv, where f l is finite. Consider the
effect produced at a point in the equatorial plane. From time t = to
^ = (r-a)/v (if the point be external) there is no disturbance. But
from time ^ to t 2 = b/v, where b is the distance from the point to the
edges of the zone, the disturbance must be identically the same as if the
harmonic distribution f^v were complete, viz. by (142),
*^*\ ...................... (175)
2
After this moment t 2 , the formula of course fails. Now narrow the
band to width adO at the equator and simultaneously increase f v so as
to make f^uLQ = e^ the strength of the shell of impressed force when
there is but one. The formula (175) will now be true only for a very
short time, and in the limit it will be true only momentarily, at the
front of the wave, viz.,
f l a/2p Q w = H = e/fy wdO, .................... (176)
going up infinitely as dO is reduced. To avoid infinities in the electric
and magnetic forces we must seemingly keep either to finite volume
or finite surface-density of vorticity of e, just as in electrostatics with
respect to electrification.
Instead of a simple shell of impressed electric force, it may be one of
magnetic force, with similar results. As a verification, calculate the
displacement through circle v on the sphere r a due to a vortex-circle
at Vj on the same surface, the latter being of unit strength. It is
,- ,
due to 2 e m Q m , through the circle v. Take then
m 2m(m+l)
which represents e m due to vortex-line of unit strength at v r Use this
in the preceding equation (177), and we obtain
as the displacement through v due to unit vortex-line at v x . Applying
this result to a circular electric current, B = /x H takes the place of
ON ELECTROMAGNETIC WAVES. PART III. 417
D = (c/4ir)B, as the flux concerned, whilst if h be the strength of the shell
of impressed magnetic force, h/4ir is the equivalent bounding electric
current. The induction through the circle v due to unit electric current
in the circle v^ is therefore obtainable from (179) by turning c to /x and
multiplying by (47r) 2 . The result agrees with Maxwell's formula for
the coefficient of mutual induction of two circles (vol. II., art. 697).
It must be noted that in the magnetic-shell application there must be
no conductivity, if the wave-formulae are to apply.
An Electromotive Impulse, m = 1.
26. Returning to the case of impressed electric force, let in a spherical
portion of an infinite dielectric a uniform field of impressed force act
momentarily. We know the result of the continued application of the
force. We have, then, to imagine it cancelled by an oppositely directed
force, starting a little later. Let ^ be the time of application of the
real force, and let it be a small fraction of 2a/v, the time the travelling
shell takes to traverse any point. The result is evidently a shell of
depth ^ at r = vt + a, in which the electromagnetic field is the same as in
the case of continued application of the force, and a similar shell situated
at r = vt- a, in which H is negative. Within this inner shell there is
no E or H. But between the two thin shells just mentioned there is a
diffused disturbance, of weak intensity, which is due to the sphericity of
the waves, and would be non-existent were they plane waves. In fact,
at time t = t v when the initial disturbance Hf^yft^ has extended itself
a small distance v^ on each side of the surface of the sphere, there is a
radial component F &t the surface itself, since, by (150),
(180)
so that the sudden removal of /j leaves two waves which do not satisfy
the condition E = p^H at their common surface of contact. On separa-
tion, therefore, there must be a residual disturbance between them.
The discontinuity in E at the moment of removing yj is abolished by
instantaneous assumption of the mean value, but it is impossible to
destroy the radial displacement which joins the two shells at the
moment they separate. Put on/j when = 0, then /j at time t t later.
The H at time t due to both is, by (142),
W-2^); (181)
which, when ^ is infinitely small, becomes
H =-t^. ........................... (182)
2/ytf- 2
First of all, at a point distant r from the centre, comes the primary
disturbance or head,
.............................. (183)
when vt = r- a, lasting for the time t r It is followed by the diffused
negative disturbance, or tail, represented by (182), lasting for the time
H.E.P. VOL. ii. 2o
418 ELECTRICAL PAPERS.
2a/v. At its end comes the companion to (183), its negative, when
vt = r + a, lasting for time t v after which it is all over. This description
applies when r > a. If r < a, the interval between the beginning and
end of the JJ-disturbance is only 2r/v. From the above follows the
integral solution expressing the effect of ^ varying in any manner with
the time.
Alternating Impressed Forces.
27. If the impressed force in the sphere, or wherever it may be, be a
sinusoidal function of the time, making p 2 = n 2 , if n = 2?r x frequency,
the complete solutions arise from (132) to (135) so immediately that we
can almost call them the complete solutions. Of course in any case in
which we have developed the connection between the impressed force
and the flux, say e = ZC, or C = Z~ 1 e ) where Z is the resistance-operator,
we may call this equation the solution in the sinusoidal case, if we state
that p 2 is to mean - n 2 . But there is usually a lot of work needed to
bring the solution to a practical form. In the present instance, how-
ever, there is scarcely any required, because u and w are simple functions
of qr, and q 2 is real. The substitution p 2 = - n 2 in u results in a real
function of nr/v, and in w in a real function x ( - 1 )*. Thus :
nr
(184)
3v 2 \ nr 3v . nr
--- sm i
nr v (185)
nr
+ _cos
n z r 2 v nr
nr)
I.
v j
In the case m = 1, if (f^cosnt is the form of / lf so that (fj represents
the amplitude, we find, writing this case fully because it is the most
important :
- sin^. (cos - -IsinY^ -
no, )v \ nr J\v
*a
- JLrin^"* . fsin + LcosV^ - *t\
)v \ nr
- 1 rin^. (sin + cosV^ - nt
nr J v \ na A /
(in) B
nr v na
ON ELECTROMAGNETIC WAVES. PART III. 419
It is very remarkable, on first acquaintance, that the impressed force
produces no external effect at all when
K.-0, or tan-.
V V
For the impressed force may be most simply taken to be a uniform field
of intensity (f } )cosnt in the sphere of radius a acting parallel to the
axis, and it looks as if external displacement must be produced. Of
course, on acquaintance with the reason, the fact that the solution is
made up of two sets of waves, those outward from the lines of vorticity
and those going inward, and then reflected out, the mystery disappears.
To show the positive and negative waves explicitly, we may write
the first of (185a) in the form
(18M)
(ant) J-
na nr
.
n 2 arj \nr na
the second line showing the primary wave out, the first the reflected
wave.* Exchange a and r within the [ ] to obtain the internal H. The
disturbance, at the surface, of the primary wave going both ways is,
from t = to 2a/v,
-
n 2 ar
The amplitude due to both waves is
The time-rate of outward transfer of energy per unit area at any
distance r is EHj^Tr. In the m th system this is
(-^)sm}^ (186)
where m is supposed odd, whilst u and - iw are the real functions of
* In reference to this formula (185rf), and the corresponding ones for other
values of m, it is not without importance to know that a very slight change
'suffices to make (lS5d) represent the solution from the first moment of starting
the impressed force. Thus, let it start when t = 0, and let the/ x in equation (139)
be (fjcosnt. Effect the two integrations thus,
6 = (/^sin nt, 4 = (/i)^ 1 ~ cos nt ^
vanishing when t = 0, and then operate with the exponentials, and we shall obtain
(185rf) thus modified: To the first line must be added
Lfife? *
2/jt.Qvr n 2 ar
and to the second line its negative. Thus modified, (185rf) is true from < = 0,
understanding that the second line begins when t = (r - a)/v, and the first when
t = (r + a)lv. The first of (185a) is therefore true up to distance r = rt-a, when
this is positive. In the shell of depth 2a beyond, it fails,
420 ELECTRICAL PAPERS.
nr/v obtained in the same way as (184). The mean value of the t
function is, by the conjugate property of u and u-, equation (114),
= -n/2v.
Using this, and integrating (186) over the complete surface of radius r,
giving
.(187)
JJ<
we find the mean transfer of energy outward per second through any
surface enclosing the sphere to be
(/J2M>2 , -..(188)
if (/m) v $m c os?i is the vorticity of the impressed force. [When in is
even substitute - w*.]
In the case m = 1 , the waste of energy per second is
due to the uniform alternating field of impressed force of intensity
(/j) cos nt within the sphere.
In reality, the impressed force must have been an infinitely long time
in operation to make the above solutions true to an infinite distance,
and have therefore already wasted an infinite amount of energy. If
the impressed force has been in operation any finite time t, however
great, the disturbance has only reached the distance r = vt + a. Of
course the solutions are true, provided we do not go further than
r = vt a. We see, therefore, that the real function of the never-ceasing
waste of energy is to set up the sinusoidal state of E and H in the
boundless regions of space which the disturbances have not yet
reached. The above outward waves are the same as in Rowland's
solutions.* Here, however, they are explicitly expressed in terms of
the impressed forces causing them.
u a makes the external field vanish when m is odd ; and w a =
when m is even ; that is, when the sinusoidal state has been assumed.
It takes only the time 2a/v to do this, as regards the sphere r = a; the
initial external disturbance goes out to infinity and is lost. This
vanishing of the external field happens whatever may be the nature of
the external medium away from the sphere, except that the initial
external disturbance will behave differently, being variously reflected
or absorbed according to circumstances.
Conducting Medium. m= 1.
28. Now consider the same problem in an infinitely extended con-
ductor of conductivity k. We may remark at once that, unless the
conductivity is low, the solution is but little different from what it
would be were the conductor not greatly larger than the spherical
*In paper referred to in 18.
ON ELECTROMAGNETIC WAVES. PART III. 421
portion within it on whose surface lie the vortex-lines of the impressed
force, owing to the great attenuation suffered by the disturbances as
they progress from the surface. In a similar manner, if the sphere be
large, or the periodic frequency great, or both, we may remove the
greater part of the interior of the sphere without much altering matters.
We have now
(190)
The realization is a little troublesome on account of this pt. The result
is that the uniform alternating field of impressed force of intensity
(/ x ) cos nt, gives rise to the internal solution
*?*}*uJr. ; see (129), 20 fl
popj 'J
(in) H={(A+)coBnt + (A-S)*mnt}, ..... (191)
where A and B are the functions of r expressed by
V OS + (_L - L + 2
rJ \2xr 2xa 2xr.2xa
i \ cos _/ i + i + 2\ 8iu -i (a+r) (192)
xrJ \2xa 2xr 2xr.2xaJ J
A = c*-'f ( i + _L - J_V OS + (_L - L + 2 n ~L (a _ r)
\__\ 2xa 2xrJ \2xr 2xa 2xr.2xaJ x
2xr
B
5 -r- o- -
2xr 2xa 2xr.2xaJ \ 2xr 2xa
1 +^_+ 2 ; \ A 1 1 \ i
2xr 2xa 2xr.2xaJ \ 2xr 2xaJ
Equation (191) showing the internal H, the external is got by exchang-
ing a and r in the functions A and B.
Now xa is easily made large, in a good conductor ; then, anywhere
near the boundary, (r = a), we have
A = e-* (a - r > cos x(a -r), -B = -*<-> sin x(a - r), ..... (194)
and (191) becomes
(in) ff=*->.cosi-z(-r)- ........ (195)
The wave-length A is
Thus, in copper, a frequency of 1600 to 1700 makes A = l centim.
Both A and the attenuation-rate depend inversely on the square roots
of the inductivity, conductivity, and frequency, whereas the amplitude
varies directly as the square root of the conductivity, and inversely as
the square roots of the others.
[The attenuation in distance X is -*X = ~ 27r ; therefore we may say
it is nearly insensible further on. If we introduce an auxiliary
422 ELECTRICAL PAPERS.
impressed force to keep the current straight, we shall, when xa is
large, just double the external H and the activity.]
To verify that very great frequency ultimately limits the disturbance
to the vortex-line of e when there is but one, we may use the last solu-
tion to construct that due to a sheet of impressed force
acting radially on the surface of the sphere. Thus,
(in) H= f^fy. e ^t--^(nt - x(a -r)- , (197)
when xa is very great. When the vorticity is confined to one line of
latitude, H in (197) vanishes everywhere except at the vortex-line.
But a further approximation is required, or a different form of solution,
to show the disturbance round the vortex-line explicitly, i.e., when n is
great, though not infinitely great.
A Conducting Dielectric. m = l.
29. Here, if k is the conductivity, c the permittivity, and /* the
inductivity, let
q = (4ir^ + fi cp 2 )* = w 1 + 2 t > ................... (198)
when p = ni. Then n^ and n z will be given by
Using this q in the general external jET-solution, but ignoring the explicit
connexion with the impressed force, we shall arrive at
(out) H = *-'vl+ ,i caa- * S m(n 2 r - nt), ( 2
where C is an undetermined constant, depending upon the magnitude
of the disturbance at r-a. So far as the external solution goes, how-
ever, the internal connexions are quite arbitrary save in the periodicity
and confinement to producing magnetic force proportional in intensity
to the cosine of the latitude. The solution (200) may be continued
unchanged as near to the centre as we please. Stopping it anywhere,
there are various ways of constructing complementary distributions in
the rest of space, from which (200) is excluded.
Wj is zero when k = 0. We then have the dielectric solution, with
% = n/v. On the other hand, c = makes
as in 28. The value of
Enormously great frequency brings us to the formulae of the non-
conducting dielectric, with a difference, thus : n 1 and n 2 become
n 2 n/Vj ..................... (202)
ON ELECTROMAGNETIC WAVES. PART III. 423
when 4:7rkfcn is a small fraction. The attenuation due to conductivity
still exists, but is independent of the frequency. We have now
(out) ff-^Vvfcos-- sinY -tA (203)
r \ nr J\v J
differing from the case of no conductivity only in the presence of the
exponential factor.
It is, however, easily seen by the form of n^ in (202) that in a good
conductor the attenuation in a short distance is very great, so that the.
disturbances are practically confined to the vortex-lines of the impressed
force, where the /^-disturbance is nearly the same as if the conductivity
were zero, as before concluded. It follows that the initial effect of the
sudden introduction of a steady impressed force in the conducting
dielectric is the emission from the seat of its vorticity of waves in the
same manner as if there were no conductivity, but attenuated at their
front to an extent represented by the factor e~ w i r , with the (202) value
of n lt in addition to the attenuation by spreading which would occur
were the medium nonconducting. This estimate of the attenuation
applies at the front only.
Current in Sphere constrained to be uniform.
30. Let us complete the solution (200) of 29 by means of a current
of uniform density parallel to the axis within the sphere of radius a,
beyond which (200) is to be the solution. This will require a special
distribution of impressed force, which we shall find. Equation (200)
gives us the normal component of electric current at r = a, by differenti-
ation. Let this be F cos 6. Then F is the density of the internal
current. The corresponding magnetic field must have the boundary-
value according to (200), and vary in intensity as the distance from the
axis, its lines being circles centred upon it, and in planes perpendicular
to it. Thus the internal H is also known. The internal E is fully
known too, being k" l T in intensity and parallel to the axis. It only
remains to find e to satisfy
curl(e-E) = /xH, (3) bis
within the sphere, and at its boundary (with the suitable surface inter-
pretation), as it is already satisfied outside the sphere. The simplest
way appears to be to first introduce a uniform field of e parallel to the
axis, of such intensity ^ as to neutralize the difference between the
tangential components of the internal and external E at the boundary,
and so make continuity there in the force of the field ; and next, to
find an auxiliary distribution e 2 , such that
curl e 2 = /zH,
and having no tangential component on the boundary. This may be
done by having e 2 parallel to the axis, of intensity proportional to
(a 2 - r 2 ) sin 6.
The result is that the internal H is got from the external by putting
r = a in (200) and then multiplying by r/a ; F from the internal H by
424 ELECTRICAL PAPERS.
multiplying by (27rrsin 0)~ l ; e 1 from the difference of the tangential
components E outside and inside is given by
(204)
Finally, the auxiliary force has its intensity given by
(205)
A remarkable property of this auxiliary force, which (or an equivalent)
is absolutely required to keep the current straight, is that it does no
work on the current, on the average ; the mean activity and waste of
energy being therefore settled by e r
Nov. 27, 1887.
PART IV.
Spherical Waves (with Diffusion) in a Conducting Dielectric.
31. In an infinitely extended homogeneous isotropic conducting
dielectric, let the surface r = a be a sheet of vorticity of impressed
electric force ; for simplicity, let it be of the first order, so that the
surface-density is represented by fv. By (127), 20, the differential
equation of H, the intensity of magnetic force is, at distance r from the
origin, outside the surface of/, (v meaning sin 6),
(206)
where / may be any function of the time. Here, in the general case,
including the unreal " magnetic conductivity " g* we have
..... (2Q7)
:ir + cp ;
if, for subsequent convenience,
......
The speed is v, and p v p 2 are the coefficients of attenuation of the parts
transmitted of elementary disturbances due to the real electric con-
ductivity k and the unreal g ; that is, e~<* is the factor of attenuation
due to conductivity. On the other hand, the distortion produced by
conductivity depends on
where the h's are functions of r, but not of p. Multiplying these
together, we convert (213) or (214) to
where the i's are functions of r, but not of p. The integrations can
now be effected. Let/ be constant, first. Then, / starting when t = 0,
we have
^- 3 (/^)=/p" 3 (^-i-^-i^ 2 ) = p~ 3 /( /) 03 sa y; ..... ( 218 )
etc., etc. Next, operating with the exponential containing p in (217)
turns ttot-(r- a)/v, and gives the required solution in the form
same function of
-a , ........... (219)
ON ELECTROMAGNETIC WAVES. PART IV. 427
where ^ = t - (r - a)/v ; the represented part beginning when ^ reaches
zero, and the rest when t - (r + a)/v reaches zero.
Fuller Development in a Special Case. Theorems involving Irrational
Operators.
36. As this process is very complex, and (219) does not admit of
being brought to a readily interpretable form, we should seek for
special cases which are, when fully developed, of a comparatively
simple nature. Write the first half of (212) thus,
(220)
Now the part in the square brackets can be finitely integrated when
ft? 1 subsides in a certain way. We can show that
....... (221)
in which, observe, the sign of a- may be changed, making no difference
on the right side (the result), but a great deal on the left side.
The simplest proof of (221) is perhaps this. First let r = a. Then
by getting the exponential to the left side, so as to operate on unity.
Next, by the binomial theorem,
^V}" ............. (223)
Now integrate, and we have (/ commencing when t = 0),
(224)
so that, finally,
(225)
It is also worth notice that, integrating in a similar manner,
< 226 >
These theorems present themselves naturally in problems relating to a
telegraph-circuit, when treated by the method of resistance-operators.
A special case of (225) is
(*Q-* ............................ (227)
428 ELECTRICAL PAPERS.
which presents itself in the electrostatic theory of a submarine
cable. *
We have now to generalize (225) to meet the case (221). The left
member of (221) satisfies the partial differential equation
t;2y2 = ^2_ -2 ) . ........................ (228)
so we have to find the solution of (228) which becomes J (vti) when
r = a. Physical considerations show that it must be an even function
of (r - a), so that it is suggested that the t in J G (o-ti) has to become, not
t-(r a)/v or t + (r - a)/v, but that t' 2 has to become their product. In
any case, the right member of (221) does satisfy (228) and the further
prescribed condition, so that (221) is correct.
If a direct proof be required, expand the exponential operator in
(221) containing r in the way indicated in .(216), and let the result
operate upon J Q (a-ti). The integrated result can be simplified down to
(221).
37. Now use (221) in (220). Let fe pt =f ~ ft t where / is constant ;
and the square bracket in (220) becomes known, being in fact the right
member of (221) multiplied by / . So, making use also of (228), we
bring (220) to
(229)
dr "
to which must be added the other part, beginning 2a/v later, got by
negativing a, except the first one. The operation (^-o- 2 )" 1 may be
replaced by two integrations with respect to r.
Let r and a be infinitely great, thus abolishing the curvature. Let
r-a = z t and/ ^a/r, which is now constant, be called e Q . Then we have
simply
(230)
showing the H produced in an infinite homogeneous conducting
dielectric medium at time t after the introduction of a plane sheet (at
2 = 0), of vorticity of impressed electric force, the surface-density of
* Thus, let an infinitely long circuit, with constants R, S, K, L, be operated
upon by impressed force at the place z = 0, producing the potential- difference V
there, which may be any function of the time. Let G be the current and V the
potential- difference at time t at distance z. Then
where q = (R + Lp)*(K + 8p)*. Take K=Q, and L=Q; then, if F be zero before
and constant after t = 0, the current at z=0 is given by
and (227) gives the solution. Prove thus : let 6 be any constant, to be finally
made infinite ; then
p\(\ ) =
by the investigation in the text. Now put 6 = 00, and (227) results.
In the similar treatment of cylindrical waves in a conductor, pi, pi, etc., occur.
We may express the results in terms of Gamma-functions.
ON ELECTROMAGNETIC WAVES. PART IV. 429
vorticity being e "W. This corroborates the solution in 8, equation
(51) [Part L, p. 383], whilst somewhat extending its meaning.
The condition to which/ is subject may be written, by (208),
f=f^, ............................... (231)
where / is constant. If, then, we desire / to be constant, p l must
vanish, which, by (208), requires k = 0, whilst g may be finite.
But we can make the problem real thus. In (229) change H to E
and pv to cv ; we have now the solution of the problem of finding the
electric field produced by suddenly magnetizing uniformly a spherical
portion of a conducting dielectric ; i.e., the vorticity of the impressed
magnetic force is to be on the surface of the sphere r = a, parallel to its
lines of latitude, and of surface-density fv, such that fve-fo 1 is constant
This makes / constant when g = and k finite, representing a real
conducting dielectric.
The Electric Force at the Origin dm to fv at r = a.
38. Eeturning to the case of impressed electric force, the differential
equation of F, the radial component of electric force inside the sphere
on whose surface r = a the vorticity of e is situated, is, by 20,
equations (136), (137),
h
qr 2 \ qaj \ qr J
At the centre, therefore, the intensity of the full force, which call
whose direction is parallel to the axis, is
= l --/ .............. (233)
Unless otherwise specified, I may repeat that the forces referred to are
always those of the fluxes, thus doing away with any consideration of
the distribution of the impressed force, and of scalar potential, of vary-
ing form, which it involves. (233) is equivalent to
(234)
Let / be constant, and p = d-a 2 /2/)i The relation of X a in (236) to the preceding
terms is explained by equations (233) or (235).
Effect of uniformly magnetizing a Conducting Sphere surrounded by a
Nonconducting Dielectric.
39. Here, of course, it is the lines of E that are circles centred upon
the axis, both inside and outside. Let h be the impressed magnetic
force, and hv the surface-density of its vorticity, at r = o> outside which
the medium is nonconducting, and inside a conducting dielectric. The
differential equation of E M the surf ace- value of the tensor of E at r = a,
is (compare (124), 19)
(239)
in which r = a, and p and q are to have the proper values on the two
sides of the surface.
Now, by (111),
W'\W= -q{l+(qr)-i(l+qr)- 1 } ............... (240)
in the case of m=l, (first order), here considered. This refers to the
external dielectric, in which q =p/v. Let v = oc , making
W\W= -ft" 1 ............................. (241)
This assumption is justifiable when the sphere has sensible conductivity,
on account of the slowness of action it creates in comparison with the
rapidity of propagation in the dielectric outside. Then (239) becomes
hv 1 flsinhq 1/1_1\ / 2 42)
cosn ( l - r *F* .................. (262)
When c = 0, (259) or (262) reduce to
where
Conducting Sphere in a Nonconducting Dielectric. Circular Vorticity
of e. Complex Reflexion. Special very Simple Case.
42. At distance r from the origin, outside the sphere of radius a,
which is the seat of vorticity of e, represented by fv, we have
r ........................ (264)
The operator < will vary according to the nature of things on both
sides of r = a. When it is a uniform conducting medium inside, and
nonconducting outside, to infinity, we shall have
when < lf depending upon the inner medium, is given by
1
47T&J + c^p cosh q-^ - (q-^a) ^sinh q : a
and < 2 , depending upon the outer medium, is given by
The solution arising from the sudden starting of/ constant is therefore
P dp
where p is now algebraical, and the summation ranges over the roots of
< = 0. There is no final H in this case, if we assume # = all over.
H.E.P. VOL. II. 2 E
434 ELECTRICAL PAPERS.
But the determinantal equation is very complex, so that this (267)
solution is not capable of easy interpretation. The wave-method is
also impracticable, for a similar reason.
In accordance, however, with Maxwell's theory of the impermeability
of a " perfect " conductor to magnetic induction from external causes,
the assumption ^ = 00 makes the solution depend only upon the
dielectric, modified by the action of the boundary, and an extraordinary
simplification results. (j>i vanishes, and the determinantal equation
becomes > 2 = 0, which has just two roots,
qa=pafi-= - Jt(j)*; ...................... (268)
and these, used in (267), give us the solution
--*3 cos - 3*(1 - 2a/r)sins N /3, ...... (269)
where z = (vt - (r - a)}/2a.
Correspondingly, the tangential and radial components of E are
............ (270)
*- / cos *l - -* cos - V 32 - sin V 3 (271)
This remarkably simple solution, considering that there is reflexion,
corroborates Prof. J. J. Thomson's investigation * of the oscillatory
discharge of an infinitely conducting spherical shell initially charged
to surface-density proportional to the sine of the latitude, for, of
course, it does not matter how thin or thick the shell may be when
infinitely conducting, so that it may be a solid sphere. (269) to (271)
show the establishment of the permanent state. Take off the im-
pressed force, and the oscillatory discharge follows. But the impressed
force keeping up the charge on the sphere need not be an external
cause, as supposed in the paper referred to. There seems no other
way of doing it than by having impressed force with vorticity fv on
the surface, but in other respects it is immaterial whether it is internal
or external, or superficial.
It may perhaps be questioned whether the sphere does reflect,
seeing that its surface is the seat of /. But we have only to shift
the seat of / to an outer spherical surface in the dielectric, to see at
once that the surface of the conductor is the place of continuous
reflexion of the wave incident upon it coming from the surface of /.
The reflexion is not, however, of the same simple character that occurs
when a plane wave strikes a plane boundary (k = oc ) flush, which
consists merely in sending back again every element of H unchanged,
but with its E reversed ; the curvature makes it much more complex.
When we bring the surface of / right up to the conducting sphere, we
make the reflexion instantaneous. At the front of the wave we have
* " On Electrical Oscillations and the Effects produced by the Motion of an
Electrified Sphere," Proc. Math. Soc., vol. xv., p. 210.
ON ELECTROMAGNETIC WAVES. PART IV. 435
by (269) and (270). This is exactly double what it would be were
the conductor replaced by dielectric of the same kind as outside, the
doubling being due to the instantaneous reflexion of the inward-going
wave by the conductor.
The other method of solution may also be applied, but is rather more
difficult. We have
H-.->-(l +1) (l -1) (l - 4- 3 ) "/. ....... (272)
pvr \ grj \ qa/ \ (fa 6 /
Expand the last factor in descending powers of (qa) s , and integrate.
The result may be written
rr_a
-
where x = a~ l (vt-r + a). Conversion to circular functions reproduces
(269).
Same Case with Finite Conductivity. Sinusoidal Solution.
4 2 A. It is to be expected that with finite conductivity, even with the
greatest at command, or ^ = (1600)~ 1 , the solution will be considerably
altered, being controlled by what now happens in the conducting sphere.
To examine this point, consider only the value of H at the boundary.
We have, by (264),
ff.-*-yv-(^+*,)-yv ..................... (274)
Let / vary sinusoidally with the time, and observe the behaviour of > x
and 2 as the frequency changes. The full development which I have
worked out is very complex. But it is sufficient to consider the case
in which k is big enough, in concert with the radius a and frequency
n/2-Tr, to make the disturbances in the sphere be practically confined to
a spherical shell whose depth is a small part of the radius. Let
s (^Tr/Xj&jtta 2 ) ; then our assumption requires e - * to be small. This
makes
............... < 275 >
and, if further, s itself be a large number, this reduces to
8^)* ............................ < 276 )
Adding on the other part of <, similarly transformed by p 2 = - n~, we
obtain
tf w*) 2 ,/wn ? f _ e* __ (J&f\, (277)
1 + (na/v)* + WV J L(*a/) + (na/v)* WV J
where the terms containing ^ show the difference made by its not being
infinite. The real part is very materially affected. Thus, copper, let
^ = (1600)-!, ^ = 1, 27T7i=1600, a = 10, .-. s = 10.
These make s large enough. Now najv is very small, but, on the
other hand,
436 ELECTRICAL PAPERS.
so that the real part of depends almost entirely on the sphere, whilst
the other part is little affected.
Now make n extremely great, say na/v = 1 ; else the same. Then
< = (| x 10 10 + 44 x 10 4 ) - 1(| x 10 10 - 44 x 10 4 ),
from which we see that the dissipation in space has become relatively
important. The ultimate form, at infinite frequency, is
t^HV + dijnl&rktfp+i)', (278)
so that we come to a third state, in which the conductor puts a stop to
all disturbance. This is, however, because it has been assumed not to
be a dielectric also, so that inertia ultimately controls matters. But if,
as is infinitely more probable, it is a dielectric, the case is quite changed.
We shall have
^ = (4^ + ^(4^ + ^)^, (279)
when the frequency is great enough, and this tends to fj^v v /^ being the
inductivity and ^ the speed in the conductor, whatever g and k may be,
provided they are finite. Thus, finally,
= H lVl +nv (280)
represents the impedance, or ratio of fv to H a , which are now in the
same phase.
At any distance outside we know the result by the dielectric-solution
for an outward wave. But there is only superficial disturbance in the
conducting sphere.
Resistance at the front of a Wave sent along a Wire.
43. In its entirety this question is one of considerable difficulty, for
two reasons, if not three. First, although we may, for practical pur-
poses, when we send a wave along a telegraph-circuit, regard it as a
plane wave, in the dielectric, on account of the great length of even the
short waves of telephony, and the great speed, causing the lateral
distribution (out from the circuit) of the electric and magnetic fields
to be, to a great distance, almost rigidly connected with the current in
the wires and the charges upon them ; yet this method of representation
must to some extent fail at the very front of the wave. Secondly, we
have the fact that the penetration of the electromagnetic field into the
wires is not instantaneous ; this becomes of importance at the front of
the wave, even in the case of a thin wire, on account of the great speed
with which it travels over the wire.* The resistance per unit length
must vary rapidly at the front, being much greater there than in the
body of the wave ; thus causing a throwing back, equivalent to electro-
static or "jar " retardation.
* The distance within which, reckoned from the front of the wave backward,
there is materially increased resistance, we may get a rough idea of by the distance
travelled by the wave in the time reckoned to bring the current-density at the
axis of the wire to, say, nine-tenths of the final value. It has all sorts of values.
It may be 1 or 1000 kilometres, according to the size of wire and material. At the
front, on the assumption of constant resistance, the attenuation is according to
6 -Rt;zL^ ft being the resistance, and L the inductance of the circuit per unit length.
Hence the importance of the increased resistance in the present question.
ON ELECTROMAGNETIC WAVES. PART IV. 437
Now, according to the magnetic theory, the resistance must be
infinitely great at the front. Thus, alternate the current sufficiently
slowly, and the resistance is practically the steady resistance. Do it
more rapidly, and produce appreciable departure from uniformity of
distribution of current in the wire, and we increase the resistance to an
amount calculable by a rather complex formula. But do it very rapidly,
and cause the current to be practically confined to near the boundary,
and we have a simplified state of things in which the resistance varies
inversely as the area of the boundary, which may, in fact, be regarded as
plane. The resistance now increases as the square root of the frequency,
and must therefore, as said, be infinitely great at the front of a wave,
which is also clear from the fact that penetration is only just
commencing.
But for many reasons, some already mentioned, it is far more probable
that the wire is a dielectric. If, as all physicists believe, the ether
permeates all solids, it is certain that it is a dielectric. Now this
becomes of importance in the very case now in question, though of
scarcely any moment otherwise. Instead of running up infinitely, the
resistance per unit area of surface of a wire tends to the finite value
4737^, This is great, but far from infinity, so that the attenuation and
change of shape of wave at its front produced by the throwing back
cannot be so great as might otherwise be expected.
Thus, in general, at such a great frequency that conduction is nearly
superficial, we have, if /A, c, k, and g belong to the wire,
BIH*(4ty+jgftM+qfF*i (281)
if E is the tangential electric force and H the magnetic force, also
tangential, at the boundary of a wire. Now let R' and L' be the
resistance and inductance of the wire per unit of its length. We must
divide H by 4?r to get the corresponding current in the wire, as ordi-
narily reckoned. So ^irA~ l times the right member of (281) is the
resistance-operator of unit length, if A is the surface per unit length ;
so, expanding (281), we get
R' or |*.3J-J(ff^Y !MgFF > (282)
where p v p 2 are as before, in (208). Here n/2r = frequency.
Disregarding , and therefore /> 2 , we have
R' or Un = (^TrnvA^{BB s >} ............. (283)
where
When c is zero, R f and L f n tend to equality, as shown by Lord Rayleigh.
But when c is finite, L'n tends to zero, and R f to vpioA~\ as indeed we
can see from (281) at once, by the relative evanescence of k and g,
when finite.
But the frequency needed to bring about an approximation towards
the constant resistance is excessive ; in copper we require trillions per
second. This brings us to the third reason mentioned ; we have no
438 ELECTRICAL PAPERS.
knowledge of the properties of matter under such circumstances, or of
ether either. The net result is that although it is infinitely more
probable that the resistance should tend to constancy than to infinity,
yet the real value is quite speculative.* Similar remarks apply to
sudden discharges, as of lightning along a conductor. The above R', it
should be remarked, is real resistance, in spite of its ultimate form,
suggestive of impedance without resistance.! The present results are
corroborative of those in Part I., and, in fact, only amount to a special
application of the same.
Reflecting Barriers.
44. Let the medium be homogeneous between r = a and r = a v where
there is a change of some kind, yet unstated. Let between them the
surface r = a be a sheet of vorticity of e of the first order. We already
know what will happen when fv is started, for a certain time, until in
fact the inward wave reaches the inner boundary, and, on the other
side, until the outward wave reaches the outward boundary ; though,
when the surface of /is not midway between the boundaries, the reflected
wave from the nearest barrier may reach into a portion of the region
beyond /, by the time the further barrier is reached by the primary
wave. The subsequent history depends upon the constitution of the
media beyond the boundaries, which can be summarized in two boundary
conditions. The expression for EjH is, in general,
by (120), extended, the extension being the introduction of y, which is
a differential operator of unstated form, depending upon the boundary
* The above was written before the publication of Professor Lodge's highly
interesting lectures before the Society of Arts. Some of the experiments
described in his second lecture are seemingly quite at variance with the magnetic
theory. I refer to the smaller impedance of a short circuit of fine iron wire than
of thick copper, as reckoned by the potential-difference at its beginning needed to
spark across the circuit between knobs. Should this be thoroughly verified, it has
occurred to me as a possible explanation that things may be sometimes so nicely
balanced that the occurrence of a discharge may be determined by the state of the
skin of the wire. A wire cannot be homogeneous right up to its boundary, with
then a perfectly abrupt transition to air ; and the electrical properties of the
transition-layer are unknown. In particular, the skin of an iron wire may be
nearly unmagnetisable, p. varying from 1 to its full value, in the transition-layer.
Consequently, in the above formula, resistance 4irfjt.v per unit surface, we may
have to take fj.= l in the extreme, in the case of an iron wire. But even then, the
explanation of Professor Lodge's results is capable of considerable elucidation.
Perhaps resonance will do it. [Professor Lodge has since examined the theory of
the apparently anomalous behaviour ; and concludes that it was due to the great
effective resistance of iron producing very rapid attenuation of the oscillations.]
*t* There is a tendency at present amongst some writers to greatly extend the
meaning of resistance in electromagnetism ; to make it signify cause/effect. This
seems a pity, owing to the meaning of resistance having been thoroughly specialized
in electromagnetism already, in strict relationship to "frictional" dissipation of
energy. What the popular meaning of " resistance" may be is beside the point.
I would suggest that what is now called the magnetic resistance be called the
magnetic reluctance ; and per unit volume, the reluctancy [or reluctivity].
ON ELECTROMAGNETIC WAVES. PART IV. 439
conditions. Let y Q and y l be the y's on the inner and outer side of the
surface of /. The differential equation of H n , the magnetic force there,
is then
fv-{(EIH)^-(EIH)^)}H t ................... (285)
as in 19. Applying (284) and the conjugate property (114) of the
functions u and w (since there is no change of medium at the surface of
f), this becomes
H _ 4vk + cp (u a - y w a )(u a - y^)^ . ...(286)
y\ - 2/0
from which the differential equation of // at any point between a and
a is obtained by changing u a - y w a to (/V)(w - y Q w) ; and at any point
between a and a l by changing u a -y l w a to (a/r)(u-y l iv).
Unless, therefore, there are singularities causing failure, the deter-
minantal equation is
2/i-2/o = > .............................. ( 287 )
and the complete solution between a and a x due to / constant may be
written down at once. Thus, at a point outside the surface of/ we have
(out) H= n-- fv = ^ f . (288)
9. r 2/i-2/o
and therefore, if / starts when i = 0,
ZT- / J av
~ +
p being now algebraic, given by (287) ; < the steady , from (288) ;
and y the common value of the (now) equal y's which identity makes
(289) applicable on both sides of the surface of/.
Construction of the Operators y x and y .
45. In order that y l and y Q should be determinable in such a way as
to render (286) true, the media beyond the boundaries must be made
up of any number of concentric shells, each being homogeneous, and
having special values of c, k, p, and g. For the spherical functions
would not be suitable otherwise, except during the passage of the
primary waves to the boundaries, or until they reached places where
the departure from the assumed constitution commenced. Assuming
the constitution in homogeneous spherical layers, there is no difficulty
in building up the forms of y Q and y l in a very simple and systematic
manner, wholly free from obscurities and redundancies. In any layer
the form of E/H is as in (284), containing one y. Now at the boundary
of two layers E is continuous, and also H (provided the physical con-
stants are not infinite), so E/H is continuous. Equating, therefore, the
expressions for EjH in two contiguous media expresses the y of one in
terms of the y of the other. Carrying out this process from the origin
up to the medium between a and a, expresses y Q in terms of the y of
the medium containing the origin ; this is zero, so that y Q is found as
an explicit function of the values of u, w, u f , w' at all the boundaries
440 ELECTRICAL PAPERS.
between the origin and . In a similar manner, since the y of the
outermost region, extending to infinity, is 1, we express y v belonging
to the region between a and a v in terms of the values of u, etc., at all
the boundaries between a and oo . Each of these four functions will
occur twice for each boundary, having different values of the physical
constants with the same value of r. I mention this method of equation
of E/H operators because it is a far simpler process than what we are
led to if we use the vector and scalar potentials ; for then the force of
the flux has three component vectors the impressed force, the slope of
the scalar potential, and the time-rate of decrease of the vector potential.
The work is then so complex that a most accomplished mathematician
may easily go wrong over the boundary conditions. These remarks
are not confined in application to spherical waves.
If an infinite value be given to a physical constant, special forms of
boundary condition arise, usually greatly simplified ; e.g., infinite con-
ductivity in one of the layers prevents electromagnetic disturbances
from penetrating into it from without ; so that they are reflected with-
out loss of energy.
Knowing y l and y Q in (288), we virtually possess the sinusoidal solu-
tion for forced vibrations, though the initial effects, which may or may
not subside or be dissipated, will require further investigation for their
determination ; also the solution in the form of an infinite series showing
the effect of suddenly starting / constant ; also the solution arising from
any initial distribution of E and H of the kind appropriate to the
functions, viz., such as may be produced by vorticity of e in spherical
layers, proportional to v (or vQ^ in general). But it is scarcely neces-
sary to say that these solutions in infinite series, of so very general a
character, are more ornamental than useful. On the other hand, the
immediate integration of the differential equations to show the develop-
ment of waves becomes excessively difficult, from the great complexity,
when there is a change of medium to produce reflexion.
Thin Metal Screens.
46. This case is sufficiently simple to be useful. Let there be at
r = a 1 a, thin metal sheet interposed between the inner and outer non-
conducting dielectrics, the latter extending to infinity. If made in-
finitely thin, E is continuous, and H discontinuous to an amount equal
to 47r times the conduction-current (tangential) in the sheet. Let K l
be the conductance of the sheet (tangential) per unit area ; then
at r = a v
Therefore by (284), when the dielectric is the same on both sides,
^u{ - w{ u{
where the functions u v etc., have the r = a 1 values. From this,
4? (290)
ON ELECTROMAGNETIC WAVES. PART IV. 441
expresses y l for an outer thin conducting metal screen, to be used in
(286). If of no conductivity, it has no effect at all, passing disturbances
freely, and y^ = 1. At the other extreme we have infinite conductivity,
making y l = u' 1 /w{, with complete stoppage of outward -going waves,
and reflexion without absorption, destroying the tangential electric
disturbance.
When the screen, on the other hand, is within the surface of /, say
at r = , of conductance K per unit area, we shall find
2/o =
where , etc., have the r = a Q values. The difference of form from y l
arises from the different nature of the r functions in the region includ-
ing the origin. As before, no conductivity gives transparency (y = 0),
and infinite conductivity total reflexion (;y = w/w). When the inner
screen is shifted up to the origin, we make y = 0, and so remove it.
Solution with Outer Screen ; K x = oo ; f constant.
47. Let there be no inner screen, and let the outer be perfectly con-
ducting. As J. J. Thomson has considered these screens,* I will be
very brief, regarding them here only in relation to the sheet of/ and
to former solutions. The determinarital equation is
w( = 0, or tar\x = x(l-x 2 )~ l , (292)
if x = ipajv. Roots nearly TT, 27r, STT, etc. ; except the first, which is
considerably less. The solution due to starting / constant, by (289), is
therefore
H= 2 ~ a ri l t pt ) (293)
which, developed by pairing terms, leads to
^=S.s-;f,Sy 2 -^ cos -^ sin )?( cos --
which of course includes the effects of the infinite series of reflexions at
the barrier. By making ^ = oo , however, the result should be the same
as if the screen were non-existent, because an infinite time must elapse
before the first reflexion can begin, and we are concerned only with
finite intervals. The result is
H=^ . ?f ^^iLYcos - JL sin Wcos - -1 sinW (295)
nvr 7rJ ^ \ Xjr J l \ Xja J l
which must be the equivalent of the simple solution (142) of 21,
showing the origin and progress of the wave.
Now reduce it to a plane wave. We must make a infinite, and
r - a = z finite. Also take fv = e, constant. We then have
* In the paper before referred to.
442 ELECTRICAL PAPERS.
showing the H at s due to a plane sheet of vorticity of e situated at
z = Q. This is the equivalent of the solution (12) of 2, indicating the
continuous uniform emission of H=e/2fj-v both ways from the plane
z = 0. [But the sign of e is changed from that of 2.]
Returning to (294), it is clear that from t = to t = (a l -a)lv, the
solution is the same as if there were no screen. Also if a is a very
small fraction of a v the electromagnetic wave of depth 2a will, when it
strikes the screen, be reflected nearly as from a plane boundary. It
would therefore seem that this wave would run to-and-fro between
the origin and boundary unceasingly. This is to a great extent true ;
and therefore there is no truly permanent state (the electric flux,
namely, alone) ; but examination shows that the reflexion is not clean,
on account of the electrification of the boundary, so that there is a
spreading of the magnetic field all over the region within the screen.
Alternating f with Reflecting Barriers. Forced Vibrations.
48. Let the medium be nonconducting between the boundaries
and a v Equation (288) then becomes
Hss va (u.-yoWa)(u-y 1 w), , 997)
/*w 2/i-2/o
giving H outside the surface of /. We see that y = and u n = make
H=Q. That is, the forced vibrations are confined to the inside of the
surface of/ only, at the frequencies given by u a = 0, provided there is no
internal screen to disturb, but independently of the structure of the
external medium (since y l is undetermined so far), with possible
exceptions due to the vanishing of y^ simultaneously. But (297),
sinusoidally realized by p 2 = - n 2 , does not represent the full final
solution, unless the nature of y and y l is such as to allow the initial
departure from this solution to be dissipated in space or killed by
resistance. Ignoring the free vibrations, let y = 0, and y l =u^/w / l ,
meaning no internal, and an infinitely conducting external screen.
Then
(out) H=(valpvr)u a {uwyui-w}f, }
- w a }f.)
(in) H= (va/pw)u {
If wtf = 0, or in full,
(v/na^t&n^iajv) = 1 - (vjna-^f^
we obtain a simplification, viz.
tf(inorout)=-(va/j*ty)(tif0 a or u a w)f; ............... (299)
and the corresponding tangential components of electric force are
^(inorout) = (va//xiT)(wX or u a w')(cp)~ l f. ........... (300)
But if u{ = 0, the result is infinite. This condition indicates that the
frequency coincides with that of one of the free vibrations possible within
the sphere r = a l without impressed force. But, considering that we may
confine our impressed force to as small a space as we please round the
origin, the infinite result is not easily understood, as regards its
development.
ON ELECTROMAGNETIC WAVES. PART V. 443
But the development of infinitely great magnetic force by a plane
sheet of/ is very easily followed in full detail, not merely with sinu-
soidal /, but with / constant. Considering the latter case, the emission
of H is continuous, as before described, from the surface of /. Now
place a plane infinitely-conducting barrier parallel to /, say on the left
side. We at once stop the disturbances going to the left and send them
back again, unchanged as regards H, reversed as regards E. The
H-disturbance on the left side of /therefore commences to be doubled
after the time a/v has elapsed, a being the distance of the reflecting
barrier from the plane of/, and on the right side after the interval 2a/v.
Next, put a second infinitely-conducting barrier on the right side of /.
It also doubles the H-disturbances as they arrive ; so that, by the
inclusion of the plane of/ between impermeable barriers, combined with
the continuous emission of H, the magnetic disturbance mounts up
infinitely, in a manner which may be graphically followed with ease.
Similarly with / alternating, at particular frequencies depending upon
the distances of the two barriers from /.
Returning to the spherical case, an infinitely-conducting internal
screen, with no external, produces
H _ KX - ^X)K - w a )fr / 301 v
/?W-wJ)
We cannot produce infinite H in this case, because the absence of an
external barrier will not let it accumulate. Shifting the surface of/
right up to the screen, or conversely, simplifies matters greatly, reducing
to the case of 42.
May 8, 1888.
PART V.
CYLINDRICAL ELECTROMAGNETIC WAVES.
49. In concluding this paper I propose to give some cases of
cylindrical waves. They are selected with a view to the avoidance
of mere mathematical developments and unintelligible solutions, which
may be multiplied to any extent ; and for the illustration of peculiarities
of a striking character. The case of vibratory impressed E.M.F. in a
thin tube is very rich in this respect, as will be seen later. At present
I may remark that the results of this paper have little application in
telegraphy or telephony, when we are only concerned with long waves.
Short waves are, or may be, now in question, demanding a somewhat
different treatment.* We do, however, have very short waves in the
* The waves here to be considered are essentially of the same nature as those
considered by J. J. Thomson, "On Electrical Oscillations in a Cylindrical Con-
ductor," Pruc. Math. Soc. vol. xvn., and in Parts I. and II. of my paper, " On the
Self-induction of Wires," Phil. Mar/., August and September, 1886 ; viz. a mixture
of the plane and cylindrical. But the peculiarities of the telegraphic problem
make it practically a case of plane waves as regards the dielectric, and cylindrical
in the wires. The " resonance " effects described in my just-mentioned paper arise
from the to-aud-fro reflexion of the plane waves in the dielectric, moving parallel
444 ELECTRICAL PAPERS.
discharge of condensers, and in vacuum-tube experiments, so that we
are not so wholly removed from practice as at first appears. But
independently of considerations of practical realization, I am strongly
of opinion that the study of very unrealizable problems may be of use
in forwarding the supply of one of the pressing wants of the present
time or near future, a practicable ether mechanically, electromagneti-
cally, and perhaps also gravitationally comprehensive.
Mathematical Preliminary.
50. On account of some peculiarities in Bessel's functions, which
require us to change the form of our equations to suit circumstances, it
is desirable to exhibit separately the purely mathematical part. This
will also considerably shorten and clarify what follows it.
Let the axis of z be the axis of symmetry, and let r be the distance of
any point from it. Either the lines of E, electric force, or of H, magnetic
force, may be circular, centred on the axis. For definiteness, choose H
here. Then the lines of E are either longitudinal, or parallel to the
axis ; or there is, in addition, a radial component of E, parallel to r.
Thus the tensor H of H, and the two components of E, say E longi-
tudinal and F radial, fully specify the field. Their connexions are these
special forms of equations (2) and (3) :
(302)
where (and always later) p stands for d/dt. This is in space where neither
the impressed electric nor the impressed magnetic force has curl, it being
understood that E and H are the forces of the fluxes, so as to include
impressed. From (302) we obtain
1 d dE
r* r *7*-'
d 1 d rr^&H
Trrdr rH+ V
the characteristics of E and H. Let now
from which all the rest follows. Merely remarking concerning k that
the realization of (316) when k is finite requires the splitting up of the
Bessel functions into real and imaginary parts, that the results are com-
plex, and that there are no striking peculiarities readily deducible ; let
us take k = at once, and keep to nonconducting dielectrics. Then,
from (316), follow the equations of E and JT, in and out ; thus
or (out) = ^(^-yg) or J^-yG^ ...... (317)
TT or _cp
-"(in) OI (out) -- - i -- : -
s same denominator
which we can now examine in detail.
Vanishing of External Field. J 0a = 0.
52. The very first thing to be observed is that J 0a = makes E and H
and therefore also F vanish outside the tube, and that this property is
independent of y, or of the nature of the external medium. We require
the impressed force to be sinusoidal or simply periodic with respect to
z and t, thus
e = e sin (mz + a) sin (nt + /?), .................... (319)
so that, ultimately, s 2 = n 2 /v 2 - m* ; ........................... ,..(320)
ON ELECTROMAGNETIC WAVES. PART V. 449
and any one of the values of s given by / 0(? = causes the evanescence
of the external field. The solutions just given reduce to
(in) E^(sjcn)^irK(JJ i J la )ie,
(321)
which are fully realized, because i signifies p/n, or involves merely a
time-differentiation performed on the e of (319).
The electrification is solely upon the inner surface of the tube. In
its substance H falls from - 4irKe inside to zero outside, and E a being
zero, the current in the tube is Ke per unit surface.
The independence of y raises suspicion at first that (321) may not
represent the state which is tended to after e is started. But since the
resistance of the tube itself is sufficient to cause initial irregularities to
subside to zero, even were there a perfectly reflecting barrier outside the
tube to prevent dissipation of these irregularities in space, there seems
no reason to doubt that (321) do represent the state asymptotically
tended to. Changing the form of y will only change the manner of the
settling down. We may commence to change the nature of the medium
immediately at the outer boundary of the tube. We cannot, however,
have those abrupt assumptions of the steady or simply periodic state
which characterize spherical waves, owing to the geometrical conditions
of a cylinder.
Case of Two Coaxial Tubes.
53. If there be a conducting tube anywhere outside the first tube,
there is no current in it, except initially. From this we may conclude
that if we transfer the impressed force to the outer tube, there will be no
current in the inner. Thus, let there be an outer tube at r = #, of con-
ductance K per unit area, containing the impressed force e r We have
....................... (322)
where 7 3 and Y 2 are the H/E operators just outside and inside the
tube, whilst E x is the E at x, on either side of the tube, resulting
from e r We have
y _cp /to-ftflk Y _cpJ lx -yG lx
~~ ~
where y l is settled by some external and y by some internal condition.
In the present case the inner tube at r = , if it contains no impressed
force, produces the condition
Yt-Yi = l*K at r = a, .................. (324)
where Y l is the internal H/E operator. Or
^7rJ\.t/(\f t /OOK\
giving ^ = - ~ ( 325 )
H.E.P. VOL. II.
450 ELECTRICAL PAPERS.
Now, using (323) in (322) brings it to
E oz-teox-ioxi-i ...(326)
2 (yi - y) - 4 - W* - y0,)(/, - W
S TTSiC
in which y is given by (325), and from (326) the whole state due to e 1
follows, as modified by the inner tube.
Now J 0a = makes y = 0j this reduces (326) to
(327)
and, by comparison with (317), we see that it is now the same as if the
inner tube were non-existent. That is, when it is situated at a nodal
surface of E due to impressed force in the outer tube, and there is
therefore no current in it (except transversely, to which the dissipation
of energy is infinitely small), its presence does nothing, or it is perfectly
transparent.
It is clearly unnecessary that the external impressed force should be
in a tube. Let it only be in tubular layers, without specification of
actual distribution or of the nature of the medium, except that it is
in layers so that c, k, and p are functions of r only ; then if the axial
portion be nonconducting dielectric, the J 0r function specifies E and
allows there to be nodal surfaces, for instance J 0a = Q, where a con-
ducting tube may be placed without disturbing the field. Admitting
this property gb initio, we can conversely conclude that e in the
tube at r = a will, when / Oa = 0, make every external cylindrical
surface a nodal surface, and therefore produce no external disturb-
ance at all.
54. Now go back to 51, equations (317), (318). There are no
external nodal surfaces of E in general (exception later). We cannot
therefore find a place to put a tube so as not to disturb the existing
field due to e in the tube at r = ft. But we may now make use of a
more general property. To illustrate simply, consider first the mag-
netic theory of induction between linear circuits. Let there be any
number of circuits, all containing impressed forces, producing a deter-
minate varying electromagnetic field. In this field put an additional
circuit of infinite resistance. The E.M.F. in it, due to the other circuits,
will cause no current in it of course, so that no change in the field
takes place. Now, lastly, close the circuit or make its resistance finite,
and simultaneously put in it impressed force which is at every moment
the negative of the E.M.F. due to the other circuits. Since no current
is produced there will still be no change, or everything will go on as
if the additional circuit were non-existent.
Applying this to our tubes, we may easily verify by the previous
equations that when there are two coaxial tubes, both containing
impressed forces, we can reduce the resultant electromagnetic field
everywhere to that due to the impressed force in one tube, provided
we suitably choose the impressed force in the second to be the negative
ON ELECTROMAGNETIC WAVES. PART V. 451
of the electric force of field due to e in the first tube when the second
is non-existent. That is, we virtually abolish the conductance of the
second tube and make it perfectly transparent.
Perfectly Reflecting Barrier. Its Effects. Vanishing of Conduction
Current.
55. To produce nodal surfaces of E outside the tube containing the
vibrating impressed force, we require an external barrier, which shall
prevent the passage of energy or its absorption, by wholly reflecting all
disturbances which reach it. Thus, let there be a perfect conductor at
r = x. This makes E = there. This requires that the y in (317),
(318) shall have the value Jo x /'Gr 0n whereas without any bound to the
dielectric it would be i. We can now choose m and n so as to make
J Qx = 0. This reduces those equations to
E=- r t
(in and out)
*"&
,(328)
This solution is now the same inside and outside the tube containing
the impressed force, and there is no current in the tube, that is, no
longitudinal current.
To understand this case, take away the impressed force and the tube.
Then (328) represents a conservative system in stationary vibration.
Now, by the preceding, we may introduce the tube at a nodal surface
of E without disturbing matters, provided there be no impressed force
in the tube. But if we introduce the tube anywhere else, where E is
not zero, we require, by the preceding, an impressed force which is at
every moment the negative of the undisturbed force of the field, in
order that no change shall occur. Now this is precisely what the
solution (328) represents, e in the tube being cancelled by the force of
the field, so that there is no conduction-current. The remarkable
thing is that it is the impressed force in the tube itself that sets up the
vibrating field, and gradually ceases to work, so that in the end it and
the tube may be removed without altering the field. That a perfect
conductor as reflector is required is a detail of no moment in its
theoretical aspect.
Shifting the tube, with a finite impressed force in it, towards a nodal
surface of E, sends up the amplitude of the vibrations to any extent.
K = and K = .
56. If the tube have no conductance, e produces no effect. This is
because the two surfaces of curl of e are infinitely close together, and
therefore cancel, not having any conductance between them to produce
a discontinuity in the magnetic force.
But if the tube have infinite conductance, we produce complete
mutual independence of the internal and external fields, except in the
452 ELECTRICAL PAPERS.
quite unessential particular that the two surfaces of curie are of
opposite kind and time together. Equations (317), (318) reduce to
(in) E=-^e, F=+ lJ > r ^ H= -l^cpe (329)
i/Oa S J 0a dZ S J 0a
(out)
J 0a - yG^ dz
(330)
Observe that (329) is the same as (328). The external solution (330)
requires y to be stated. When y = i, for a boundless dielectric, the
realization is immediate.
s = 0. Vanishing of E all over, and of F and H also internally.
57. This is a singularity of quite a different kind. When n = mv, we
make ,s- = 0. Of course there is just one solution with a given wave-
length along z; a great frequency with small wave-length, and con-
versely.
E vanishes all over, that is, both inside and outside the tube contain-
ing e, provided s/y is zero. The internal J^and therefore also Evanish.
Thus within the tube is no disturbance, and outside, (317) (318)
reduce to
(out) H=4irKe, F~--4arK ...(331)
r en r dz
Observe that H and F do not fluctuate or alternate along r, but that
H has the same distribution (out from the tube) as if e were steady and
did not vary along z.
A special case is in = 0. Then also n = 0, or e is steady and indepen-
dent of z. F vanishes, and the first of (331) expresses the steady state.
Without this restriction, the current in the tube is Ke per unit
surface, owing to the vanishing of the opposing longitudinal E of the
field. This property was, by inadvertence, attributed by me in a
former paper * to a wire instead of a tube. The wave-length must be
great in order to render it applicable to a wire, because instantaneous
penetration is assumed.
I mentioned that s/y must vanish. This occurs when y i t or the
external dielectric is boundless. But it also occurs when E = at r = x,
produced by a perfectly conductive screen. This is plainly allowable
because it does not interfere with the E =-0 all-over property. What
the screen does is simply to terminate the field abruptly. Of course it
is electrified.
s = and H x = 0.
58. But with other boundary conditions, we do not have the solutions
(331). Thus, let H X = Q, instead of E x = 0. This makes y = J lx /G lx in
* "On Resistance and Conductance Operators," Phil. Mag., Dec. 1887, p. 492,
Ex. . [vol. n. t p. 366].
ON ELECTROMAGNETIC WAVES. PART V. 453
(317), (318). There are at least two ways (theoretical) of producing
this boundary condition. First, there may be at r = v a screen made of
a perfect magnetic conductor (g = ao). Or, secondly, the whole medium
beyond r = x may be infinitely elastive and resistive (c = 0, k = 0) to an
infinite distance.
Now choose 5 = in addition, and reduce (317), (318). The results
are
Ess _e_ F= 1 dH
l+frfyf&Za' cpdz*
.
(in)or < out > *
which are at once realized by removing p from the denominator to the
numerator.
Although E is not now zero, it is independent of ?', only varying
with t and z.
When s 2 is negative, or n < m/v, the solutions (317), (318) require
transforming in part because some of the Bessel functions are unreal.
Use (312), because (^ is now real. There are no alternations in E or H
along r. They only commence when n > mv.
Separate A dims of the Two Surfaces of curl e.
59. Since all the fluxes depend solely upon the curl of e, and not
upon its distribution, and there are two surfaces of curl e in the tube
problem, their actions, which are independent, may be separately
calculated. The inner surface may arise from e in the - direction in
the inner dielectric, or by the same in the + direction in the tube and
beyond it. The outer may be due to e in the - direction beyond the
tube, or in the + direction in the tube and inner dielectric.
We shall easily find that the inner surface of curl of e, say of surface-
density / 15 produces
(333)
(OUt) E= "la^Or-^Or/ /
same denominator
from which H may be got by the E/H operator.
The external sheet, say / 2 , produces
(in) E = ^ Or ^ la ~ yl*)f
(334)
(out)
where the unwritten denominators are as in the first of (333). Observe
that when J la = 0, / x produces no external field (in tube or beyond it).
It is then only / 2 that operates in the tube and beyond.
454 ELECTRICAL PAPERS.
Now take f 2 = e and /j = - e in (333) and (334) and add the
results. We then obtain (317), (318); and it is now J 0a = that
makes the external field vanish, instead of J la = when / x alone is
operative.
Having treated this problem of a tube in some detail, the other
examples may be very briefly considered, although they too admit of
numerous singularities.
Circular Impressed Force in Conducting-Tube.
60. The tube being as before, let the impressed force e (per unit
length) act circularly in it instead of longitudinally, and let e be a
function of t only, so that we have an inner and an outer cylindrical
surface of longitudinally directed curl of e. H is evidently longitudinal
and E circular, so that we now require to use the (314) operator.
At the tube E a is continuous, this being the tensor of the force of the
flux on either side, and H is discontinuous thus,
............ (335)
Substituting the (314) operator, with y = inside, and y undetermined
outside, and using the conjugate property (307), we obtain
H m or , out ,= -j(4.-yguV r W-\ ...... (336)
or l = ,- - or
_.
same denominator
When e is simply periodic, J la = makes the external E and H vanish
independent of the nature of y. The complete solution is then
(338)
The conduction-current in the tube is Ke per unit area of surface.
To make the conduction-current vanish by balancing the impressed
force against the electric force of the field that it sets up, put an
infinitely-conducting screen at r-x outside the tube, and choose the
frequency to make J lx = 0, since we now have y = J-^JG^ We shall
then have the same solution inside and outside, viz.
H= --^ie, E=- J *e-,. ...(339)
so that at the tube itself, E = - e. This case may be interpreted as in
55, the tube being at a nodal surface of E.
A special case of (338) is when n = 0, or e is steady. Then there is
merely the longitudinal H inside the tube, given by H=4=irKe.
ON ELECTROMAGNETIC WAVES. PART V. 455
Cylinder of Longitudinal cwrl of e in a Dielectric.
61. In a nonconductive dielectric let the impressed electric force be
such that its curl is confined to a cylinder of radius a, in which it is
uniformly distributed, and is longitudinal. Let / be the tensor of curl e,
and let it be a function of t only. Since E is circular and H longi-
tudinal, we have (314) as operator, in which k is to be zero. This is
outside the cylinder. Inside, on the other hand, on account of the
existence of curie, the equation corresponding to (314) is
J ........................... (340)
At the boundary r = a both E and H are continuous ; so, by taking
r = a in (340) and in the corresponding (314) with k = 0, and eliminating
E a or H a between them, we obtain the equation of the other. We
obtain
(out) = ulr - lr , .. ..(341)
--
in which y, as usual, is to be fixed by an external boundary condition,
or, if the medium be boundless, y = i.
We see at once that J la = 0, with / simply-periodic, makes the exter-
nal fluxes vanish. We should not now say that it makes the external
field vanish, though the statement is true as regards H, because the
electric force of the field does not vanish ; it cancels the impressed
force, so that there is no flux. This property is apparently independent
of y. But, since there is no resistance concerned, except such as may
be expressed in y, it is clear that (341), sinusoidally realized, cannot
represent the state which is tended to after starting/, unless there be
either no barrier, so that initial disturbances can escape, or else there
be resistance somewhere, to be embodied in y, so that they can be
absorbed, though only through an infinite series of passages between
the boundary and the axis of the initial wave and its consequences.
Thus, with a conservative barrier producing E = at r~x, and
y -Jix/@ixi there is no escape for the initial effects, which remain in the
form of free vibrations, whilst only the forced vibrations are got by
taking s' 2 = + constant in (341). The other part of the solution must
be separately calculated. If t/ liB = 0, E and H run up infinitely. If
J la = also, the result is ambiguous.
With no barrier at all, or y = i,vre have
(out) / E = H2a)-VU0ir + ^)/l ...(342)
\ J ET=(2^)-i/ la (/ 0r -^ 0r )/ ,J
which are fully realized. Here / =/7ra 2 , which may be called the
strength of the filament. We may most simply take the impressed
force to be circular, its intensity varying as r within, and inversely as r
outside the cylinder. Then/= 2eja, if e a is the intensity at r = a.
When nr/v is large, (342) becomes, by (308), writing / sin nt for/ ,
(out) E = f *vH=^(sm(nt- + ?\ ........... (343)
4# \irnrj \ v 4/
456 ELECTRICAL PAPERS
approximately. 2?rr should be a large multiple, and 2ira a small frac-
tion of the wave-length along r.
Filament of curl e. Calculation of Wave.
62. In the last, let / be constant, whilst a is made infinitely small.
It is then a mere filament of curl of e at the axis that is in operation.
We now have, bythe second of (342), with J la = ^na/v,
|-(/ fr -iffJ/ = If= -(
which may be regarded as the simply-periodic solution or- as the
differential equation of H. In the latter case, put in terms of W by
(311), then
^=(2^)-%/2 7 r)-i/F/ ; ......................... (345)
or, expanding by (309),
........ (346)
in which / may be any function of the time. Let it be zero before, and
constant after t = 0. Then, first,
Next effect the integrations of this function indicated by the inverse
powers of q or p/v, thus
2r)]- .......... (348)
Lastly, operating on this by t~ 9r turns vt to vt r, and brings (346) to
H=(f /2irpv)(W -**)-*, .................... (349)
which is ridiculously simple. Let Z be the time-integral of H, then
?- 1 )*] .................. (350)
from which we may derive E ; thus
curl Z = cE, or E=- ldZ =- ___ *#* (351)
c dr 27rr^2-r 2 *
The other vector-potential A, such that E= -^>A, is obviously
...................... (352)
All these formulae of course only commence when vt reaches r. The
infinite values of E and H at the wave-front arise from the infinite con-
centration of the curl of e at the axis.
Notice that E = .............................. (353)
ON ELECTROMAGNETIC WAVES. PART VI. 457
everywhere. It follows from this connexion between E and H (or from
their full expressions) that
= ce* = c(/ /27ir) 2 ; .................. (354)
where e denotes the intensity of impressed force at distance r, when it
is of the simplest type, above described. That is, the excess of the
electric over the magnetic energy at any point is independent of the
time. Both decrease at an equal rate ; the magnetic energy to zero,
the electric energy to that of the final steady displacement ce/4?r.
6 2 A. The above E and H solutions are fundamental, because all
electromagnetic disturbances due to impressed force depend solely upon,
and come from, the lines of curl of the impressed force. From them, by-
integration, we can find the disturbances due to any collection of recti-
linear filaments of f. Thus, to find the H due to a plane sheet of parallel
uniformly distributed filaments, of surface-density /, we have, by (349),
at distance a from the plane, on either side,
H- f
J
y
where the limits are (vW - a 2 )*. Therefore
after the time t = a/v ; before then, H is zero. [Compare with 2,
equation (12).]
62 B. Similarly, a cylindrical sheet of longitudinal f produces
H _ fa f dO .
where b is the distance of the point where H is reckoned from the
element adO of the circular section of the sheet, a being its radius. The
limits have to be so chosen as to include all elements of / which have
had time to produce any effect at the point in question. When the
point is external and vt exceeds a + r the limits are complete, viz. to
include the whole circle. The result is then, at distance r from the axis
of the cylinder,
n f a l^ v f^l-3 x 1-3.5.7 .r 2 4.3 1.3.6.7.9.11 a* 6.5.4 1 ,.
r= ' ''' " " ' (
where x = (2ar) 2 (i^ 2 - a 2 - r 2 )~ 2 .
This formula begins to operate when x= 1, or vt = a + r. As time goes
on, x falls to zero, leaving only the first term.
PART VI.
Cylindrical Surface of Circular curl e in a Dielectric.
63. Let the curl of the impressed electric force be wholly situated on
the surface of a cylinder, of radius a, in a nonconducting dielectric. The
458 ELECTRICAL PAPERS.
impressed force e to correspond may then be most conveniently imagined
to be either longitudinal, within or without the cylinder, uniformly dis-
tributed in either case (though oppositely directed), and the density of
curie will be e; or, the impressed force may be transferred to the sur-
face of the cylinder, by making e radial, but confined to an infinitely
thin layer. The measure of the surface-density of curl e will now be
/== M - (OTItl ...................... -( 356 )
where e is the total impressed force (its line-integral through the layer).
The second form of this equation shows the effect produced on the
electric force E of the flux, outside and inside the surface. This E is,
as it happens, also the force of the field ; but in the other case, when e
is uniformly distributed within the cylinder, producing f=e, we have
the same discontinuity produced by /.
H being circular, we use the operator (313). Applying it to (356),
we obtain
n . ................... (357)
from which, by the conjugate property (307), and the operator (313),
we derive
or J^-yGf, (358)
or J la (Ji r -yG lr )f, (359)
in which / is a function of t, and it may be also of z. If so, then we
have the radial component F of electric force given by
F w or (out) = - /i,W. - yG^ or / la (/ lr - *A,) (360)
From these, by the use of Fourier's theorem, we can build up the
complete solutions for any distribution of / with respect to z; for
instance, the case of a single circular line of curl e.
Jja = 0. Vanishing of External Field.
64. Let/ be simply-periodic with respect to t and z ; then J la 0, or
v 2 -m 2 }=0, ........................ (361)
produces evanescence of E and H outside the cylinder. The indepen-
dence of this property of y really requires an unbounded external
medium, or else boundary-resistance, to let the initial effects escape or
be dissipated, because no resistance appears in our equations except in
y. The case s = or n = mv is to be excepted from (361) ; it is treated
later.
ON ELECTROMAGNETIC WAVES. PART VI. 459
y = i. Unbounded Medium.
65. When n/v > m, s is real, and our equations give at once the fully
realized solutions in the case of no boundary, by taking y = i,
,i or (out) =
or =
ir(Jia - iG la ) or J la (J lr - iG lr )jf,
J^(G la + iJ la ) or J^G^ + i/) V,
n, or (out) = 4 (j lr (G la + i/ la ) or J la (G lr + i/ lr ))
(362)
in which i means pjn.
The instantaneous outward transfer of energy per unit length of
cylinder is (by Poynting's formula)
and the mean value with respect to the time comes to
....................... (363)
if / is the maximum value of /, [thus, f=f cos mz sin nt]. This may of
course be again averaged to get rid of the cosine.
s = 0. Vanishing of External E.
66. When n = mv, we make s = 0, and then (362) reduce to the singular
solution
a f v n P i a 2 df "
' Cp/, (out) = 0, ^ (out) = - i - - .
Observe that the internal longitudinal displacement is produced entirely
by the impressed force (if it be internal), though there is radial displace-
ment also, on account of the divergence of e (if internal). Outside the
cylinder, the displacement is entirely perpendicular to it.
H and F do not alternate along r. This is also true when s 2 is nega-
tive, or n lies between and mv. Then, q 2 being positive, we have
(365)
as the rational form of the equation of the external E when the fre-
quency is too low to produce fluctuations along r.
The system (364) may be obtained directly from (358) to (360) on
the assumption that s/y is zero when s is zero. But (364) appears to
require an unbounded medium. Even in the case of the boundary
condition E = at r = z, which harmonizes with the vanishing ' of E
externally in (364), there will be the undissipated initial effects con-
tinuing.
If, on the other hand, H X =Q, making y = J lx /G lx) we shall not only
have the undissipated initial effects, but a different form of solution for
460 ELECTRICAL PAPERS.
the forced vibrations. Thus, using this expression for ?/, and also s = 0,
in (358) to (360), we obtain
(366)
representing the forced vibrations.
Effect of suddenly Starting a Filament of e.
67. The vibratory effects due to a vibrating filament we find by taking
a infinitely small in (362), that is J la = ^sa. To find the wave produced
by suddenly starting such a filament, transform equations (358), (359)
by means of (311). We get [e being intensity of longitudinal e]
(367)
where W is given by (309) ; the accent means differentiation to r, and
the suffix a means the value at r - a.
In these, let e = ira 2 e, which we may call the strength of the filament,
and let a be infinitely small. We then obtain
Now if e is a function of t only, it is clear that there is no scalar electric
potential involved. We may therefore advantageously employ (and for
a reason to be presently seen) the vector-potential A, such that
E=-pA, or A=-p~ l E; and /x#= -^. (369)
The equation of A is obviously, by the first of (369) applied to second
of (368),
A = \(pl%mP)*W% (370)
Comparing this equation with that of H in (345) (problem of a filament
of curl of e), we see that / there becomes e here, and pH there becomes
A here. The solution of (370) may therefore be got at once from the
solution of (345), viz. (349). Thus
A = % ; (371)
from which, by (369),
E = W* ?> #=-- W _, (372)
the complete solution. It will be seen that
(373)
ON ELECTROMAGNETIC WAVES. PART VI. 461
whilst the curious relation (353) in the problem, of a filament of curie
is now replaced by A=rpZ/t, (374)
where Z is the time-integral of the magnetic force ; so that
H=pZ, and curlZ = cE, (375)
Z being merely the vectorised Z. It is the vector-potential of the
magnetic current.
The following reciprocal relation is easily seen by comparing the
differential equations of an infinitely fine filament e and a finite fila-
ment. The electric current-density at the axis due to a longitudinal
cylinder of e (uniform) of radius a is numerically identical with the
total current through the circle of radius a due to the same total
impressed force (that is, rfe) concentrated in a filament at the axis, at
corresponding moments.
68. "Having got the solutions (372) for a filament e^ it might appear
that we could employ them to build up the solutions in the case of, for
instance, a cylinder of longitudinal impressed force of finite radius a.
But, according to (372), E would be positive and H negative every-
where and at every moment, in the case of the cylinder, because the
elementary parts are all positive or all negative. This is clearly a
wrong result. For it is certain that, at the first moment of starting
the longitudinal impressed force of intensity e in the cylinder, E just
outside it is negative ; thus
E ^e, in or out, at r = a, t = ;
and that H is positive ; viz.
H=e/2fMV at r = a, t = 0.
We know further that, as E starts negatively just outside the cylinder,
E will be always negative at the front of the outward wave, and H
positive; thus _ E . faH ^ x (a/r) (37 6)
the variation in intensity inversely as the square root of the distance
from the axis being necessitated in order to keep the energy constant
at the wave-front. The same formula with + E instead of - E will
express the state at the front of the wave running in to the axis.
There is thus a momentary infinity of E at the axis, viz., when t = a/v.
So far we can certainly go. Less securely, we may conclude that
during the recoil, E will be settling down to its steady value e within
the cylinder, and therefore the force of the field there will be positive,
and, by continuity, also positive outside the cylinder. Similarly, H
must be negative at any distance within which E is decreasing. We
conclude, therefore, that the filament-solutions (372) only express the
settling down to the final state, and are not comprehensive enough to
be employed as fundamental solutions.
Sudden Starting of e longitudinal in a Cylinder.
69. In order to fully clear up what is left doubtful in the last para-
graph, I have investigated the case of a cylinder of e comprehensively.
462 ELECTRICAL PAPERS.
The following contains the leading points. We have to make four inde
pendent investigations: viz., to find (1), the initial inward wave; (2),
the initial outward wave ; (3), the inside solution after the recoil ; (4),
the outside solution ditto. We may indeed express the whole by a
definite integral, but there does not seem to be much use in doing so,
as there will be all the labour of finding out its solutions, and they are
what we now obtain from the differential equations.
Let E l and E 2 be the E's of the inward and outward waves. Their
equations are
e t ....................... (377)
- ....................... (378)
where U and W are given by (309), the accent means differentiation to
r, and the suffix indicates the value at r = a. To prove these, it is
sufficient to observe that U and W involve qr and e~ qr respectively, so
that (377) expresses an inward and (378) an outward wave; and
further that, by (310), we have
E l -E z = e at r = a, always; ............... (379)
which is the sole boundary condition at the surface of curl of e.
Expanding (377), we get
3-- + - + ... e> (380)
where B + S is given by (309), and y = Sqa. Now, e being zero before
and constant after t = 0, effect the integrations indicated by the inverse
powers of p, and then turn t to t v where
The result is
.... 4
"~
. _.
"T^ a+ 1
....... (3
the structure of which is sufficiently clear. Here z : = vt-JSa.
This formula, when vt < a, holds between r = a and r = a - vt. But
when vt > a though < 2a, it holds between r = a and vt - a. Except
within the limits named, it is only a partial solution.
70. As regards E% it may be obtained from (381) by the following
changes. Change E : to - E 2 on the left, and on the right change z l
to - z 2l where
It is therefore unnecessary to write out E 2 . This E 2 formula will
hold from r = a to r = vt + a, when vt<2a; but after that, when the
front of the return wave has passed r = a, it will only hold between
r = vt-a and vt + .
ON ELECTROMAGNETIC WAVES. PART VI. 463
71. Next to find E 3 , the E in the cylinder when vt> a and the solu-
tion is made up of two oppositely going waves, and E the external E
after vt = 2a, when it is made up of two outward going waves. I have
utterly failed to obtain intelligible results by uniting the primary waves
with a reflected wave. But there is another method which is easier,
and free from the obscurity which attends the simultaneous use of U
and W. Thus, the equations of E B and E are
(382)
............. (383)
by (367) ; and a necessity of their validity is the presence of two waves
inside the cylinder, because of the use of / and J^ ; it is quite inad-
missible to use J when only one wave is in question, because J^ = 1
when r = 0, and being a differential operator in rising powers of p, the
meaning of (382) is that we find E^ at r by differentiations from E B at
r = 0; thus (382) only begins to be valid when vt = a.
To integrate (382), (383), it saves a little trouble to calculate the
time-integrals of E 3 and E 4 , say
A 3 =-p-*E 3 , A,= -p-^E, ........ . ....... (384)
The results are - A s = /<. e -(vW-arf, ........................ (385)
^' 1 ................ (386)
From these derive E 3 and E by time-differentiation, and H 3 , H 4 by
space-differentiation, according to
pK, or H=---^ (387)
We see that the value of E 3 at the axis, say E , is
E Q = evt(v*P-a 2 )~*; (388)
and by performing the operation J 0r in (385) we produce, if u = (v*t 2 - a 2 )*,
-Ao = -\ u + ~(- -
- (3
from which we derive
(390)
These formulae commence to operate when vt = a at the axis, and when
i)t = a + r at any point r < a, and continue in operation for ever after.
464 ELECTRICAL PAPERS.
72. Lastly, perform the operation (2/sa)J la in (386), and we obtain
A = a2 *ri -( - l ^-\ !*(*. 3(W 35 * ;4 * 4 \
4 2v [_u + 8 V u 3 + u* ) + 64\M 5 u 7 ~~?~/
45a 6 / 5 135v 2 / 2 315^ 231 tW\ ~1
+ 4. 36.64V' ^ + ^ 9 ~ "~^~ P / + "__]'
from which we derive
15?4)
. (392)
1
These begin to operate at r = a when vt = 2a ; and later, the range is
from r = a to r = vt - a.
This completes the mathematical work. As a check upon the
accuracy, we may test satisfaction of differential equations, and of
the initial condition, and that the four solutions join together with
the proper discontinuities.
73. The following is a general description of the manner of establish-
ing the steady flux. We put on e in the cylinder when t = 0. The
first effect inside is E 1 = \e, at the surface, and H^ = EJpv. This
primary disturbance runs in to the axis at speed v, varying at its
front inversely as the square root of the distance from the axis, thus
producing a momentary infinity there. At this moment t = a/v, E l is
also very great near the axis. In the meantime, E l ,has been increasing
generally all over the cylinder, so that, from being \& initially at the
boundary, it has risen to '77 e, whilst the simultaneous value at r - ^a
is about *95 e.
Now consider E 3 within the cylinder, it being the natural con-
tinuation of Ey The large values of E 1 near the axis subside with
immense rapidity. But near the boundary E l still goes on increasing.
The result is that when vt = 2a, and the front of the return-wave reaches
the boundary, E z has fallen from oo to l'154e at the axis; at r = Jo.
the value is 1-183 e; at r = fa it is 1-237 e; and at the boundary the
value has risen to 1-71 e, which is made up thus, 1-21 e + ^e; the first
of these being the value just before the front of the return-wave arrives,
the second part the sudden increase due to the wave-front. E s is now
a minimum at the axis and rises towards the wave-front, the greater
part of the rise being near the wave-front.
Thirdly, go back to = and consider the outward wave. First,
-^2 - ~~ % e at r = a - This runs out at speed v, varying at the front
inversely as ri As it does so, the E 2 that succeeds rises, that is, is
less negative. Thus when vt = a, and the front has got to r = 2, the
values of E. 2 are - '232 e at r = a and - -353 e at r = 2a. Still later, as
this wave forms fully, its hinder part becomes positive. Thus, when
fully formed, with front at r = 3a, we have E z = - -288 e at r = 3a ;
- '14:5 e at r = 2a; and '21 e at r = a. This is at the moment when
the return-wave reaches the boundary, as already described.
ON ELECTROMAGNETIC WAVES. PART VI. 465
The subsequent history is that the wave E 2 moves out to infinity,
being negative at its front and positive at its back, where there is
a sudden rise due to the return-wave E^ behind which there is a rapid
fall in E^ not a discontinuity, but the continuation of the before-
mentioned rapid fall in E 3 near its front. The subsidence to the
steady state in the cylinder and outside is very rapid when the front
of E 4 has moved well out. Thus, when vt = 5a, we have E 3 = 1-022 e
at r = a, and of course, just outside, we have E 4 = '022e; and when
vt = l(k, we have E 3 = 1 -005 e, E 4 = -005 e, at r = a.
As regards H, starting when t = with the value e/2pv at r = a only,
at the front of the inward or outward wave it is E = pvH, as usual.
It is positive in the cylinder at first, and then changes to negative.
Outside, it is first positive for a short time, and then negative for ever
after.
74. We can now see fully why the solution for a filament e Q of e can
not be employed to build up more complex solutions in general, whilst
that for a filament / of curl e can be so employed. For, in the latter
case, the disturbances come, ab initio, from the axis, because the lines
of curl e are the sources of disturbance, and they become a single line
at the axis. But in the former case it is not the body of the filament,
but its surface only, that is the real source, however small the filament
may be, producing first E negative (or against e) just outside the
filament, and, immediately after, E positive. Now when the diameter
of the filament is indefinitely reduced, we lose sight altogether of the
preliminary negative electric and positive magnetic force, because their
duration becomes infinitely small, and our solutions (372) show only
the subsequent state of positive electric and negative magnetic force
during the settling down to the final state, but not its real commence-
ment, viz., at the front of the wave.
75. The occurrence of momentary infinite values of E or of H, in
problems concerning spherical and cylindrical electromagnetic waves,
is physically suggestive. By means of a proper convergence to a point
or an axis, we should be able to disrupt the strongest dielectric, starting
with a weak field, and then discharging it. Although it is impossible
to realize the particular arrangements of our solutions, yet it might be
practicable to obtain similar results in other ways.*
It may be remarked that the solution worked out for an infinitely
* If we wish the solution for an infinitely long cylinder to be quite unaltered,
when of finite length I, let at z = Q and z = l infinitely conducting barriers be
placed. Owing to the displacement terminating upon them perpendicularly, and
the magnetic force being tangential, no alteration is required. Then, on taking
off the impressed force, we obtain the result of the discharge of a condenser
consisting of two parallel plates of no resistance, charged in a certain portion
only ; or, by integration, charged in ?ny manner.
To abolish the momentary infinity at the axis, in the text, substitute for the
surface distribution of curl of e a distribution in a thin layer. The infinity will
be replaced by a large finite value, without other material change. Of course the
theory above assumes that the dielectric does not break down. If it does, we
change the problem, and have a conducting (or resisting) path, possibly with
oscillations of great frequency if the resistance be not too great, as Prof. Lodge
believes to be the case in a lightning discharge.
H.E.P. VOL. II. 2o
466 ELECTRICAL PAPERS.
long cylinder of longitudinal e is also, to a certain extent, the solution
for a cylinder of finite length. If, for instance, the length is 21, and
the radius ft, disturbances from the extreme terminal lines of f (or curie)
only reach the centre of the axis after the time (a 2 + l 2 )*/v, whilst from
the equatorial line of f the time taken is a/v, which may be only a little
less, or very greatly less, according as I/a is small or large. If large, it
is clear that the solutions for E and H in the central parts of the
cylinder are not only identical with those for an infinitely long cylinder
until disturbances arrive from its ends, but are not much different
afterwards.
Cylindrical Surface of Longitudinal f, a Function of and t.
76. When there is no variation with 0, the only Bessel functions con-
cerned are J Q and J r The extension of the vibratory solutions to
include variation of the impressed force or its curl as cos 6, cos 20, etc.,
is so easily made that it would be inexcusable to overlook it. Two
leading cases will be very briefly considered. Let the curl of the
impressed force be wholly upon the surface of a cylinder of radius a,
longitudinally directed, and be a function of t and 6, its tensor being/,
the measure of the surface-densitj'. H is also longitudinal, of course,
whilst E has two components, circular E and radial F. The connections
are
from which the characteristic of H is
1 d dH
-0 ..................... (394)
r \ r /
if s 2 = -^2/v 2 and w 2 = - d 2 /d0 2 . Consequently
H= (J mr -yG mr ) cos mB x function of t ............ (395)
when m? is constant, and the E/H operator is
* IJLrlgk ...(396)
H cpJ mr -yGj
if J mr or J m (sr) is the m th Bessel-function, and G mr its companion,
whilst the ' means d/dr.
The boundary condition is
Ei = E*-f at r = a, .................... (397)
E-i being the inside, E 2 the outside value of the force of the flux.
Therefore, using (396) with ?/ = inside, we obtain
where x is a constant, being ?r/2 when m = 0, according to (307), and
always ?r/2 if G^ has the proper numerical factor to fix its size.
We see that if
ON ELECTROMAGNETIC WAVES. PART VI. 467
where / is constant, the boundary H, and with it the whole external
field, electric and magnetic, vanishes when
/..-o.
If 77i = 0, or there is no variation with 0, the impressed force may be
circular, outside the cylinder, and varying as r~ l .
If m= 1, the impressed force may be transverse, within the cylinder,
and of uniform intensity.
Conducting Tube, e Circular, a Function of 6 and t.
77. This is merely chosen as the easiest extension of the last case.
In it let there be two cylindrical surfaces of f, infinitely close together.
They will cancel one another if equal and opposite, but if we fill up the
space between them with a tube of conductance K per unit area, we get
the case of e circular in the tube, e varying with 6 and t, and produce a
discontinuity in H (which is still longitudinal, of course). Let E a be
the common value of E just outside and inside the tube ; e + E a is then
the force of the flux in the substance of the tube, and
........................ (399)
the discontinuity equation, leads, by the use of (396) and the conjugate
property of J m and G- m as standardized* in the last paragraph, through
to the equation of E M viz.,
...(400)
ira Lt/maVt/tna # w w_
from which we see that it is J^ = that now makes the external field
vanish.
78. This concludes my treatment of electromagnetic waves in relation
to their sources, so far as a systematic arrangement and uniform method
is concerned. Some cases of a more mixed character must be reserved.
It is scarcely necessary to remark that all the dielectric solutions may
be turned into others, by employing impressed magnetic instead of
electric force. The hypothetical magnetic conductor is required to
obtain full analogues of problems in which electric conductors occur.
August 10, 1888.
* [If we take Stokes's formula for J mt thus
then the substitution of sin for cos and - cos for sin will give the O m function
standardized as in the text. Also note that the infiniteness of G when /3 is
omitted, referred to in footnote p. 445, arises when q* is + ].
468 ELECTRICAL PAPERS.
XLIV. THE GENERAL SOLUTION OF MAXWELL'S ELECTRO-
MAGNETIC EQUATIONS IN A HOMOGENEOUS ISO-
TROPIC MEDIUM, ESPECIALLY IN REGARD TO THE
DERIVATION OF SPECIAL SOLUTIONS, AND THE
FORMULAE FOR PLANE WAVES.
[Phil. Mag., Jan. 1889, p. 30.]
Equations of the Field.
1. ALTHOUGH, from the difficulty of applying them to practical problems,
general solutions frequently possess little practical value, yet they may
be of sufficient importance to render their investigation desirable, and
to let their applications be examined as far as may be practicable. The
first question here to be answered is this. Given the state of the whole
electromagnetic field at a certain moment, in a homogeneous isotropic
conducting dielectric medium, to deduce the state at any later time,
arising from the initial state alone, without impressed forces.
The equations of the field are, if p stand for d/dt,
(1)
......................... (2)
the first being Maxwell's well-known equation defining electric current
in terms of the magnetic force H, k being the electric conductivity and
c/4?r the electric permittivity (or permittance of a unit cube condenser),
and E the electric force ; whilst the second is the equation introduced
by me* as the proper companion to the former to make a complete
system suitable for practical working, g being the magnetic conductivity
and //, the magnetic inductivity. This second equation takes the place
of the two equations
E=-A-W, curlA = /xH, .................. (3)
of Maxwell, where A is the electromagnetic momentum at a point, and
Mf the scalar electric potential. Thus ^ and A are murdered, so to
speak, with a great gain in definiteness and conciseness. As regards g,
however, standing for a physically non-existent quality, such that the
medium cannot support magnetic force without a dissipation of energy
at the rate = 4irg/p. But a circuital state of /xH disappears at once, by
instantaneous transference to infinity. Thus any varying impressed
force h is accompanied without delay by the corresponding steady flux,
the magnetic induction.
When the inertia associated with //, is considered, the result is rather
striking and difficult to understand. It appears, however, to belong to
the same class of (theoretical) phenomena as the following. If a coil in
which there is an electric current be instantaneously shunted on to a
second coil in which there is no current, then, according to Maxwell,
the first coil instantly loses current and the second gains it, in such a
way as to keep the momentum unchanged. Now we cannot set up a
current in a coil instantly, so that we have a contradiction. But the
disagreement admits of easy reconciliation. We cannot set up current
instantly with a finite impressed force, but if it be infinite we can. In
the case of the coils there is an electromotive impulse, or infinite electro-
motive force acting for an infinitely short time, when the coils are con-
nected, with corresponding instantaneous changes in their momenta.
A loss of energy is involved.
It is scarcely necessary to remark that the true physical theory
involves other considerations, on account of the dielectric not being
infinitely elastive, and on account of diffusion in the wires ; so that
we have sparking and very rapid vibrations in the dielectric. The
energy which is not wasted in the spark, and which would go out to
GENERAL SOLUTION OF ELECTROMAGNETIC EQUATIONS. 473
infinity were there no conducting obstacles, is probably all wasted
practically in the heat of conduction-currents in them.
Impressed Forces.
8. Given initially E and H , we know that the diverging parts
must either remain constant or subside, and are, in a manner, self-
contained; but the circuital parts, which would give rise to waves,
may be kept from changing by means of impressed forces e and h .
Thus, let E and H be circuital. To keep them steady we have, in
equations (1), (2), to get rid of^E and^?H. Thus
curl (H - h ) = 47rE , \
curl(e -E ) -fcftyj"
are the equations of steady fields E and H , these being the forces of
the fluxes. Or
curl h = curl H - 47rE , 1
curl e = curl EO + ^H,,,/''
give the curls of the required impressed forces in terms of the given
fluxes, and any impressed forces having these curls will suffice.
Now, on the sudden removal of e , h , the forces E , H , which had
hitherto been the forces of the fluxes, become, instantaneously, the
forces of the field as well. That is, the fluxes themselves do not change
suddenly, except in such a case as a tangential discontinuity in a flux
produced at a surface of curl of impressed force, when, at the surface
itself, the mean value will be immediately assumed on removal of
the impressed force. We know, therefore, the effects due to certain
distributions of impressed force when we know the result of leaving
the corresponding fluxes to themselves without impressed force. It is,
however, the converse of this that is practically useful, viz., to find the
result of leaving the fluxes without impressed force by solving the
problem of the establishment of the steady fluxes when the impressed
forces are suddenly started ; because this problem can often be attacked
in a comparatively simple manner, requiring only investigation of the
appropriate functions to suit the surfaces of curl of the impressed
forces. The remarks in this paragraph are not limited to homogeneity
and isotropy.
s Primitive Solutions for Plane Waves.
9. If we take z normal to the plane of the waves, we may suppose
that both E and H have x and y components. This is, however, a
wholly unnecessary mathematical complication, and it is sufficient to
suppose that E is everywhere parallel to the -axis, and H to the y-axis.
The specification of an initial state is therefore E Q , jET , the tensors
of E and H, given as functions of z; and the circuital equations (1), (2)
become
-dH/dz = (7rk + cp)E, -dE/dz=(47rg + w)H. (15)
Now the operator q 2 in (5) becomes
474 ELECTRICAL PAPERS.
where by V we may now understand d/dz simply. Therefore, by (7),
the solutions of (15) are
When the initial states are such as ae te , or acosbz, the realization is
immediate, requiring only a special meaning to be given to q in (17).
But with more useful functions-, as ae~ 62 ' 2 , etc., etc., there is much work
to be performed in effecting the differentiations, whilst the method
fails altogether if the initial distribution is discontinuous.
But we may notice usefully that when E and H Q are constants the
solutions are
E = c-WE Q , tf=-WJEr oi ................. (18)
which are quite independent of one another. Further, since disturb-
ances travel at speed v, (IS) represents the solutions in any region in
which E Q and H are constant, from i = up to the later time when a
disturbance arrives from the nearest plane at which E or H Q varies.
Fourier-Integrals.
10. Now transform (17) to Fourier-integrals. We have Fourier's
theorem,
f(a)cosm(z-a)dmda, ............. (19)
and therefore <{>(V 2 )f(z) = - [ f f(a)( - m 2 ) cos m(z - a) dm da ; (20)
^J o J -*
applying which to (17) we obtain
E = I dm da\ E Q cos m(z - a)( cosh - -sinhW
"" JoJ- L 2
= I dm da\ H cos m(z - ft)^cosh + -sinh \g
(21)
in which, by (16), f = )
where
"2T" 1 " 2.4.6.8' 8 2 4 2. 4.6 ^2.4.6.8.10
^6= --T* + T /i = 00 + ^1* / 2 = #i + ^2> et c-; ......... (31)
and the properties of the/'s corresponding to (28), (29) are
* + ( ^fr+,+ ...-" when r-0,
= when r is even, except ;
and
= 1 . 3. 5 . . . (r -
\ /
when r is odd, with the + sign for r=l, 5, 9, ..., and the - sign for
the rest. The first case in (32), of r = 0, is very important. But in
case r= 1, the coefficient in (33) is + 1 ; thus,
Special Initial States.
1 2. Now let there be an initial distribution of H only, so that, by (17),
(34)
by (17). Let H Q be zero on the right side and constant on the left
side of the origin, and let us find H and E at a point on the right side.
The operator e vfv is inoperative, so that, by (30),
- S f 1+ s 2 / 2 -s 3 / 3 + ...)#< 1 / 35)
'
* These fs are the same as in my paper "On Electromagnetic Waves," 8
[vol. ii., p. 384] ; but s there is a here.
GENERAL SOLUTION OF ELECTROMAGNETIC EQUATIONS. 477
the immediate integration of which gives
K-/i( 1 ~ -J * f
...(36)
To obtain the E due to E constant from z = - co to 0, use the first
of (36) ; change H to E, H to E^ and change the sign of o-, not
forgetting it in the f's. To obtain the corresponding H due to E , use
the second of (36) ; change E to H, H Q to E , and /* to c. So
where the accent means that the sign of a- is changed in the f's.
From these, without going any further, we can obtain a general idea
of the growth of the waves to the right and left of the origin, because
the series are suitable for small values of vt. But, reserving a descrip-
tion till later, notice that E in (36) and H in (37) must be true on
both sides of the origin ; on expanding them in powers of z we con-
sequently find that the coefficients of the odd powers of z vanish, by
the first of (28), and what is left may be seen to be the expansion of
.................. (38)
the complete solution for E due to H . Similarly,
.................. (39)
is the complete solution for H due to E . In both cases the initial
distribution was on the left side of the origin; but, if its sign be
reversed, it may be put on the right side, without altering these
solutions.
Similarly, by expanding the first of (36) and first of (37) in powers
of z we get rid of the even powers of z, and produce the solutions
given by me in a previous paper,* which, however, it is needless to
write out here, owing to the complexity.
Arbitrary Initial States.
13. Knowing the solutions due to the above distributions, we find
those due to initial E da at the origin, or H Q da, by differentiation to z:
* " Electromagnetic Waves," 8 [vol. n., p. 383],
478 ELECTRICAL PAPERS.
and for this we do not need the firsts of (36) and (37), but only the
seconds. The results bring the Fourier-integrals (21) to
E . f-
H=
where p = d/dt, V = d/dz,
z+vt
(40)
Another interesting form is got by the changes of variables
These lead to
T P* ( TT ^ " W \ T JY V \*\t? \
w >0 ~ I I ^fJin tytl ^/^l')) I 'I
...(42)
The connexions and partial characteristic ofUorW are
dW a- dU a- r
r
(43)
and this characteristic has a solution
(44)
where m is any + integer, and in which the sign of the exponent may
be reversed. We have utilized the case m = only.
Evaluation of "Fourier-Integrals.
14. The effectuation of the integration (direct) of the original Fourier-
integrals will be found to ultimately depend upon
q v
provided vt > z, where, as before,
(f- = o- 2 - m 2 v 2 .
By equating coefficients of powers of z 2 in (45) we get
(45)
2 fsinh gt^fa _ ^1 .3.5.(2r - 1) J>^ .............. (46)
except with r = ; then =v~ l J Q (a-fi).
GENERAL SOLUTION OF ELECTROMAGNETIC EQUATIONS. 479
To prove (45), expand the ^-function in powers of o- 2 . Thus,
symbolically written,
sin mvi (47)
q \ mv
the operand being in the brackets, and p~ l meaning integration from
to t with respect to t. Thus, in full,
"sin mvt er 2 (\ sinm^
2j,'-
mv
~. ...(48)
2 4 o
Now the value of the first term on the right is
v~ l , or 0, when z is <, or >vt.
Thus, in (48), if z>vt, since the first term vanishes, so do all the rest,
because their values are deduced from that of the first by integrations
to t, which during the integrations is always vt. In another form,
disturbances cannot travel faster than at speed v.
But when z < vt in (48), it is clear that whilst if goes from to t or
from to z/v, and then from z/v to t, the first integral is zero from to
z/v t so that the part z/v to t only counts. Therefore the second term is
ma mv
The third is, similarly,
and so on, in a uniform manner, thus proving that the successive terms
of (48) are the successive terms of the expansion of (45) (right member)
in powers of o- 2 ; and therefore proving (45).
The following formulse occur when the front of the wave is in
question, where caution is needed in evaluations :
, A *\
(49)
m
sinh a-t _ 2 rsinmvt sinh qt^
o- 7rJ m q
Interpretation of Results.
15. Having now given a condensation of the mathematical work, we
may consider, in conclusion, the meaning and application of the formulae.
480 ELECTRICAL PAPERS.
In doing so, we shall be greatly assisted by the elementary theory of a
telegraph circuit. It is not merely a mathematically analogous theory,
but is, in all respects save one, essentially the same theory, physically,
and the one exception is of a remarkable character. Let the circuit
consist of a pair of equal parallel wires, or of a wire with a coaxial tube
for the return, and let the medium between the wires be slightly
conducting. Then, if the wires had no resistance, the problem of the
transmission of waves would be the above problem of plane waves in a
real dielectric, that is, with constants //., c, and k, but without the
magnetic conductivity; i.e. g = Q in the above.
The fact that the lines of magnetic and electric force are no longer
straight is an unessential point. But it is, for convenience, best to take
as variables, not the forces, but their line-integrals. Thus, if V be the
line-integral of E across the dielectric between the wires, V takes the
place of E. Then JcE, the density of the conduction-current, is replaced
by KVj where K is the conductance of the dielectric per unit length of
circuit ; and cE/4:7r, the displacement, becomes SV, where S is the per-
mittance per unit length of circuit. The density of electric current
cpE/^Tr is then replaced by SpV. Also SV\& the charge per unit length
of circuit.
Next, take the line-integral of H/4ir round either conductor for
magnetic variable. It is (7, usually called the current in the wires.
Then /*//", the induction, becomes LC ; where LC is the momentum
er unit length of circuit, L being the inductance, such that
A more convenient transformation (to minimize the trouble with
47r's) is
E to V, E to C, p to L, c to S, hrk to K.
Now, lastly, the wires have resistance, and this is without any repre-
sentation whatever in a real dielectric. But, as I have before shown,
the effect of the resistance of the wires in attenuating and distorting
waves is, to a first approximation (ignoring the effects of imperfect
penetration of the magnetic field into the wires), representable in the
same manner exactly as the corresponding effects due to #, the
hypothetical magnetic conductivity of a dielectric.* Thus, in addition
to the above,
becomes R,
R being the resistance of the circuit per unit length.
16. In the circuit, if infinitely long and perfectly insulated, the total
charge is constant. This property is independent of the resistance of
the wires. If there be leakage, the charge Q at time t is expressed in
terms of the initial charge Q b} 7
independently of the way the charge redistributes itself.
In the general medium, the corresponding property is persistence of
* " Electromagnetic Waves," 6 [p. 379, vol. n.].
GENERAL SOLUTION OF ELECTROMAGNETIC EQUATIONS. 481
displacement, no matter how it redistributes itself, provided k be zero,
whatever g may be. And, if there be electric conductivity,
E/f
D~.
rVo . --v.
i
2
2
^^^_l
3
-* ,3 _^
3 1_^^|
Now introduce leakage to make RjL = K/S. Then 2 2 2 2 1 Z shows
the curve of F, provided e~ /s = J. We have F=iF on the left, and
P^JFlintherest.
Thirdly, let the leakage be in excess. Then, when F has fallen, by
leakage only, to JF on the left, the curve 3 3 3 3 1 Z shows F; it is
^Fo a t the origin, - ^F at the back, and ^F at the front.
[The third case is numerically wrong. Thus, at the front we have
j7/F = j- (/5l+ ^, at origin Je~ 2/)1 ', and behind e- 2 "'. Now take p 2 = 0.
Then, when e~ flit = J, we have F/F = J at front, \ at origin, at back,
and j behind. It is later on that V becomes negative at the back.
Thus, when c~ pit = J, we have F/F = -| at front, ^V at origin, -^ at
back, and y 1 ^ behind. And when e~ Pit = ^, we have F/F = T 1 g - at front,
T |- at origin, - / T at back, and -^ behind.]
Of course there has to be an adjustment of constants to make
-(Riu+s/is)t j^ t h e game | j n a ]| casegj v j z ? the attenuation at the front.
18. Precisely the same applies when it is C that is initially given
instead of FQ, provided we change the sign of a-. That is, we have the
curve 1 when the leakage is in excess, and the curve 3 when the leakage
is smaller than that required to produce distortionless transmission.
19. Now transferring attention to the general medium, if we make
the substitution of magnetic conductivity for the resistance of the wires,
the curve 1 would apply when it is E that is the initial state and g in
excess, and 3 when it is deficient ; whilst if H is the initial state, 1
GENERAL SOLUTION OF ELECTROMAGNETIC EQUATIONS. 483
applies when g is deficient, and 3 when in excess. But g is really zero,
so we have the curve 1 for that of H and 3 for that of E.
This forcibly illustrates the fact that the diffusion of charge in a sub-
marine cable and the diffusion of magnetic disturbances in a good con-
ductor, though mathematically analogous, are physically quite different.
They are both extreme cases of the same theory ; but they arise by
going to opposite extremities; with the peculiar result that, whereas
the time-constant of retardation in a submarine cable is proportional to
the resistance of the wire, that in the wire itself is proportional to its
conductivity.
20. Going back to the diagram, if we shift the curves bodily through
unit distance to the left, and then take the difference between the new
and the old curves, we shall obtain the curves showing how an initial
distribution of V or C through unit-distance at the origin divides and
spreads. In the case of curve 2, we have clean splitting without a
trace of diffusion. In the other cases there is a diffused disturbance
left behind between the terminal waves, positive in case 1, negative in
case 3. But I have sufficiently described this matter in a former
paper. *
October 18, 1888.
POSTSCRIPT.
On the Metaphysical Nature of the Propagation of the Potentials.
At the recent Bath Meeting of the British Association there was con-
siderable discussionf in Section A on the question of the propagation of
electric potential. I venture therefore to think that the following
remarks upon this subject may be of interest.
According to the way of regarding the electromagnetic quantities I
have consistently carried out since January 1885, the question of the
propagation of, not merely the electric potential ^, but the vector
potential A, does not present itself as one for discussion ; and, when
brought forward, proves to be one of a metaphysical nature.
We make acquaintance, experimentally, not with potentials, but with
forces, and we formulate observed facts with the least amount of
hypothesis, in terms of the electric force E and magnetic force H. In
Maxwell's development of Faraday's views, E and H actually represent
the state of the medium anywhere. (It comes to the same thing if we
consider the fluxes, but less conveniently in general.) Granting this,
it is perfectly obvious that in any case of propagation, since it is a
physical state that is propagated, it is E and H that are propagated.
Now, in a limited class of cases, E is expressible as -V*". Con
siderations of mathematical simplicity alone then direct the mathe-
matician's attention to " and its investigation, rather than to that of
E directly. But when this is possible the field is steady, and no
question of propagation presents itself (except in the very artificial form
* "Electromagnetic Waves," 7 [vol. n., p. 382].
tSee Prof. Lodge's "Sketch of the Electrical Papers read in Section A," The
Electrician, September 21 and 28, 1888.
484 ELECTRICAL PAPERS.
of balanced exchanges). When there is propagation, and H is involved,
we have
Now this is not an electromagnetic law specially, but strictly a truism,
or mathematical identity. It becomes electromagnetic by the definition
ofA '
leaving A indeterminate as regards a diverging part, which, however,
we may merge in - V*". Supposing, then, A and to become fixed in
this or some other way, the next question in connection with propaga-
tion is, Can we, instead of the propagation of E and H, substitute that
of >F and A, and obtain the same knowledge, irrespective of the
artificiality of W and A ? The answer is perfectly plain we cannot do
so. We could only do it if ^ A, given everywhere, found E and H.
But they cannot. A finds H, irrespective of *P, but both together will
not find E. We require to know a third vector, A. Thus we have M*,
A, and A, required, involving seven scalar specifications to find the six
in E and H. Of these three quantities, the utility of A is simply to
find H, so that we are brought to a highly complex way of representing
the propagation of E in terms of and A, giving no information about
H, which is, it seems to me, as complex and artificial as it is useless and
indefinite.
Again, merely to emphasize the preceding, the variables chosen should
be capable of representing the energy stored. Now the magnetic
energy may be expressed in terms of A, though with entirely erroneous
localization ; but the electric energy cannot be expressed in terms of "SK
Maxwell (chap. XI. vol. II.) did it, but the application is strictly limited
to electrostatics ; in fact, Maxwell did not consider electric energy
comprehensively. The full representation in terms of potentials
requires M* and Z, the vector-potential of the magnetic current. (This
is developed in my work " On Electromagnetic Induction and its
Propagation " [vol. L, p. 507].) This inadequacy alone is sufficient to
murder ^ and A, considered as subjects of propagation.
Now take a concrete example, leaving the abstract mathematical
reasoning. Let there be first no E or H anywhere. To produce any,
impressed force is absolutely needed. Let it be impressed e, and of the
simplest type, viz , an infinitely extended plane sheet of e of uniform
intensity, acting normally to the plane. What happens ? Nothing at
all. Yet the potential on one side of the plane is made greater by the
amount e (tensor of e) than on the other side. Say ^f = \e and - \e.
Thus we have instantaneous propagation of ^ to infinity. I prefer,
however, to say that this is only a mathematical fiction, that nothing is
propagated at all, that the electromagnetic mechanism is of such a
nature that the applied forces are balanced on the spot, that is, in the
sheet, by the reactions.
To emphasize this again, let the sheet be not infinite, but have a
circular boundary. Let the medium be of uniform inductivity p, and
permittivity c. Then, irrespective of its conductivity, disturbances are
GENERAL SOLUTION OF ELECTROMAGNETIC EQUATIONS. 485
propagated at speed fl = (/xc)"i, and their source is the vortex-line of e,
on the edge of the disk. At any time t less than a/v, where a is the
radius of the disk, the disturbance is confined within a ring whose axis
is the vortex-line. Everywhere else, E = and H = 0. On the surface
of the ring, E = pvH, and E and H are perpendicular ; there can be no
normal component of either.
Now, we can naturally explain the absence of any flux in the central
portion of the disk, by the applied forces being balanced by the
reactions on the spot, until the wave arrives from the vortex-line.
But how can we explain it in terms of *P, seeing that *P has now to
change by the amount e at the disk, and yet be continuous everywhere
else outside the ring ? We cannot do it, so the propagation of fails
altogether. Yet the actions involved must be the same whether the
disk be small or infinitely great. We must therefore give up the idea
altogether of the propagation of a *F to balance impressed force. In
the ring itself, however, we may regard the propagation of "*" (a different
one), A, and A ; or, more simply, of E and H.
If there be no conductivity, the steady electric field is assumed any-
where the moment the two waves from opposite ends of a diameter of
the disk coexist ; that is, as soon as the wave arrives from the more
distant end.* But this simplicity is quite exceptional, and seems to be
confined to plane and spherical waves. In general there is a subsidence
to the steady state after the initial phenomena.
If it be remarked that incompressibility (or something equivalent or
resembling it) is needed in order that the medium may behave as described
(i.e., no flux except at the vortex-line initially), and that if the medium
be compressible we shall have other results (a pressural wave, for
example, from the disk generally), the answer is that this is a wholly
independent matter, not involved in Maxwell's dielectric theory, though
perhaps needing consideration in some other theory. But the moment
we let the electric current have divergence (the absence of which makes
the vortex-lines of e to be the sources of disturbances), we at once (in
my experience) get lost in an almost impenetrable fog of potentials.
Maxwell's theory unamended, on the other hand, works perfectly and
without a trace of indefiniteness, provided we regard E and H as the
variables, and discard his " equations of propagation " containing the
two potentials.!
October 22, 1888.
* "Electromagnetic Waves," 25 [p. 415, vol. n.].
t [March 20, 1889. Referring to the example given above of a circular disk, I
strangely overlooked the fact that the absence of flux initially can be expressed by
infinitely rapid propagation of both a ^ and an A. In the disk itself we must have
- V^ - A= - impressed force, so that there is no flux there, and outside we must
have - V* - A = 0. This makes it go. But as regards propagation, it only makes
matters worse. It is a reductio ad absurdum to have an electrostatic field pro-
pagated infinitely rapidly, and, simultaneously, the electric force of induction,
its exact negative, merely to cancel the former, itself quite hypothetical.
In my paper "On the Electromagnetic Effects due to Moving Electrification,"
Phil. May., April, 1889 (vol. IL, Art. L.), is an explicit example showing the
absurdity of the thing.]
486 ELECTRICAL PAPERS.
XLV. LIGHTNING DISCHARGES, ETC.
[The Electrician, Aug. 17, 1888, p. 479.]
THE gap between the electrical phenomena of common practice and
those concerned in the transmission of light and heat, a gap that it
once seemed almost impossible to bridge, is being gradually filled up,
both from the theoretical and the experimental side ; both from above,
by the observation of dark heat and in other ways ; and from below,
by electrical means, as condenser-discharges, vacuum-tube experiments,
etc. Dr. Lodge's recent work on lightning discharges, especially the
experiments described in his second lecture, deserves the most careful
attention, as a substantial addition to our knowledge of the subject, and
also because it is, so far as I know, the first serious attempt to treat the
subject electromagnetically.
The fluids are played out ; they are fast evaporating into nothingness.
The whole field of electrostatics must be studied from the electro-
magnetic point of view to obtain an adequately comprehensive notion
of the facts of the case ; and it is here that Dr. Lodge's experiments are
also useful.
Independently of this, I should not be surprised to find that a new
fact is contained in some of the experiments. Now a new fact is a
serious matter, and its existence can only be granted upon the most
conclusive evidence, of varied nature. There is already some inde-
pendent evidence, viz., in Kundt's. recent paper on the speed of light in
metals. But it is scarcely sufficient.
There is the plainest possible evidence that with waves of telephonic
frequency the magnetic force and the flux induction are proportionate,
and that their ratio is a large number in iron. I have observed, and I
read that Ayrton and Perry have also observed, decrease of the
inductivity with increased wave-frequency. But, at least with me, it
went only a little way, and I had not the opportunity to extend the
experiments.
Now a conducting wire at the first moment of receiving a wave (in
the dielectric, of course) performs the important function of guiding it
and preventing its dissipation in space ; and besides that, the nature of
the conductor partly determines what impedance the wave suffers,
causing a reflection back, with heaping up behind, so to speak, of the
electric disturbance. But at first the conduction-current is purely super-
ficial. It is clear then that at the very front of a wave, where con-
duction is just commencing on the surface, the conductor cannot be
treated as if it had the same properties (conductivity, inductivity,
permittivity) as if it were material in bulk, for only a thin layer of
molecules is concerned. We therefore do not know what the true
boundary condition is when pushed to the extreme. And yet it may
be that this unknown condition may sometimes serve to determine a
choice of paths.
Thus, iron may behave, superficially, as if it were non-magnetic.
(This does not mean that the inductivity of an iron wire is unity.) In
LIGHTNING DISCHARGES, ETC. 487
Kundt's experiments, electromagnetically interpreted, the inductivity
of iron is nowhere ; the conductivity, too, must, in other cases as well,
be less than the steady value. This corroborates Maxwell's remarks
concerning gold-leaf. Of course the application of electromagnetic
principles to the passage of light through material substances is at
present in a very tentative state ; so that too much importance should
not be attached to the speculations one may be led to make in these
matters.
(If a conductor could be treated as homogeneous right up to its
surface, the initial resistance of unit of surface I calculate to be kirpv,
where //, is the inductivity and v the speed of transmission in the con-
ductor. But neither p nor r can be considered to be known in the case
of iron.) [See p. 437, vol. IL]
Another matter I wish to direct attention to is this. Dr. Lodge has
described some experiments relating to the reflection of waves sent
along a circuit. It will also be in the knowledge of some readers that
Sections XL. to XLVI. of my " Electromagnetic Induction and its
Propagation," Electrician, June to September, 1887 (and a straggler,
XLVII., December 31, 1887), deal with the subject of the transmission
of waves along wires, their reflection, absorption, etc., by a new method.
Now I find that there is an idea prevalent that it is only possible for
very advanced mathematicians to understand this subject. It is true
that when it is comprehensively considered it is by no means easy.
But I desire to call attention to the fact (as I did in one or more of the
articles referred to) that all the main features of the transmission,
reflection, absorption, etc., of waves can be worked out (as done there
by me) by elementary algebra.
I was informed (substantially) that no one read my articles. Possibly
some few may do so now, with Dr. Lodge's experiments in practical
illustration of some of the matters considered.
My next communication, I may add (written in September, 1887), is
on the important subject of the measure of the inductance of circuits,
and its true effects, in amplification of preceding matter. It has also
special reference to some experimental observations. It has also some
valuable annotations by an eminent authority. [Art. xxxviu., vol. II.,
p. 160.]
P.S. In connection with lightning discharges, I may remark that it
is usual, and seems very natural, to assume that the discharge is
initiated' at the place of the visible spark the crack, so to speak. But
my recent investigations lead me to conclude that this is by no means
necessary, and that the strongest dielectric can be disrupted by a suit-
able convergence of a wave to a centre or an axis, starting with any
steady field.
For instance, if in a cylindrical portion of a dielectric the displacement
be uniform, and parallel to the axis, and it be allowed to discharge, the
convergence of the resulting wave to the axis causes the electric force
to mount up infinitely there, momentarily ; hence disruption.
But I do not pretend to give a complete theory of the thundercloud.
It is only a detail.
488 ELECTRICAL PAPERS.
P.P.S. In Dr. Fleming's recent articles on the theory of alter-
nating currents, I observe that he calls the component Ln of the imped-
ance (E 2 + L 2 n' 2 )* the "inductive resistance."
I should myself have scarcely thought that it deserved a name, for of
course we must draw the line somewhere. But the fact that Dr.
Fleming has given it a name is evidence that he found it convenient to
do so. Taking it, then, for granted that it should have a special name,
I can only object to the one chosen that it creates two kinds of resist-
ance. I desire to recognise but one the resistance. I might, for
instance, call Ln the hindrance. Thus, in the case of a coil, R is the
electric resistance, Ln the magnetic hindrance, and their resultant the
impedance. But in any case it would not be a term for popular use,
August 13, 1888.
XL VI. PRACTICE VERSUS THEORY. ELECTROMAGNETIC
WAVES.
[The Electrician, Oct. 19, 1888, p. 772.]
THE remarkable leader in The Electrician for Oct. 12, 1888, states very
lucidly some of the ways in which theory and practice seem to become
antagonistic. There is, however, one point which does not, I think,
receive the attention it deserves, which is, that it is the duty of the
theorist to try to keep the engineer who has to make the practical
applications straight, if the engineer should plainly show that he is
behind the age, and has got shunted on to a siding. The engineer
should be amenable to criticism.
Another point is this. It might appear from the concluding para-
graph of the article to which I have referred that the points at issue
between Mr. Preece's views and my own were mere matters of com-
plicated corrections, not affecting the main argument much. But the
case is far different. A complete change of type is involved.
Now, I shall have great pleasure, when opportunity offers, in en-
deavouring to demonstrate that such is the case, and that the despised
self-induction is the great moving agent ; that although Mr. Preece, in
the presence of some distinguished mathematicians, recently boasted *
that he made mathematics his slave, yet it is not wholly improbable
that he is a very striking and remarkable example of the opposite pro-
cedure ; that although Mr. Preece, who, as a practical engineer, knows
all about electromagnetic inertia and throttling, does not see the
use of inductance, impedance, and all that sort of thing, yet there is
not wanting evidence to make it not wholly unbelievable that Mr.
Preece is not quite fully acquainted with the subject as generally
* [The Discussion on Lightning Conductors at the Bath meeting of the B.A.,
reported at length in The Electrician, Sept. 21 and 28, 1888, is interesting reading,
and is made quite amusing by Mr. Preece's attack upon mathematicians to his own
exaltation, and the rejoinders thereto.]
PRACTICE VERSUS THEORY. ELECTROMAGNETIC WAVES. 489
understood; that, for example, his coefficient of self-induction is of
very different size, and has very different properties, from the theo-
retical one ; and that Mr. Preece's knowledge of the manner of trans-
mission of signals, though it may not be "extensive," is certainly
" peculiar."
I may take the opportunity of adding that on account of a certain
peculiar concurrence and concatenation of circumstances last year
rendering it impossible for me to communicate the practical applications
of my theory (based upon Maxwell's views, so far as the higher de-
velopments are concerned), either vid the S. T.-E. and E. or four other
channels, the resultant effect of which was to screen Mr. Preece from
criticism, combined with the fact that Mr. Preece, in his papers to the
Royal Society, British Association, and S. T.-E. and E. has taken his
stand upon Sir W. Thomson's celebrated theory of the submarine cable,
I have been forced, with great reluctance, to assume what may have
appeared to be, superficially, an apparently unnecessarily aggressive
attitude towards the said theory. But those who are acquainted with
the subject will know that there is no antagonism whatever between
the electrostatic theory and the wider theory ; and those, further, who
may be acquainted with the peculiar concurrence I have mentioned
will understand the meaning of the apparent aggressiveness.
In addition, it seems to me to be almost mathematically certain that
Sir W. Thomson would emphatically repudiate the very notion of apply-
ing his theory of the diffusion of potential to cases to which it does not
apply, and to which it was never meant to apply ; and I cannot find
any evidence in his writings that he ever would have made such a
misapplication.
p.S. Is self-induction played out? I think not. What is played
out is what we may call (uniting the expressions of Ayrton, Preece,
Thomson, and Lodge) the British engineer's self-induction, which stands
still, and won't go. But the other self-induction, in spite of strenuous
efforts to stop it, goes on moving; nay, more, it is accumulating
momentum rapidly, and will, I imagine, never be stopped again. It is,
as Sir W. Thomson is reported to have remarked, with a happy union
of epigrammatic force and scientific precision, " in the air." Then
there are the electromagnetic waves. Not so long ago they were
nowhere ; now they are everywhere, even in the Post Office. Mr.
Preece has been advising Prof. Lodge to read Prof. Poynting's paper
on the transfer of energy. This is progress, indeed ! Now these waves
are also in the air, and it is the " great bug " self-induction that keeps
them going.
On this question of waves I take the opportunity of referring to a
point mentioned at the Bath meeting by Prof. Fitzgerald. That phy-
sicist, in directing attention to Hertz's recent experiments, considered
that they demonstrated the truth of the propagation of waves in time
through the ether ; but that, on the other hand, the waves sent along
a circuit did not do so, because they might be explained by action at a
distance.
It seems to me, however, that the more closely we look at the matter
490 ELECTRICAL PAPERS.
the less distinction there is between the two cases, and that to an
unbiassed mind the experiments of Prof. Lodge, sending waves of short
length into a miniature telegraph circuit, with consequent "resonance"
effects, are equally conclusive to those of Hertz on the point named ; in
one respect, perhaps, more so, because their theory is simpler, and can
be more closely followed.
But, after all, has it been demonstrated that we cannot explain the
propagation of electromagnetic waves in time by action at a distance,
pure and simple ? I suggest the following as evidence to the contrary.
Take the case of Maxwell's non-conducting dielectric. Let the electric-
current element cause magnetic force at a distance according to Ampere's
law, and let the magnetic current element cause electric force at a
distance according to the same law with sign reversed. Then
curl H = cE, and - curl E = /xH
follow, and propagation of waves in time follows. That is, by instant-
aneous mutual action at a distance between electric-current elements,
and also between magnetic-current elements, we get propagation in
time. Of course the currents may be oppositely moving electric or
magnetic fluids or particles.
Whether there is any flaw here or not, it is scarcely necessary for me
to remark that I do not believe in action at a distance. Not even
gravitational.
XLVII. ELECTROMAGNETIC WAVES, THE PROPAGATION
OF POTENTIAL, AND THE ELECTROMAGNETIC EFFECTS
OF A MOVING CHARGE.
[The Electrician-, Part L, Nov. 9, 1888, p. 23 ; Part II., Nov. 23, 1888, p. 83 ;
Part. III., Dec. 7, 1888, p. 147 ; Part IV., Sept. 6, 1889, p. 458.]
PART I.
IN connection with the letters of Profs. Poynting and Lodge in The
Electrician, Nov. 2, 1888, I believe that the following extract from a
letter from Sir William Thomson (which I have permission to publish)
will be of interest [see Postscript, p. 483, vol. IL, to elucidate] :
" I don't agree that velocity of propagation of electric potential is a
merely metaphysical question. Consider an electrified globe, A, moved
to and fro, with simple harmonic motion, if you please, to fix the ideas.
Consider very quickly-acting electroscopes B, B', at different distances
from A. If the indications of B, B' were exactly in the same phase,
however their places are changed, the velocity of propagation of electric
potential would be infinite ; but if they showed differences of phase,
they would demonstrate a velocity of propagation of electric potential.
" Neither is velocity of propagation of ' vector-potential ' meta-
physical. It is simply the velocity of propagation of electromagnetic
force the velocity of * electromagnetic waves/ in fact."
ELECTROMAGNETIC WAVES, ETC. 491
Taking the second point first, it is, I think, clear that if by the pro-
pagation of vector-potential is to be understood that of electric and
magnetic disturbances, it is merely the mode of expression that is in
question. I am myself accustomed to mentally picture the electric and
magnetic forces or fluxes, arid their propagation, which takes place at
the speed of light or thereabouts, because they give the most direct
representation of the state of the medium, which, I think, must be
agreed is the real physical subject of propagation. But if we regard
the vector-potential directly, then we can only get at the state of the
medium by complex operations, and we really require to know the
vector-potential both as a function of position and of time, for its space-
variation has to furnish the magnetic force, and its time-variation the
electric force ; besides which, there is sometimes the space-variation of a
scalar potential in addition to be regarded, before we can tell what the
electric force is. Besides this roundaboutness, it implies a knowledge
of the full solution, and if we do not possess it, it is much simpler to
think of the propagation of the electric and magnetic disturbances, and
I find that this method works out much more easily in the solution of
problems.
The other question will, I believe, be found to be ultimately of pre-
cisely the same nature. Start with the sphere A at rest, and the field
steady, and consider two external points, P and P', at different distances.
The electric force at them has different values, and the whole field has
a potential. But now give the sphere a displacement, and bring it to
rest again in a new position. Is the readjustment of potential instan-
taneous 1 I should say, Certainly not, and describe what happens thus.
When the sphere is moved, magnetic force is generated at its boundary
(lines circles of latitude, if the axis be the line of motion), and with it
there is necessarily disturbance of electric force. The two together
make an electromagnetic wave, which goes out from the sphere at the
speed of light, and at the front of the wave we have E = f^vH, where E
is the electric and H the magnetic force intensity. Before the front
reaches P or P' we have the electric field represented by the potential
function, but after that it cannot be so represented until the magnetic
force has wholly disappeared, when again we have a steady field repre-
sentable by a potential function. It is difficult to see how to plainly
differentiate any propagation of potential per se.
If the motion is simple-harmonic, there is a train of outward waves
and no potential. I imagine that an electroscope, if infinitely sensitive
and without reactions, would register the actual state of the electric
field, irrespective of its steadiness. By an electroscope, as this is a
purely theoretical question, I understand the very simplest one, a very
small charge at a point ; or, say, the unit charge, the force on which is
the electric force of the field.
When these things are closely examined into, if the facts as regards
the propagation of disturbances (electric and magnetic) are agreed on,
the only subject of question is the best mode of expressing them, which
I believe to be in terms of the forces, not potentials.
But there really is infinite speed of propagation of potential sometimes ;
492 ELECTRICAL PAPERS.
on examination, however, it is found to be nothing more than a mathe-
matical fiction, nothing else being propagated at the infinite speed.
It will be understood that I preach the gospel according to my inter-
pretation of Maxwell, and that any modification his theory of the
dielectric may receive may involve a fresh kind of propagation at pre-
sent not in question.
Nov. 5, 1888.
PART II.
The question raised by Prof. S. P. Thompson (in The Electrician,
Nov. 16, 1888, p. 54) as to whether the motion of an uncharged
dielectric through a field of electric force produces magnetic effects
must, I think, be undoubtedly answered in the affirmative. As the
distribution of displacement varies, its time-variation is the electric
current, with determinable magnetic force to match. When the speed
of motion is a small fraction of that of light, we may regard the
displacement as having at every moment its proper steady distribution,
so that there is no difficulty in estimating the magnetic effects, except,
it may be, of a merely mathematical character. For instance, the case
of a sphere moving in a field which would be uniform were the sphere
absent, may be readily attacked, and does perfectly well to illustrate
the general nature of the action.
But if the moved dielectric have the same electric permittivity as
the surrounding medium, so that there is no difference made in the
steady distribution, the question which may be now raised as to the
possible production of transient disturbances is one to which the above
theory does not present any immediate answer. I believe that the
body will be magnetized transversely to the electric displacement and
the velocity. [The motional magnetic force is referred to.]
Another question, somewhat connected, is contained in Prof. Poynt-
ing's suggestion (in letter to Prof. Lodge, The Electrician, p. 829, vol.
xxi.) that electric displacement may possibly be produced without
magnetic force by the agency of pyroelectricity. But, whatever the
agency, it would, I conceive, be a new fact quite outside Maxwell's
theory legitimately developed. We may have subsidence of electric
displacement without magnetic force; but I cannot see any way to
produce it.
But the main subject of this communication is the electromagnetic
effect of a moving charge. That a moving charge is equivalent to an
electric current-element is undoubted, and to call it a convection-
current. as Prof. S. P. Thompson does, seems reasonable. The true
current has three components, thus,
where H is the magnetic force, C the conduction-current, D the dis-
placement, and p the volume-density of electrification moving with
velocity u. The addition of the term pu is, I presume, the extension
made by Prof. Fitzgerald to which Prof. S. P. Thompson refers. At
any rate, I can at present see no other.
ELECTROMAGNETIC WAVES, ETC. 493
There are several ways of arriving at the conclusion that a moving
charge must be regarded as an electric current; but, when that is
admitted, we are very far from knowing what its magnetic effect is. No
cut-and-dried statement of it can be made, because it varies according to
circumstances. The magnetic field, whatever it be in a given case, is
not that of a current-element (supposing the charge to be at a point),
for that is anti-Maxwellian, but is that of the actual system of electric
current, which is variable.
Thus, in the case of motion at a speed which is a small fraction of
that of light, the magnetic field (as found by Prof. J. J. Thomson) is
the same as that of Ampere's current-element represented by pn ; that
is, a current-element whose direction is that of u and whose moment is
pu, if u is the tensor of u (understanding by "moment," current-density
x volume) ; but the true current to correspond bears the same relation
to the current-element as the induction of an elementary magnet bears
to its magnetic moment. The magnetic energy due to the motion of
a charge q upon a sphere of radius a in a medium of inductivity /*,
at a speed u which is only a very small fraction of that of light, is
expressed by J/^ 2 w' 2 /a. But if the speed be not a small fraction of
that of light, the result is very different. Increasing the speed of
the charge causes not merely greater magnetic force but changes its
distribution altogether, and with it that of the electric field. It is no
use discussing the potential. There is not one. The magnetic field
tends to concentrate itself towards the equatorial plane, or plane
through the charge perpendicular to the line of motion. When the
speed equals that of light itself this process is complete, and the
is simply a plane wave (electromagnetic).
Since a charge at a point gives infinite values,
it is more convenient to distribute it. Let it be,
first, of linear density q along a straight line AB,
moving in its own line at the speed of light. Then
the field is contained between the parallel planes
through A and B perpendicular to AB, and is
completely given by
where E and H are the intensities of the electric
and magnetic forces at distance r from AB. The
lines of E radiate uniformly from AB in all direc-
tions parallel to the planes ; those of H are every-
where perpendicular to those of E, or are circles
centred upon AB. Outside this electromagnetic
wave there is no disturbance. I should remark that the above is a
description of the exact solution. It is, of course, nothing like the
supposed field of a current-element AB.
To still further realize, we may substitute a cylindrical distribution
for the linear, and then, again, terminate the lines of E on another
cylindrical surface between the bounding planes. To find the resulting
distributions of E and H (always perpendicular) may be done by super-
494 ELECTRICAL PAPERS.
imposition of the elementary solutions, or by solving a bidimensional
problem in a well-known manner.
Those who are acquainted with my papers in this journal will
recognise that what we have arrived at is simply the elementary
plane wave travelling along a distortionless circuit. All roads lead
to Rome !
Returning to the case of a charge q at a point moving through a
dielectric, if the speed of motion exceeds that of light, the disturbances
are wholly left behind the charge,
and are confined within a cone,
A v, and here
I have so-far failed to find any solution which will satisfy all the neces-
sary conditions without unreality. The description at the close of
Part II. must therefore be received as a suggestion, at present uncon-
firmed. I hope to consider the matter in a future communication.
P.S. In a recent number Mr. W. P. Granville raised the question of
action through a medium being only action at a short distance instead
of a long one, and asked for instruction. His inquiry has elicited no
response. This is not, however, because there is nothing to be said
about it. The matter did not escape the notice of the " anti-distance-
action sage." My own opinion is that the question involved is, if not
metaphysical, dangerously near to being so ; consequently, whole books
might be devoted to it. At present, however, I think it is more useful
to try to find out what happens, and to construct a medium to make it
happen ; after that, perhaps, the matter referred to may be more
advantageously discussed. The well of truth is bottomless.
PART IV.
In previous communications [above] I have discussed this matter.
Referring to the case of steady rectilinear motion, I gave a description
of the result when the speed of the charge exceeds that of light, obtained
mainly by general reasoning, and stated my inability to find a solution
to represent it. The displacement cannot be outside a certain cone of
semi-vertical angle whose sine equals the ratio v/u of the speed of light
to that of the charge, which is at the apex.
In the Phil. Mag. for July, 1889, Prof. J. J. Thomson has examined
this question. Like myself, he fails to find a solution within the cone ;
but concludes that the displacement is confined to its surface. If so, it
must form, along with the magnetic induction, an electromagnetic wave.
But it may be readily seen that such a wave is impossible, having no
stability.
For as the charge moves from A to B, a given surface-element, C,
would move to D. In doing so its area would vary directly as its
distance from the apex, and the energy in the element would therefore
ELECTROMAGNETIC WAVES, ETC. 497
vary inversely as its distance from the apex, and the forces, electric
and magnetic, would therefore vary inversely as the square root of the
distance from the apex, instead of inversely as the distance, which is
obviously necessary in order that the
displacement may be confined to the
surface. This conflict of conditions
constitutes instability. In the Phil.
Mag. for April, 1889, I suggested
that whilst there must be a solution
of some kind, one representing a
stead)/ state was impossible. This
conclusion is confirmed by the failure of Prof. Thomson's proposed
surface- wave to keep itself going.
Prof. Thomson, who otherwise confirms my results, has also extended
the matter by supposing that the medium itself is set in motion, as well
as the electrification. This is somewhat beyond me. I do not yet
know certainly that the ether can move, or its laws of motion if it can.
Fresnel thought the earth could move through the ether without dis-
turbing it ; Stokes, that it carried the ether along with it, by giving
irrotational motion to it. Perhaps the truth is between the two. Then
there is the possibility of holes in the ether, as suggested by a German
philosopher. When we get into one of these holes, we go out of
existence. It is a splendid idea, but experimental evidence is much
wanting.
But if we consider that the medium supporting the electric and
magnetic fluxes is really set moving when a body moves, and assume a
particular kind of motion, it is certainly an interesting scientific ques-
tion to ask what influence the motion exerts on the electromagnetic
phenomena. I do not, however, think that any new principles are
involved.
The general connections of E and H, referred to fixed space without
conductivity, being
curl(e-E) = /^H, .............................. (1)
curl(H-h)=cpE, .............................. (2)
where p stands for d/dt and e and h are the impressed parts of E and H ;
if there is also motion of electrification, we have to consider it to con-
stitute a convection-current, a part of the true current, and so make (2)
become
........................ (3)
where p is the density of electrification, whose velocity is u. [See Part
II.] It now remains to specify e and h. They are zero when the
medium supporting the fluxes is at rest. But if it moves, and its
velocity is w, there is, first, the electric force due to motion in a
e-fVwH, ................................ (4)
which is well known : and next the magnetic force due to motion in an
electric field, h = oVEw, ............................... (5)
H.E.P. VOL. II. 2 I
498 ELECTRICAL PAPERS.
which is not so well known. (First, I believe, given by me in the third
Section of "Electromagnetic Induction and its Propagation," The
Electrician, January 24, 1885 [vol. I., p. 446] ; again, obtained in a
different way in Section XXIL, January 15, 1886 [vol. I., p. 546]; see
also Phil. Mag., August, 1886 [vol. II., Art. L.], and an example of the
use of (4) and (5) in The Electrician, April 12, 1889, p. 683 [vol. II.,
Art. LI.].)
The mechanical force called by Maxwell the "electromagnetic force"
is VCB, where C is the true current and B the induction. It is the
force on the matter supporting electric current. Let it move. If w is
its velocity, the activity of the force is
wVCB = CVBw= -eC (6)
Similarly, as I obtained in Section xxn. above referred to, there is a
mechanical force (the magneto-electric) on matter supporting magnetic
current G = /xpH/47r, expressed by 4?rVDG, and its activity is
47TWVDG = 47rGVwD = -hG (7)
Of course e and h. are reckoned as impressed forces, which is the reason
of the change of sign. Their activities are eC and hG.
It should be remarked further, that the above expressions for e
and h are not certain. For I have shown that the sources of all
disturbances are the lines of curl of the impressed forces (Phil. Mag.,
Dec., 1887) [vol. n., p. 362], and that the fluxes produced depend
solely upon the curls of e and h, both as regards the steady fluxes
and the variable ones leading to them. We may, therefore, use any
other expressions for e and h which have the same curls as the
above. And, in fact, we see that equations (1) and (2) only contain
their curls.
Equations (1) and (3), with e and h defined by (4) and (5), therefore
enable us to determine the effect of the moving medium. Prof.
Thomson also arrives at (4) and (5), and at the " magneto-electric
force," in his paper to which I have referred, by an entirely different
method. And to show how well things fit together, he concludes, from
the consideration of the moving medium, that a moving electrified
surface is a current-sheet, which is another way of saying that a convec-
tion current is a part of the true current, as expressed in (3). I must,
however, disagree with Prof. Thomson's assumption that the motion
must be irrotational. It would appear, by the above, that this limita-
tion is unnecessary.
As an example, and to introduce a new point, take the case of a charge
q moving at speed u along the axis of z. It will come to the same thing
if we keep the charge at rest, and move the medium the other way.
We then use the equations (1) and (2), and in them use (4) and (5)
with w = - u. Now when the steady state is arrived at, we have p = 0,
so (1) and (2) become
curl(/>iVHu-E) = 0, (8)
curl(H-cVuE) = (9)
ELECTROMAGNETIC WAVES, ETC. 499
In addition, the divergence of D must be q at the origin, and the
divergence of B must be zero. The latter gives, applied to (9),
H = cVuE, (10)
which gives H fully in terms of E. Eliminate H from (8) by means of
(10), and we get
curl(/xcVuVEu-E) = 0, (11)
or curl nC(E- s k > )-En=0, (12)
where E$ is the ^-component of E and k a unit vector along z ; or, inte-
grating, and writing the three components,
dP dP /, u*\dP
where P is a scalar potential. Here is the new point. There is a
potential, of a peculiar kind. The displacement due to the moving
charge is distributed in precisely the same way as if it were at rest in an
eolotropic medium, whose permittivity is c in all directions transverse to
the line of motion, but is smaller, viz., c(l -v?/v 2 ), along that line and
parallel to it. The potential P is given by
(H)
It is a particular case of eolotropy. In general, c lt c 2 , c 3 , the prin-
cipal permittivities, are all unequal. Then, with q at the origin, the
potential is
..... (15)
Observe that although the electric force in the substituted problem
of a charge at rest in an eolotropic medium is the slope of a potential ;
yet it is not so when the medium is isotropic, and moves past the fixed
charge, or vice versa, although the distributions of displacement are the
same.
When u = v, we abolish the permittivity along the 2-axis in the
substituted case, so that the displacement must be wholly transverse.
We then have the plane electromagnetic wave. When u is greater than
v it makes the permittivity negative along z ; this is an impossible
electrical problem, and furnishes another reason for supposing that
there can be no steady state in the corresponding electromagnetic
problem.
It now remains to find what would happen if electrification were con-
veyed through a medium faster than the natural speed of propagation
of disturbances. There is the cone ; but what takes place within it ?
Aug. 25, 1889.
500 ELECTRICAL PAPERS.
XLVIII. THE MUTUAL ACTION OF A PAIR OF RATIONAL
CURRENT-ELEMENTS.
[The Electrician, Dec. 28, 1888, p. 229.]
STRICTLY speaking, there is no such thing, from the Maxwellian point
of view, as mutual action between current elements. Suppose, however,
we have the well-known Amperian field of magnetic force usually
ascribed to a current-element at one place, and a similar one centred at
another place, it is clear that the forces concerned are quite definite,
according to Maxwell's theory. The electric current of such an arrange-
ment is closed. It is related to the nominal current, viz., in the
element, in the same way as the induction of an elementary magnet is
related to its magnetic moment, as regards the space-distribution. We
may term the arrangement a rational current-element. If we take any
number of equal rational current-elements and put them in line, with
opposite poles in contact, only the terminal poles are left free, so that
the current consists of a straight or curved line or tube of current,
joining two points, A and B, with external continuity produced by
means of an equal current diverging
from the positive pole B in all directions
uniformly, and converging to the nega-
tive pole A in a similar manner. Of
course the tubes of current from B join
on to those at A, and are curved ; but
it would only confuse matters to super-
impose the two systems of polar current,
which are much better kept separate.
The rational current-element itself is
to be regarded as an infinitely small
volume with a uniform current distributed in it, and of the com-
plementary currents from and to the poles. The moment is current-
density multiplied by volume, ignoring the complementary currents
altogether for the moment. What the actual current in the element
may be does not matter much. It depends on the shape of the element.
Thus, if spherical, the nominal strength of current, reckoned by its
moment, is half as great again as the real, owing to the back action of
the polar current. We need only consider the moment, which is fully
representative of the external magnetic field, which, it should be
remembered, is that due to the moment, according to Ampere's rule.
To further illustrate, take the case of a charge, q, moving at speed u,
small compared with that of light [p. 495, vol. II.], through a dielectric.
The moment is qu ; the magnetic force is qu/r 2 at distance r in the
equatorial plane, and elsewhere proportional to the cosine of the
latitude. The actual state of things in the element may require very
complex calculations to discover, but is of little importance.
The mutual action of two German or irrational current-elements is
indeterminate, and so we get a large number of so-called theories of
electrodynamics. But the mutual action of a pair of rational current-
THE MUTUAL ACTION OF CURRENT-ELEMENTS. 501
elements is a legitimate subject of inquiry, is determinate, and does not
involve any action at a distance. The quantity from which, by
dynamical methods, we derive the forces (mechanical) on the elements,
is the mutual magnetic energy (leaving out of consideration the electro-
static force, if any), that part of the magnetic energy due to both rational
current-elements. If I have correctly calculated it, the mutual energy
M of elements whose distance apart is r, in the medium of inductivity
/*, is expressed by
where u^ u 2 , u 3 are the components of G v the moment of the first
element, and v v v 2 , v 3 those of the second, C 2 , on the understanding
that the axis of x is the line
joining the elements, whilst
the y and z axes are, as usual,
perpendicular to it and to each
other. In another form,
KT / COS i i cPr
\ r 2 ds l ds^
where e is the inclination of
the elements Cj and C 2 , parallel
to 8 l and S 2 .
If we substitute for r, in the
differential coefficient, an arbitrary function R, we obtain the most
general formula which will lead to Neumann's result for closed circuits.
It is this R that is, by German methods, indeterminate, nationalize
the elements, and we fix it to be r. Clausius took R = 0, I believe. It
does not matter at all, so far as closed circuits are concerned, what
formula we use, provided Neumann's result is complied with ; but it is
interesting to observe that the problem as stated by me has no un-
certainty about it (except any possible working errors) and makes M
definite, whilst it is not a mere mathematical abstraction (i.e., the
problem), but representative of (under certain circumstances) a reality.
It is for these reasons that I mention the matter. For, as a matter of
fact, I believe the whole method is fundamentally wrong, and of little
practical service in the investigation of electromagnetism from the
physical side, i.e., with propagation in time through a medium. What
does it matter about the current-elements ? They are not in it. Still,
such formulas are sometimes of service, as, for instance, in the calcula-
tion of inductances.
It has been stated, on no less authority than that of the great
Maxwell, that Ampere's law of force between a pair of current-elements
is the cardinal formula of electrodynamics. If so, should we not be
always using it ? Do we ever use it ? Did Maxwell, in his treatise ?
Surely there is some mistake. I do not in the least mean to rob
Ampere of the credit of being the father of electrodynamics ; I would
only transfer the nameTof cardinal formula to another due to him,
502 ELECTRICAL PAPERS.
expressing the mechanical force on an element of a conductor support-
ing current in any magnetic field ; the vector product of current and
induction. There is something real about it ; it is not like his force
between a pair of unclosed elements ; it is fundamental ; and, as every-
body knows, it is in continual use, either actually or virtually (through
electromotive force) both by theorists and practicians.
Nov. 25, 1888.
XLIX. THE INDUCTANCE OF UNCLOSED CONDUCTIVE
CIRCUITS.
IN my communication on "The Mutual Action of Rational Current-
Elements" [the last Art. XLVIIL] I described 'the meaning of, and gave
the formula for, the mutual energy M of a pair of rational current-
elements.
Thus, let G^ and C 2 be their moments, r their distance apart, e the
angle between their directions Sj and S 2 , ^ the magnetic inductivity of
the medium (uniform), and M the mutual energy. Then,
(1)
It follows immediately from this that the mutual inductance of any
two linear circuits is
M being now the mutual inductance. If the circuits are closed the
second part contributes nothing, and we have
(3)
the common form of Neumann's equation, with the /x prefixed to adapt
it to Maxwell's theory.
But if the lines are unclosed, then, according to my description of the
nature of a rational current-element, the linear currents become closed
by means of currents uniformly diverging from their positive ends, and
uniformly converging to their negative ends. The second part of (2) is
now finite. Let P t and P 2 be the positive poles, Nj and N 2 the negative
poles of the linear currents, and let the value of the second part of (2)
be M v It is given by
P^VN^), ................. (4)
where P X N 2 means the length of the straight line joining P T to N 2 , and
similarly for the rest. We may, therefore, calculate M by Neumann's
formula, applied to the linear circuits, and then add the correction (4)
to obtain the complete expression.
INDUCTANCE OF UNCLOSED CONDUCTIVE CIRCUITS. 503
A practical application is to the theory of a Hertzian oscillator, at
least of a certain kind. Let a straight wire join two conducting spheres,
or discs, etc. Imagine an impressed force to act in the wire, and to
vary in any not too rapid manner. The current will leak out (or in)
from (or to) the wire as well as the terminal conductors, but if they are
relatively large nearly all the current will go across the air from one
terminal conductor to the other, and we may ignore the wire-leakage.
The permittance S is then that of the dielectric between the two spheres
(say), and is quite definite. Also, if the changes of current are not too
rapid, as mentioned, the current in the air will follow the lines or tubes
of displacement. The inductance L is therefore also quite definite, in
accordance with Maxwellian principles, so that the natural frequency of
oscillation of the condenser-conductor circuit can be calculated with
considerable precision from the dimensions.
If, as an illustrative approximation, we suppose the current to come
from the centre of one sphere and go to that of the other, and then
diverge or converge uniformly, we have to find the inductance L of &
straight wire or tube of length / and radius a, with terminal continua-
tions as before specified. In the Phil. Mag., July, 1888, Prof. Lodge
calculates L without any allowance for the current in the dielectric, viz.,
by Neumann's formula (3). We have therefore only to examine what
the correction (4) amounts to.
In the case of two very close parallel lines, we may put
PjPa^ = ]^, and P^, = P^ = /,
so that the correction is simply - //A. That is, if the dielectric current
is ignored, (3) overestimates M by the amount pi. The same applies
when it is the inductance of a straight tube or solid wire that is in
question. Deduct its length in centimetres from the uncorrected to
obtain the true value, in c.g.s. electromagnetic units, i.e., centimetres.
Prof. Lodge (loc. cit.) also gives the formula which Hertz says Max-
well's theory gives. On making the comparison, I find it is equivalent
to adding, instead of deducting I, from the result of Neumann's formula.
It should be remarked, as an essential condition of the validity of the
process described above, when practically applied, that the changes of
current must not be too rapid. When the changes are slow the im-
mense speed of propagation of disturbances through the air causes the
electric displacement at any moment in the neighbourhood of the
vibrator to be very nearly that which would obtain according to electro-
static principles, and the current to follow the tubes of displacement.
But go to the other extreme, and imagine the changes to be so rapid
that waves, whose length is a fractional part of the length of the
vibrator, are produced. It is then clear that the theory would not
apply at all, either as regards the inductance or the permittance. Now
Hertz, in that series of brilliant experiments which have gone far
towards practically establishing the truth of Maxwell's inimitable
theory of the ether considered as a dielectric, sometimes employs waves
which are not very much longer than the vibrator itself. Only close to
the vibrator, therefore, do we have the electrostatic field (approximately)
504 ELECTRICAL PAPERS.
predominant, and we may expect a sensible error in applying the electro-
static theory. It is, however, quite easy in fact, easier to use longer
waves. But in any case, the exact calculation of the permittance and
inductance of a vibrator involves a good deal of mathematics to find
relatively small corrections.
July 21, 1889.
L. ON THE ELECTROMAGNETIC EFFECTS DUE TO THE
MOTION OF ELECTRIFICATION THROUGH A DIELECTRIC.
[Phil. Mag., April, 1889, p. 324.]
Theory of the Slow Motion of a Charge.
1. THE following paper consists of, First, a short discussion of the
theory of the slow motion of an electric charge through a dielectric,
having for object the possible correction of previously published results.
Secondly, a discussion of the theory of the electromagnetic effects due
to motion of a charge at any speed, with the development of the com-
plete solution in finite form when the motion is steady and rectilinear.
Thirdly, a few simple illustrations of the last when the charge is
distributed.
Given a steady electric field in a dielectric, due to electrification. It
is sufficient to consider a charge q at a point, as we may readily extend
results later. If this charge be shifted from one position to another,
the displacement varies. In accordance, therefore, with Maxwell's
inimitable theory of a dielectric, there is electric current produced. Its
time-integral, which is the total change in the displacement, admits of
no question ; but it is by no means an elementary matter to settle its
rate of change in general, or the electric current. But should the speed
of the moving charge be only a very small fraction of that of the pro-
pagation of disturbances, or that of light, it is clear that the accommo-
dation of the displacement to the new positions which are assumed by
the charge during its motion is practically instantaneous in its neighbour-
hood, so that we may imagine the charge to carry about its stationary
field of force rigidly attached to it. This fixation of the displacement
at any moment definitely fixes the displacement-current. We at once
find, however, that to close the current requires us to regard the moving
charge itself as a current-element, of moment equal to the charge
multiplied by its velocity ; understanding by moment, in the case of a
distributed current, the product of current-density and volume. The
necessity of regarding the moving charge as an element of the "true
current" may be also concluded by simply considering that when a
charge q is conveyed into any region, an equal displacement simul-
taneously leaves it through its boundary.
Knowing the electric current, the magnetic force to correspond
becomes definitely known if the distribution of inductivity be given ;
MOTION OF ELECTRIFICATION THROUGH A DIELECTRIC. 505
and when this is constant everywhere, as we shall suppose now and
later, the magnetic force is simply the circuital vector whose curl
is 4 TT times the electric current; or the vector-potential of the curl of
the current; or the curl of the vector-potential of the current, etc., etc.
Thus, as found by J. J. Thomson,* the magnetic field of a charge
moving at a speed which is a small fraction of that of light is that
which is commonly ascribed to a current-element itself. I think it,
however, preferable to regard the magnetic field as the primary object
of attention ; or else to regard the complete system of closed current
derived from it by taking its curl as the unit, forming what we may
term a rational current-element, inasmuch as it is not a mere mathe-
matical abstraction, but is a complete dynamical system involving
definite forces and energy.
2. Let the axis of z be the line of motion of the charge q at the speed
u ; then the lines of magnetic force H are circles centred upon the axis,
in planes perpendicular to it, and its tensor H at distance r from the
charge, the line r making an angle 6 with the axis, is given by
........................... (1)
where v = sin 0, E the intensity of the radial electric force, c the per-
mittivity such that /x cv 2 = l, if /x is the other specific quality of the
medium, its inductivity, and v is the speed of propagation.
Since, under the circumstance supposed of u/v being very small, the
alteration in the electric field is insensible, and the lines of E are radial,
we may terminate the fields represented by (1) at any distance r = a
from the origin. We then obtain the solution in the case of a charge q
upon the surface of a conducting sphere of radius a, moving at speed u.
This realization of the problem makes the electric and magnetic energies
finite. Whilst, however, agreeing with J. J. Thomson in the funda-
mentals, I have been unable to corroborate some of his details; and
since some of his results have been recently repeated by him in another
place,! it may be desirable to state the changes I propose, before pro-
ceeding to the case of a charge moving at any speed.
The Energy and Forces in the Case of Slow Motion.
3. First, as regards the magnetic energy, say T. This is the space-
summation 2/x JT 2 /87r; or, by
The limits are such as include all space outside the sphere r = a. The
coefficient | replaces T 2 .
4. Next, as regards the mutual magnetic energy M of the moving
charge and any external magnetic field. This is the space-summation
* Phil. Mag., April, 1881.
t " Applications of Dynamics to Physics and Chemistry," chap, iv., pp. 31 to 37.
J The Electrician, Jan. 24, 1885, p. 220 [vol. i., p. 446].
506 ELECTRICAL PAPERS.
2 /A H H/47r, if H is the external field ; and, by a well-known trans-
formation, it is equivalent to 2A F, if A is any vector whose curl is
^ H , whilst F is the current-density of the moving system. Further,
if we choose A to be circuital, the polar part of T will contribute
nothing to the summation, so that we are reduced to the volume-
integral of the scalar product of the circuital A of the one system
and the density of the convection- current in the other. Or, in the
present case, with a single moving charge at a point, we have simply
the scalar product A u to represent the mutual magnetic energy ; or
^~=A u?, ................................. (3)
which is double J. J. Thomson's result.
5. When, therefore, we derive from (3) the mechanical force on the
moving charge due to the external magnetic field, we obtain simply
Maxwell's "electromagnetic force" on a current-element, the vector
product of the moment of the current and the induction of the external
field ; or if F is this mechanical force,
F = M VuH , .............................. (4)
which is also double J. J. Thomson's result. Notice that in the appli-
cation of the "electromagnetic force" formula, it is the moment of the
convection-current that occurs. This is not the same as the moment of
the true current, which varies according to circumstances ; for instance,
in the case of a small dielectric sphere uniformly electrified throughout
its volume, the moment of the true current would be only f of that of
the convection-current.
The application of Lagrange's equation of motion to (3) also gives
the force on q due to the electric field so far as it can depend on M ;
that is, a force _ ^
where the time-variation due to all causes must be reckoned, except
that due to the motion of q itself, which is allowed for in (4). And
besides this, there may be electric force not derivable from A , viz.
where ^ is the scalar potential companion to A .
6. Now if the external field be that of another moving charge, we
shall obtain the mutual magnetic energy from (3) by letting A be the
vector-potential of the current in the second moving system, constructed
so as to be circuital. Now the vector-potential of the convection-
current qu is simply qu/r ; this is sufficient to obtain the magnetic force
by curling; but if used to calculate the mutual energy, the space-
summation would have to include every element of current in the other
system. To make the vector-potential circuital, and so be able to
abolish this work, we must add on to qu/r the vector-potential of the
displacement current to correspond. Now the complete current may be
considered to consist of a linear element qu having two poles ; a radial
current outward from the + pole in which the current-density is qu/4:irr?;
and a radial current inward to the - pole, in which the current-density
is - qu/^Trrj ; where r l and r z are the distances of any point from the
MOTION OF ELECTRIFICATION THROUGH A DIELECTRIC. 507
poles. The vector-potentials of these currents are also radial, and their
tensors are \qu and - \qu. We have now merely to find their resultant
when the linear element is indefinitely shortened, add on to the former
v is
there any difficulty. We must first settle upon what basis to work.
First the Faraday-law (p standing for d/dt),
-curlE = /v?H, (11)
requires no , change when there is moving electrification. But the
analogous law of Maxwell, which I understand to be really a definition
of electric current in terms of magnetic force, (or a doctrine), requires
modification if the true current is to be
C+pD + /ou; (12)
viz., the sum of conduction-current, displacement-current, and convec-
tion-current pu, where p is the volume-density of electrification. The
addition of the term />u was, I believe, proposed by G. F. Fitzgerald.*
(This was not meant exactly for a new proposal, being in fact after
Rowland's experiments; besides which, Maxwell was well acquainted
with the idea of a convection-current. But what is very strange is that
Maxwell, who insisted so strongly upon his doctrine of the quasi-
incompressibility of electricity, never formulated the convection-current
in his treatise. Now Prof. Fitzgerald pointed out that if Maxwell, in
his equation of mechanical force,
F = VCB - eW - raVft,
had written E for - V*P, as it is obvious he should have done, then the
inclusion of convection-current in the true current would have followed
naturally. (Here C is the true current, B the induction, e the density
of electrification, m that of imaginary magnetic matter, "*" the electro-
static and ft the magnetic potential, and E the real electric force.)
Now to this remark I have to add that it is as unjustifiable to derive
H from ft as E from * ; that is, in general, the magnetic force is not
the slope of a scalar potential ; so, for - Vft we should write H, the real
magnetic force.
* Brit. Assoc., Southport, 1883.
MOTION OF ELECTRIFICATION THROUGH A DIELECTRIC. 509
But this is not all. There is possibly a fourth term in F, expressed
by 47rVDG, where D is the displacement and G the magnetic current ;
I have termed this force the " magneto-electric force," because it is the
analogue of Maxwell's "electromagnetic force," VCB. Perhaps the
simplest way of deriving it is from Maxwell's electric stress, which was
the method I followed.*
Thus, in a homogeneous nonconducting dielectric free from electri-
fication and magnetization, the mechanical force is the sum of the
"electromagnetic" and the "magnetoelectric," and is given by
F _ 1 dW
?~3P
where W = VEH/4?r is the transfer-of-energy vector.
It must, however, be confessed that the real distribution of the
stresses, and therefore of the forces, is open to question. And when
ether is the medium, the mechanical force in it, as for instance in a
light-wave, or in a wave sent along a telegraph-circuit, is not easily to
be interpreted.)
The companion to (11) in a nonconducting dielectric is now
curlH = cpE + 47rpu ................... . ........ (13)
Eliminate E between (11) and (13), remembering that H is circuital,
because /x is constant, and we get
Q?> 2 -V 2 )H = curl4izy>u, ........................ (14)
the characteristic of H. Here V 2 = d 2 /dx 2 + ..., as usual.
Comparing (14) with the characteristic of H when there is impressed
force e instead of electrification />, which is
we see that />u becomes cpe/47r. We may therefore regard convection-
current as impressed electric current. From this comparison also, we
may see that an infinite plane sheet of electrification of uniform density
cannot produce magnetic force by motion perpendicular to its plane.
Also, we see that the sources of disturbances when p is moved are the
places where /ou has curl ; for example, a dielectric sphere uniformly
filled with electrification (which is imaginable), when moved, starts the
magnetic force solely upon its boundary.
The presence of "curl" on the right side tells us, as a matter of
mathematical simplicity, to make H/curl the variable. Let
H = curlA, ................................. (15)
and calculate A, which may be any vector satisfying (15). Its
characteristic is
Q? 2 /*> 2 -V 2 )A = 47r / >u ............................ (16)
The divergence of A is of no moment, and it is only vexatious compli-
cation to introduce ^F. The time-rate of decrease of A is not the real
*"E1. Mag. Ind. and its Prop." xxn. The Electrician, Jan. 15, 1886, p. 187
[vol. i., p. 545].
510 ELECTRICAL PAPERS.
distribution of electric force, which has to be found by the additional
datum
divcE = 47i7>, ............................... (17)
where E is the real force.
9. " Symbolically " expressed, the solution of (16) is
47T P U _-47T / )U/V 2
Here the numerator of the fraction to the right is the vector-potential of
the convection-current. Calling it A , we have
Inserting in (18) and expanding, we have
.................. (20)
Given then /ou as a function of position and time, A is known by (19),
and (20) finds A, whilst (15) finds H.
Complete Solution in the Case of Steady Rectilinear Motion. Physical
Inanity of "*&.
10. When the motion of the electrification is all in one direction, say
parallel to the s-axis, u, A , and A are all parallel to this axis, so that
we need only consider their tensors. When there is simply one charge
q at a point, we have
A = ur
and (20) becomes
(21)
at distance r from q. When the motion is steady, and the whole electro-
magnetic field is ultimately steady with respect to the moving charge,
we shall have, taking it as origin,
p = -u(d/dz) = -uD,
for brevity ; so that
(22)
Now the property W +a = (n + 2)(w + 3)r" ........................ (23)
brings (22) to ^ = ^i + g^ + ^ + ...}; ................. (24)
and the property D Zn r 2n ~ l = l*.3 2 .5*...(2w- l)V/r, ............... (25)
where v = sin 6, being the angle between r and the axis, brings (24) to
' ....... < 26 >
MOTION OF ELECTRIFICATION THROUGH A DIELECTRIC. 51 1
which, by the Binomial Theorem, is the same as
A = (qulr){l-u*v*l<#Y\ ....................... (27)
the required solution.
11. To derive //, the tensor of the circular H, let rv = h, the distance
from the axis. Then, by (15),
.- T (28)
dh dr r dp r 2 \ r dp) V v 2 )
by (27), if /z = cos#. Performing the differentiation, and also getting
out E, the tensor of the electric force, we have the final result that the
electromagnetic field is fully given by *
cE =*. l-*/** t> H=cEuv, ............... (29)
r' 2 (1 -**/*)*
with the additional information that E is radial and H circular.
Now, as regards ^, if we bring it in, we have only got to take it out
again. When the speed is very slow we may regard the electric field as
given by - VM* plus a small correcting vector, which we may call the
electric force of inertia. But to show the physical inanity of "*P, go to
the other extreme, and let u nearly equal v. It is now the electric force
of inertia (supposed) that equals + V^ nearly (except about the equa-
torial plane), and its sole utility or function is to cancel the other - V^
of the (supposed) electrostatic field. It is surely impossible to attach
any physical meaning to and to propagate it, for we require two TF's,
one to cancel the other, and both propagated infinitely rapidly.
As the speed increases, the electromagnetic field concentrates itself
more and more about the equatorial plane, 6 = \TC. To give an idea of
the accumulation, let tt 2 /0 2 = -99. Then cE is -01 of the normal value
q/r 2 at the pole, and 10 times the normal value at the equator. The
latitude where the value is normal is given by
~ (30)
Limiting Case of Motion at the Speed of Light. Application to a
Telegraph Circuit.
12. Whentt = fl, the solution (29) becomes a plane electromagnetic
wave, E and H being zero everywhere except in the equatorial plane.
As, however, the values of E and H are infinite, distribute the charge
along a straight line moving in its own line, and let the linear-density
be q. The solution is then f
H=Ecv = 2qv/r ............................. (31)
at distance r from the line, between the two planes through the ends of
the line perpendicular to it, and zero elsewhere.
To further realize, let the field terminate internally at r = a, giving a
cylindrical-surface distribution of electrification, and terminate the tubes
* The Electrician, Dec. 7, 1888, p. 148 [p. 495, vol. n.].
tlbid., Nov. 23, 1888, p. 84 [p. 493, vol. 11.].
512 ELECTRICAL PAPERS.
of displacement externally upon a coaxial cylindrical surface ; we then
produce a real electromagnetic plane wave with electrification, and of
finite energy. We have supposed the electrification to be carried through
the dielectric at speed v, to keep up with the wave, which would of course
break up if the charge were stopped. But if perfectly-conducting
surfaces be given on which to terminate the displacement, the natural
motion of the wave will itself carry the electrification along them. In
fact, we now have the rudimentary telegraph-circuit, with no allowance
made for absorption of energy in the wires, and the consequent
distortion. If the conductors be not coaxial, we only alter the distri-
bution of the displacement and induction, without affecting the
propagation without distortion.*
If we now make the medium conduct electrically, and likewise
magnetically, with equal rates of subsidence, we shall have the same
solutions, with a time-factor e~^ producing ultimate subsidence to zero ;
and, with only the real electric conductivity in the medium the wave is
running through, it will approximately cancel the distortion produced
by the resistance of the wires the wave is passing over when this resist-
ance has a certain value. f We should notice, however, that it could
not do so perfectly, even if the magnetic retardation in the wires due to
diffusion were zero ; because in the case of the unreal magnetic con-
ductivity its correcting influence is where it is wanted to be, in the
body of the wave ; whereas in the case of the wires, their resistance,
correcting the distortion due to the external conductivity, is outside the
wave ; so that we virtually assume instantaneous propagation laterally
from the wires of their correcting influence, in the elementary theory of
propagation along a telegraph-circuit which is symbolized by the
equations
(32)
where R, L, K, and S are the resistance, inductance, leakage-conduct-
ance, and permittance per unit length of circuit, C the current, and V
what I, for convenience, term the potential-difference, but which I have
expressly disclaimed^: to represent the electrostatic difference of
potential, and have shown to represent the transverse voltage or line-
integral of the electric force across the circuit from wire to wire,
including the electric force of inertia. Now in case of great distortion,
as in a long submarine cable, this /^approximates towards the electro-
static potential-difference, which it is in Sir W. Thomson's diffusion
theory ; but in case of little distortion, as in telephony through circuits
of low resistance and large inductance, there may be a wide difference
between my V and that of the electrostatic force. Consider, for
instance, the extreme case of an isolated plane-wave disturbance with no
spreading-out of the tubes of displacement. At the boundaries of the
* The Electrician, Jan. 10, 1885 [p. 440, vol. i.]. Also "Self-Induction of
Wires," Part IV. Phil. Mag., Nov. 1886 [p. 221, vol. n.].
t " Electromagnetic Waves," 6, Phil. Mag., Feb. 1888 [p. 379, vol. n.]. The
Electrician, June, 1887 [p. 123, vol. n.].
t " Self-induction of Wires," Part. II., Phil. Mag., Sept. 1886 [vol. n., p. 189].
MOTION OF ELECTRIFICATION THROUGH A DIELECTRIC. 513
disturbance the difference between V and the electrostatic difference of
potential is great.
But it is worth noticing, as a rather remarkable circumstance, that
when we derive the system (32) by elementary considerations, viz., by
extending the diffusion-system by the addition of the E.M.F. of inertia
and leakage-current, we apparently as a matter of course take V to
mean the same as in the' diffusion-system. The resulting equations are
correct, and yet the assumption is certainly wrong. The true way
appears to be that given by me in the paper last referred to, by con-
sidering the line-integral of electric force in a closed curve [vol. II.,
p. 187. Also p. 87]. We cannot, indeed, make a separation of the
electric force of inertia from - VP" without some assumption, though
the former is quite definite when the latter is suitably defined, But,
and this is the really important matter, it would be in the highest
degree inconvenient, and lead to much complication and some confusion,
to split V into two components, in other words, to bring in "^f and A.
In thus running down Mf, I am by no means forgetful of its utility in
other cases. But it has perhaps been greatly misused. The clearest
course to pursue appears to me to invariably make E and H the primary
objects of attention, and only use potentials when they naturally suggest
themselves as labour-saving appliances.
Special Tests. The Connecting Equations.
13. Returning to the solutions (29), the following are the special tests
of their accuracy. Let E l and JE 2 be the z and h components of E.
Then, by (11) and (13), with the special meaning assumed by^?, we have
7,77 /
r -==- tin. - CU
hdh
_^L-cA or
dz az
dE l dE dH
---
- r -=-
,(33)
In addition to satisfying these equations, the displacement outward
through any spherical surface centred at the charge may be verified to
be q ; this completes the test of the accuracy of (29).
But (33) are not limited to the case of a single point-charge, being
true outside the electrification when there is symmetry with respect to
the z-axis, and the electrification is all moving parallel to it at speed u.
When u = , E 1 = Q, and E 2 = E = [j.vH, so that we reduce to
Aff =' ........................... - (34)
outside the electrification. Thus, if the electrification is on the axis of z,
we have
E/nv = H=2qv/r, ........................... (35)
differing from (31) only in that q, the linear density, may be any
function of z.
H.E.P. VOL. II. 2K
514 ELECTRICAL PAPERS.
The Motion of a Charged Sphere. The Condition at a Surface of
Equilibrium (Footnote).
14. If, in the solutions (29), we terminate the fields internally at
r = a, the perpendicularity of E and the tangentiality of H to the surface
show that (29) represents the solutions in the case of a perfectly con-
ducting sphere of radius a, moving steadily along the 2-axis at the speed
u, and possessing a total charge q. The energy is now finite. Let U
be the total electric and T the total magnetic energy. By space-
integration of the squares of E and H we find that they are given by
Z7=JL.
2ca
2ca
(36)
in which %<#. When ii = v, with accumulation of the charge at the
equator of the sphere, we have infinite values, and it appears to be
only possible to have finite values by making a zone at the equator
cylindrical instead of spherical. The expression for T in (37) looks
quite wrong ; but it correctly reduces to that of equation (2) when u/v
is infinitely small.*
* [I am indebted to Mr. G. F. C. Searle, of Cambridge, for the opportunity of
making a somewhat important correction before going to press. In a private
communication (August 19, 1892) he informed me that he had verified the accuracy
of the solution for a point-charge, which he had also obtained in another way,
from equations equivalent to (33), without the use of the function A of 8 to 10 ;
but he cast doubt upon the validity of the extension made in 14, from a point-
charge to a charged conducting sphere, and asked the plain question (in effect),
What justification is there for terminating the displacement perpendicularly, to
make a surface of equilibrium ?
On examination, I find that there is no justification whatever, exceptions
excepted. The true boundary condition may, however, be found without a fresh
investigation. On p. 499 the problem of uniform motion of electrification through
a dielectric medium, or conversely, of the uniform motion of. the whole medium
past stationary electrification, is reduced to a case of eolotropy in electrostatics.
The eS'ect of the motion of the isotropic medium on the displacement emanating
from stationary electrification is there shown to be identical with the effect of
keeping the medium stationary and reducing its permittivity in lines parallel to
the (abolished) motion from c to c(l -w 2 /^ 2 ), whilst keeping the transverse permit-
tivity the same. The transverse concentration of the displacement is obvious.
Now the function P (equation (14), p. 499) is the electrostatic potential in the
stationary eolotropic problem, so that its slope - VP, which call F, is the electric
force, and the displacement D is a linear function thereof, say D = XF, where X is
the permittivity operator. The condition of equilibrium is that F is perpendicular
to the surface where it terminates, this being required to make curl F = 0, or the
voltage zero in every circuit. Now, in the corresponding problem of the same
electrification in a moving isotropic medium, we have the same function P (no
longer the electrostatic potential) and the same derived vector F, whilst the
displacement D is also derived from F in the same way. But whilst the meaning
of D is the same in both cases, that of F is not. In the eolotropic case, F is the
MOTION OF ELECTRIFICATION THROUGH A DIELECTRIC. 515
The State when the Speed of Light is exceeded.
15. The question now suggests itself, What is the state of things
when u>v1 It is clear, in the first place, that there can be no dis-
turbance at all in front of the moving charge (at a point, for simplicity).
Next, considering that the spherical waves emitted by the charge in its
motion along the -axis travel at speed v, the locus of their fronts is a
conical surface whose apex is at the charge itself, whose axis is that of
z, and whose semiangle is given by
smO = v/u (38)
The whole displacement, of amount q, should therefore lie within this
cone. And since the moving charge is a convection-current qu, the
displacement-current should be towards the apex in the axial portion of
the cone, and change sign at some unknown distance, so as to be away
from the apex either in the outer part of the cone or else upon its
boundary. The pulling back of the charge by the electric stress would
require the continued application of impressed force to keep up the
motion, and its activity would be accounted for by the continuous addi-
tion made to the energy in the cone ; for the transfer of energy on its
boundary is perpendicularly outward, and the field at the apex is being
continuously renewed.
The above general reasoning seems plausible enough, but I cannot
find any solution to correspond that will satisfy all the necessary condi-
tions. It is clear that (29) will not do when u > v. Nor is it of any
use to change the sign of the quantity under the radical, when needed,
to make real. It is suggested that whilst there should be a definite
solution, there cannot be one representing a steady condition of E
and H with respect to the moving charge. As regards physical
electric force, and is not parallel to D. In the moving isotropic medium, on the
other hand, F is not the electric force, which is E, parallel to D. Nevertheless,
the same condition formally obtains, for we have curlF = in the moving medium,
requiring that F shall be perpendicular to a surface of equilibrium, not the
electric force or displacement. P = constant is therefore the equation to a
surface of equilibrium. That is, in the case of a point-charge, the surfaces of
equilibrium are not spheres, but are concentric oblate spheroids, whose principal
axes are proportional to the square roots of c, c, and c(l-w 2 /v 2 ), the principal
permittivities in the eolotropic problem. In the extreme case of u = v, the
spheroid reduces to a flat circular disc, with a single circular line of electrification
on its edge. It would seem, however, to be a matter of indifference, in this
extreme case, whether the conductor be a disc or a solid sphere.. Bearing in
mind the conditions assumed to prevail in the problem of motion of sources of
displacement in a uniform medium, we see that if we introduce conductors, say by
filling up spaces void of electric force with conducting matter, this should not
interfere with the assumed motions. (See also " Electromagnetic Theory," 164.)
Equations (36), (37) express the electric and magnetic energy outside a sphere
of radius a, within which is either a point-source at the origin, or any equivalent
spheroidal electrified surface.
In the corresponding bidimensional problem of 17 in the text, with the
solution (43), it is clear from the above that the surface of equilibrium is an
elliptic cylinder, the shorter axis being in the direction of motion, and the axes
themselves in the ratio 1 to ( 1 - M 2 /^ 2 )*. This surface degenerates to a flat strip
when u = v. ]
516 ELECTRICAL PAPERS.
possibility, in connexion with the structure of the ether, that is not
in question.*
A Charged Straight Line moving in its own Line.
16. Let us now derive from (29), or from (27), the results in some
cases of distributed electrification, in steady rectilinear motion. The
integrations to be effected being all of an elementary character, it is
not necessary to give the working.
First, let a straight line AB be
charged to linear density q, and be in
motion at speed u in its own line
from left to right. Then a-t P we
shall have
... (39)
2 /x 2 + -v 2 )
from which H= - dAjdh gives
H^gufl - ^f Vl _ T -same f n of r 2 , /* 2 , v 2 "l, (40
where /x = cos 0, v = sin 0.
When P is vertically over B, and A is at an infinite distance, we shall
................................ (41)
which is one half the value due to an infinitely long (both ways) straight
current of strength qu. The notable thing is the independence of the
ratio u/v.
* [The difficulty about the above method and solution (29) is that it is not
explicit enough when u > v, and does not indicate the limits of application. It
gives a real solution for the hinder cone, a real solution for the forward cone, and
an unreal solution in the rest of space, but we have no instruction to reject the
part for the forward cone and the unreal part, nor have we any means of testing
that the remainder, confined to the hinder cone, is the proper solution, viz., by
the test of divergence, to give the right amount of electrification. The integral
displacement comes to - GO . Now this may require to be supplemented by
+ oo + q on the boundary of the cone, but we have no way of testing it.
But certain considerations led me to the conclusion that the problem of u>v
was really quite as definite a one as that of u < v, and that a correct method of
a general character (independent of the magnitude of u) would show this explicitly.
I therefore (in 1890) attacked the problem from a different point of view, employ-
ing the method of resistance-operators (or an equivalent method). Form the
complete differential equation D = 0u, connecting the displacement D associated
with a moving point-charge with its velocity u, which is any function of the
time t. Here is a differential operator, a function of p or djdt. The solution of
this equation gives D explicitly in terms of u, whether steady or variable, and its
structure indicates the limits of application.
Taking u = constant, we obtain the result (29) when u < v. But when u > v, the
formula tells us to exclude all space except the hinder cone, and that in it, the
solution is not (29), but double as much. That is, double the right member of the
first of (29) when u > v. The boundary of the cone is also a displacement sheet.
The displacement is to the charge in the cone, and from the charge on its surface.
Being so near the end of the second volume, I regret that there is no space
here for the mathematical investigation, which cannot be given in a few words,
and must be reserved.]
MOTION OF ELECTRIFICATION THROUGH A DIELECTRIC. 517
But if u = v in (40), the result is zero, unless ^ = 1, when we have
the result (41). But if P be still further to the left, we shall have to
add to (41) the solution due to the electrification which is ahead of P.
So when the line is infinitely long both ways, we have double the result
in (41), with independence of u/v again.
But should q be a function of z, we do not have independence of u/v
except in the already-considered case of u = v, with plane waves, and no
component of electric force parallel to the line of motion.
A Charged Straight Line moving Transversely.
17. Next, let the electrified line be in
steady motion perpendicularly to its length.
Let q be the linear density (constant), the
2-axis that of the motion, the z-axis coin-
cident with the electrified line, and that of
y upward on the paper. Then the A at
P will be
. (42)
(1 - U*lv*) X 2 + {Xl + yS + 38(1 - tf/rlp '
where y and z belong to P, and x v x 2 are the limiting values of x in the
charged line. From this derive the solution in the case of an infinitely
long line. It is
where v = sin 9 ; understanding that E is radial, or along qP in the
figure, and H rectilinear, parallel to the charged line.
Terminating the fields internally at r = a, we have the case of a per-
fectly conducting cylinder of radius a, charged with q per unit of length,
moving transversely. When u = v there is disappearance of E and H
everywhere except in the plane 6 = JTT, as in the case of the sphere, with
consequent infinite values. It is the curvature that permits this to
occur, i.e. producing infinite values ; of course it is the self-induction
that is the cause of the conversion to a plane wave, here and in the
other cases. There is some similarity be-
tween (43) and (29). In fact, (43) is the
bidimensional equivalent of (29).
A Charged Plane moving Transversely.
18. Coming next to a plane distribution
-y*
of electrification, let q be the surface-density,
and the plane be moving perpendicularly
to itself. Let it be of finite breadth and
of infinite length, so that we may calculate H from (43).
at Pis
The result
H
a 1 **;
,(44)
518 ELECTRICAL PAPERS.
When P is equidistant from the edges, H is zero. There is therefore
no H anywhere due to the motion of an infinitely large uniformly
charged plane perpendicularly to itself. The displacement-current is
the negative of the convection-current and at the same place, viz. the
moving plane, so there is no true current.
Calculating E v the ^-component of E, z being measured from left to
right, we find
(45)
The component parallel to the plane is H/cu. Thus, when the plane is
infinite, this component vanishes with H, and we are left with
cE 1 = cE = 2Trq, ............................. (46)
the same as if the plane were at rest.
A Charged Plane moving in its own Plane.
19. Lastly, let the charged plane be moving in its own plane. Refer
to the first figure, in which let AB now be the trace of the plane when
of finite breadth. We shall find that
(47)
z l and z 2 being the extreme values of z, which is measured parallel to
the breadth of the plane.
Therefore, when the plane extends infinitely both ways, we have
H=2Trqu . ................................ (48)
above the plane, and its negative below it. This differs from the previous
case of vanishing displacement-current. There is H, and the convection-
current is not now cancelled by coexistent displacement-current.
The existence of displacement-current, or changing displacement, was
the basis of the conclusion that moving electrification constitutes a part
of the true current. Now in the problem (48) the displacement-current
has gone, so that the existence of H appears to rest merely upon the
assumption that moving electrification is true current. But if the plane
be not infinite, though large, we shall have (48) nearly true near it, and
away from the edges ; whilst the displacement-current will be strong
near the edges, and almost nil where (48) is nearly true.
But in some cases of rotating electrification, there need be no dis-
placement anywhere, except during the setting up of the final state.
This brings us to the rather curious question whether there is any
difference between the magnetic field of a convection-current produced
by the rotation of electrification upon a good nonconductor and upon a
good conductor respectively, other than that due to diffusion in the
conductor. For in the case of a perfect conductor, it is easy to imagine
that the electrification could be at rest, and the moved conductor merely
slip past it. Perhaps Professor Rowland's forthcoming experiments on
convection-currents may cast -some light upon this matter.
December 27, 1888.
DEFLECTION OF AN ELECTROMAGNETIC WAVE. 519
LI. DEFLECTION OF AN ELECTROMAGNETIC WAVE BY
MOTION OF THE MEDIUM.
[The Electrician, April 12, 1889, p. 663.]
THIS subject is of interest in connection with theories of Aberration,
which requires to be explained electromagnetically. A plane wave in
a nonconducting dielectric is carried on at speed v = (/*c)~*, where p is
the inductivity and c the permittivity, and is not altered in any way,
according to the rudimentary theory, that is to say, which overlooks
dispersion. But if the medium be moving through the ether, it is
altered in a manner depending upon the speed of motion and the angle
it makes with the undisturbed direction of propagation.
Thus, let E Q = ^vH^ specify a plane wave in a medium at rest,
E being the tensor of the electric and H Q of the magnetic force.
Next set the medium in motion with velocity u, changing E to E and
H to H, thus
E = e + E , H = h + H , (A)
where e and h are the auxiliary electric and magnetic forces due to the
motion. To find them, we have, first, the electric force due to motion
of matter in a magnetic field, or
e = /*VuH, (B)
which formula is well known, and is included in Maxwell's treatise.
Next, the magnetic force due to motion in an electric field, or
h = cVEu (C)
This equation, which is as necessary as (B), was, so far as I am at
present aware, first given by me in Section III. of " Electromagnetic In-
duction and its Propagation," January 24, 1885 [vol. I., p. 446], and was
again considered later on in connection with the " magneto-electric
force," which is as necessary as Maxwell's " electromagnetic force."
We require one more relation, viz., between E and H , viz.,
H = cVvE , (D)
the property of a plane wave, due to Maxwell ; and we can now fully
find the auxiliaries e and h in terms of the originals E and H . Here
v is the vectorized v of the wave when undisturbed.
In the above V is the symbol of vector product. Thus VuH is the
vector perpendicular to u and to H, whose tensor equals the product of
their tensors, u and H, into the sine of the angle between their directions.
But this is merely used to state the general relations in a compact and
intelligible form, instead of with Cartesian circumlocutions.
Instead of taking the general case, it is convenient to divide into
three, viz., (1), u parallel to v ; (2), u parallel to E ; (3), u parallel to
H . By putting the results together we shall obtain the mixed-up
general case.
(1). u parallel to v. Here the medium is moving in the same direc-
tion as that of undisturbed propagation, and there is no alteration of
520 ELECTRICAL PAPERS.
direction of either E or H , so that it is only necessary to specify the
tensors of the auxiliaries e and h. Thus :
e=- En fc=-_^ ' H Q ................... (1)
u+v c u+v (
If, for example, the medium be moving at half the speed v, and with
it, the displacement and induction in a given length are spread over a
space half as great again as
if the medium were at rest,
so that their intensities are
reduced to two-thirds of the
undisturbed values. There
is no discontinuity when u is
equal to or greater than v.
But if the medium move
the other way there is com-
pression into half the space,
so that the intensities are
doubled. As it is increased
up to i\ the compression in-
creases infinitely. After that, with u>v, there is reversal of sign of E
and H as compared with E and H .
(2). u and E parallel. Here h is parallel to H , but e is parallel to
V. Their tensors are given by .
(3). Lastly, u and H parallel. Now e is parallel to E , whilst h is
parallel to v. Their tensors are
In either case, (2) or (3), the angle of deflection 6 is given by
n UV
consequently the deflection is wholly independent of the plane of
polarization.
Thus, let a slab of (say) glass move in its own plane at speed u, and
a plane-wave from the upper medium strike the glass flush. The trans-
mitted rays are deflected as shown in Fig. 1, the deflection being given
by the above formula, where, observe, v is the speed in the glass when
at rest, and u the speed of the glass with respect to the external medium.
The above working out of the effect of moving matter on a plane
electromagnetic wave is (if done properly) strictly in accordance with
electromagnetic principles. But it will be observed that Fresnel's result,
relating to the alteration in the speed of light produced by moving a
transparent medium through which it is passing, is not accounted for.
It is said to have been thoroughly confirmed by Michelson. I should
like to direct the attention of electromagneticians to this question, with
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 521
a view to the discovery of a modification of the above data, or correction
of the working, in order to explain Fresnel arid Michelson, which must
be done electromagnetically. Mr. Glazebrook has made Sir W. Thom-
son's extraordinary contractile ether do it by an auxiliary hypothesis ;
surely, then, Maxwell's ether equations could be appropriately modified.
LII. ON THE FORCES, STRESSES, AND FLUXES OF ENERGY
IN THE ELECTROMAGNETIC FIELD.
[Royal Society. Received June 9, Read June 18, 1891.* Abstract in
Proceedings, vol. 50, 1891 ; Paper in Transactions, A. 1892.]
(ABSTRACT.)
THE abstract nature of this paper renders its adequate abstraction
difficult. The principle of conservation of energy, when applied to a
theory such as Maxwell's, which postulates the definite localization of
energy, takes a more special form, viz., that of the continuity of energy.
Its general nature is discussed. The relativity of motion forbids us to
go so far as to assume the objectivity of energy, and to identify energy,
like matter ; hence the expression of the principle is less precise than
that of the continuity of matter (as in hydrodynamics), for all we can
say in general is that the convergence of the flux of energy equals the
rate of increase of the density of the energy ; the flux of the energy
being made up partly of the mere convection of energy by motion of
the matter (or other medium) with which it is associated localizably, and
partly of energy which is transferred through the medium in other
ways, as by the activity of a stress, for example, not obviously (if at
all) representable as the convection of energy. Gravitational energy is
the chief difficulty in the way of the carrying out of the principle. It
must come from the ether (for where else can it come from ?), when it
goes to matter ; but we are entirely ignorant of the manner of its dis-
tribution and transference. But, whenever energy can be localized, the
principle of continuity of energy is (in spite of certain drawbacks con-
nected with the circuital flux of energy) a valuable principle which
should be utilized to the uttermost. Practical forms are considered.
In the electromagnetic application the flux of energy has a four-fold
make-up, viz., the Poynting flux of energy, which occurs whether the
medium be stationary or moving; the flux of energy due to the
activity of the electromagnetic stress when the medium is moving ; the
convection of electric and magnetic energy ; and the convection of other
energy associated with the working of the translational force due to the
stress.
As Electromagnetism swarms with vectors, the proper language for
its expression and investigation is the Algebra of Vectors. An account
* Typographical troubles have delayed the publication of this paper. The foot-
notes are of date May 11, 1892.
522 ELECTRICAL PAPERS.
is therefore given of the method employed by the author for some
years past. The quaternionic basis is rejected, and the algebra is based
upon a few definitions of notation merely. It may be regarded "as
Quaternions without quaternions, and simplified to the uttermost ; or
else as being merely a conveniently condensed expression of the Cartesian
mathematics, understandable by all who are acquainted with Cartesian
methods, and with which the vectorial algebra is made to harmonize.
It is confidently recommended as a practical working system.
In continuation thereof, and preliminary to the examination of
electromagnetic stresses, the theory of stresses of the general type, that
is, rotational, is considered ; and also the stress activity, and flux of
energy, and its convergence and division into translational, rotational,
and distortional parts; all of which, it is pointed out, maybe associated
with stored potential, kinetic, and wasted energy, at least so far as the
mathematics is concerned.
The electromagnetic equations are then introduced, using them in
the author's general forms, i.e., an extended form of Maxwell's circuital
law, defining electric current in terms of magnetic force, and a com-
panion equation expressing the second circuital law ; this method
replacing Maxwell's in terms of the vector-potential and the electro-
static potential, Maxwell's equations of propagation being found im-
possible to work and not sufficiently general. The equation of activity
is then derived in as general a form as possible, including the effects of
impressed forces and intrinsic magnetization, for a stationary medium
which may be eolotropic or not. Application of the principle of con-
tinuity of energy then immediately indicates that the flux of energy in
the field is represented by the formula first discovered by Poynting.
Next, the equation of activity for a moving medium is considered. It
does not immediately indicate the flux of energy, and, in fact, several
transformations are required before it is brought to a fully significant
form, indicating (1), the Poynting flux, the form of which is settled ;
(2), the convection of electric and magnetic energy; (3), a flux of energy
which, from the form in which the velocity of the medium enters,
represents the flux of energy due to a working stress. Like the
Poynting flux, it contains vector products. From this flux the stress
itself is derived, and the form of translational force, previously tentatively
developed, is verified. It is assumed that the medium in its motion
carries its properties with it unchanged.
A side matter which is discussed is the proper measure of "true"
electric current, in accordance with the continuity of energy. It has a
four-fold make-up, viz., the conduction-current, displacement-current,
convection-current (or moving electrification), and the curl of the
motional magnetic force.
The stress is divisible into an electric and a magnetic stress. These
are of the rotational type in eolotropic media. They do not agree with
Maxwell's general stresses, though they work down to them in an
isotropic homogeneous stationary medium not intrinsically magnetized
or electrized, being then the well-known tensions in certain lines with
equal lateral pressures.
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 523
Another and shorter derivation of the stress is then given, guided by
the previous, without developing the expression for the flux of energy.
Variations of the properties permittivity and inductivity with the strain
can be allowed for. An investigation by Professor H. Hertz is referred
to. His stress is not agreed with, and it is pointed out that the
assumption by which it is obtained is equivalent to the existence of
isotropy, so that its generality is destroyed. The obvious validity of
the assumption on which the distortional activity of the stress is
calculated is also questioned.
Another form of the stress-vector is examined, showing its relation
to the fictitious electrification and magnetic current, magnetification
and electric current, produced on the boundary of a region by termi-
nating the stress thereupon ; and its relation to the theory of action at
a distance between the respective matters and currents.
The stress-subject is then considered statically. The problem is now
perfectly indeterminate, in the absence of a complete experimental
knowledge of the strains set up in bodies under electric and magnetic
influence. Only the stress in the air outside magnets and conductors
can be considered known. Any stress within them may be superadded,
without any difference being made in the resultant forces and torques.
Several stress-formulae are given, showing a transition from one extreme
form to another. A simple example is worked out to illustrate the
different ways in which Maxwell's stress and others explain the
mechanical actions. Maxwell's stress, which involves a translational
force on magnetized niatter (even when only inductively magnetized),
merely because it is magnetized, leads to a very complicated and un-
natural way of explanation. It is argued, independently, that no stress-
formula should be allowed which indicates a translational force of the
kind just mentioned.
Still the matter is left indeterminate from the statical standpoint.
From the dynamical standpoint, however, we are led to a certain
definite stress-distribution, which is also, fortunately, free from the
above objection, and is harmonized with the flux of energy. A pecu-
liarity is the way the force on an intrinsic magnet is represented. It
is not by force on its poles, nor on its interior, but on its sides, referring
to a simple case of uniform longitudinal magnetization ; i.e., it is done
by a ^wasi-electromagnetic force on the fictitious electric current which
would produce the same distribution of induction as the magnet does.
There is also a force where the inductivity varies. This force on
fictitious current harmonizes with the conclusion previously arrived at
by the author, that when impressed forces set up disturbances, such
disturbances are determined by the curl of the impressed forces, and
proceed from their localities.
In conclusion it is pointed out that the determinateness of the stress
rests upon the assumed localization of the energy and the two laws of
circuitation, so that with other distributions of the energy (of the same
proper total amounts) other results would follow ; but the author has
been unable to produce full harmony in any other way than that
followed.
524 ELECTRICAL PAPERS.
General Remarks, especially on the Flux of Energy.
1. The remarkable experimental work of late years has inaugurated
a new era in the development of the Faraday-Maxwellian theory of the
ether, considered as the primary medium concerned in electrical pheno-
mena electric, magnetic, and electromagnetic. Maxwell's theory is no
longer entirely a paper theory, bristling with unproved possibilities.
The reality of electromagnetic waves has been thoroughly demonstrated
by the experiments of Hertz and Lodge, Fitzgerald and Trouton, J. J.
Thomson, and others ; and it appears to follow that, although Maxwell's
theory may not be fully correct, even as regards the ether (as it is
certainly not fully comprehensive as regards material bodies), yet the
true theory must be one of the same type, and may probably be merely
an extended form of Maxwell's.
No excuse is therefore now needed for investigations tending to
exhibit and elucidate this theory, or to extend it, even though they be
of a very abstract nature. Every part of so important a theory deserves
to be thoroughly examined, if only to see what is in it, and to take note
of its unintelligible parts, with a view to their future explanation or
elimination.
2. Perhaps the simplest view to take of the medium which plays
such a necessary part, as the recipient of energy, in this theory, is to
regard it as continuously filling all space, and possessing the mobility
of a fluid rather than the rigidity of a solid. If whatever possess the
property of inertia be matter, then the medium is a form of matter.
But away from ordinary matter it is, for obvious reasons, best to call
it as usual by a separate name, the ether. Now, a really difficult and
highly speculative question, at present, is the connection between
matter (in the ordinary sense) and ether. When the medium trans-
mitting the electrical disturbances consists of ether and matter, do they
move together, or does the matter only partially carry forward the ether
which immediately surrounds it 1 Optical reasons may lead us to con-
clude, though only tentatively, that the latter may be the case ; but at
present, for the purpose of fixing the data, and in the pursuit of investi-
gations not having specially optical bearing, it is convenient to assume
that the matter and the ether in contact with it move together. This
is the working hypothesis made by H. Hertz in his recent treatment of
the electrodynamics of moving bodies; it is, in fact, what we tacitly
assume in a straightforward and consistent working out of Maxwell's
principles without any plainly-expressed statement on the question of
the relative motion of matter and ether ; for the part played in Maxwell's
theory by matter is merely (and, of course, roughly) formularized by
supposing that it causes the etherial constants to take different values,
whilst introducing new properties, that of dissipating energy being the
most prominent and important. We may, therefore, think of merely
one medium, the most of which is uniform (the ether), whilst certain
portions (matter as well) have different powers of supporting electric
displacement and magnetic induction from the rest, as well as a host
of additional properties; and of these we can include the power of
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 525
supporting conduction-current with dissipation of energy according to
Joule's law, the change from isotropy to eolotropy in respect to the
distribution of the several fluxes, the presence of intrinsic sources of
energy, etc.*
3. We do not in any way form the equations of motion of such a
medium, even as regards the uniform simple ether, away from gross
matter ; we have only to discuss it as regards the electric and magnetic
fluxes it supports, and the stresses and fluxes of energy thereby necessi-
tated. First, we suppose the medium to be stationary, and examine
the flux of electromagnetic energy. This is the Poynting flux of
energy. Next we set the medium into motion of an unrestricted kind.
We have now necessarily a convection of the electric and magnetic
energy, as well as the Poynting flux. Thirdly, there must be a similar
convection of the kinetic energy, etc., of the translation al motion ; and
fourthly, since the motion of the medium involves the working of
ordinary (Newtonian) force, there is associated with the previous a flux
of energy due to the activity of the corresponding stress. The question
is therefore a complex one, for we have to properly fit together these
various fluxes of energy in harmony with the electromagnetic equations.
A side issue is the determination of the proper measure of the activity
of intrinsic forces, when the medium moves ; in another form, it is the
determination of the proper meaning of "true current" in Maxwell's
sense.
4. The only general principle that we can bring to our assistance in
interpreting electromagnetic results relating to activity and flux of
energy, is that of the persistence of energy. But it would be quite
inadequate in its older sense referring to integral amounts ; the definite
localization by Maxwell, of electric and magnetic energy, and of its
waste, necessitates the similar localization of sources of energy ; and in
the consideration of the supply of energy at certain places, combined
with the continuous transmission of electrical disturbances, and there-
fore of the associated energy, the idea of a flux of energy through space,
and therefore of the continuity of energy in space and in time, becomes
forced upon us as a simple, useful, and necessary principle, which
cannot be avoided.
When energy goes from place to place, it traverses the intermediate
space. Only by the use of this principle can we safely derive the
electromagnetic stress from the equations of the field expressing the
two laws of circuitation of the electric and magnetic forces ; and this
* Perhaps it is best to say as little as possible at present about the connection
between matter and ether, but to take the electromagnetic equations in an abstract
manner. This will leave us greater freedom for future modifications without con-
tradiction. There are, also, cases in which it is obviously impossible to suppose
that matter in bulk carries on with it the ether in bulk which permeates it.
Either, then, the mathematical machinery must work between the molecules ; or
else, we must make such alterations in the equations referring to bulk as will be
practically equivalent in effect. For example, the motional magnetic force VDq
of equations (88), (92), (93) may be modified either in q or in D, by use of a smaller
effective velocity q, or by the substitution in D or cE of a modified reckoning
of c for the effective permittivity.
526 ELECTRICAL PAPERS.
again becomes permissible only by the postulation of the definite
localization of the electric and magnetic energies. But we need not go
so far as to assume the objectivity of energy. This is an exceedingly
difficult notion, and seems to be rendered inadmissible by the mere
fact of the relativity of motion, on which kinetic energy depends. We
cannot, therefore, definitely individualize energy in the same way as is
done with matter.
If p be the density of a quantity whose total amount is invariable,
and which can change its distribution continuously, by actual motion
from place to place, its equation of continuity is
convq/D = /3, (1)
where q is its velocity, and q/> the flux of />. That is, the convergence
of the flux of p equals the rate of increase of its density. Here p may
be the density of matter. But it does not appear that we can apply
the same method of representation to the flux of energy. We may,
indeed, write
convX = J, (2)
if X be the flux of energy from all causes, and T the density of localiz-
able energy. But the assumption X = Tq would involve the assumption
that T moved about like matter, with a definite velocity. A part of T
may, indeed, do this, viz., when it is confined to, and is carried by
matter (or ether) ; thus we may write
conv(qr+X) = r, (3)
where T is energy which is simply carried, whilst X is the total flux of
energy from other sources, and which we cannot symbolize in the form
Tq ; the energy which comes to us from the Sun, for example, or
radiated energy. It is, again, often impossible to carry out the principle
in this form, from a want of knowledge of how energy gets to a certain
place. This is, for example, particularly evident in the case of gravita-
tional energy, the distribution of which, before it is communicated to
matter, increasing its kinetic energy, is highly speculative. If it come
from the ether (and where else can it come from ?), it should be possible
to symbolize this in X, if not in <\T ; but in default of a knowledge of
its distribution in the ether, we cannot do so, and must therefore turn
the equation of continuity into
S + conv(qr+X) = T, (4)
where S indicates the rate of supply of energy per unit volume from
the gravitational source, whatever that may be. A similar form is
convenient in the case of intrinsic stores of energy, which we have
reason to believe are positioned within the element of volume concerned,
as when heat gives rise to thermoelectric force. Then S is the activity
of the intrinsic sources. Then again, in special applications, T is con-
veniently divisible into different kinds of energy, potential and kinetic.
Energy which is dissipated or wasted comes under the same category,
because it may either be regarded as stored, though irrecoverably, or
passed out of existence, so far as any immediate useful purpose is
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 527
concerned. Thus we have as a standard practical form of the equation
of continuity of energy referred to the unit volume,
S + conv{X + C((U + T)} = Q+U+f, (5)
where S is the energy supply from intrinsic sources, U potential energy
and T kinetic energy of localizable kinds, (\[U-\-T) its convective flux,
Q the rate of waste of energy, and X the flux of energy other than
convective, e.g., that due to stresses in the medium and representing
their activity. In the electromagnetic application we shall see that
U and T must split into two kinds, and so must X, because there is
a flux of energy even when the medium is at rest.
5. Sometimes we meet with cases in which the flux of energy is
either wholly or partly of a circuital character. There is nothing
essentially peculiar to electromagnetic problems in this strange and
apparently useless result. The electromagnetic instances are paralleled
by similar instances in ordinary mechanical science, when a body is in
motion and is also strained, especially if it be in rotation. This result
is a necessary consequence of our ways of reckoning the activity of
forces and of stresses, and serves to still further cast doubt upon the
" thinginess " of energy. At the same time, the flux of energy is going
on all around us, just as certainly as the flux of matter, and it is
impossible to avoid the idea ; we should, therefore, make use of it and
formnlarize it whenever and as long as it is found to be useful, in spite
of the occasional failure to obtain readily understandable results.
The idea of the flux of energy, apart from the conservation of energy,
is by no means a new one. Had gravitational energy been less obscure
than it is, it might have found explicit statement long ago. Professor
Poynting* brought the principle into prominence in 1884, by making
use of it to determine the electromagnetic flux of energy. Professor
Lodgef gave very distinct and emphatic expression of the principle
generally, apart from its electromagnetic aspect, in 1885, and pointed
out how much more simple and satisfactory it makes the principle
of the conservation of energy become. So it would, indeed, could we
only understand gravitational energy ; but in that, and similar respects,
it is a matter of faith only. But Professor Lodge attached, I think,
too much importance to the identity of energy, as well as to another
principle he enunciated, that energy cannot be transferred without being
transformed, and conversely; the transformation being from potential
to kinetic energy or conversely. This obviously cannot apply to the
convection of energy, which is a true flux of energy ; nor does it seem
to apply to cases of wave-motion in which the energy, potential and
kinetic, of the disturbance, is transferred through a medium unchanged
in relative distribution, simply because the disturbance itself travels
without change of type ; though it may be that in the unexpressed
internal actions associated with the wave-propagation there might be
found a better application.
* Poynting, Phil. Trans., 1884.
t Lodge, Phil. Mag., June, 1885, " On the Identity of Energy."
528 ELECTRICAL PAPERS.
It is impossible that the ether can be fully represented, even merely
in its transmissive functions, by the electromagnetic equations. Gravity
is left out in the cold; and although it is convenient to ignore this
fact, it may be sometimes usefully remembered, even in special electro-
magnetic work ; for, if a medium have to contain and transmit gravita-
tional energy as well as electromagnetic, the proper system of equations
should show this, and, therefore, include the electromagnetic. It seems,
therefore, not unlikely that in discussing purely electromagnetic specu-
lations, one may be within a stone's throw of the explanation of gravita-
tion all the time. The consummation would be a really substantial
advance in scientific knowledge.
On the Algebra and Analysis of Vectoi's without Quaternions. Outline of
Author's System.
6. The proper language of vectors is the algebra of vectors. It is,
therefore, quite certain that an extensive use of vector-analysis in
mathematical physics generally, and in electromagnetism, which is
swarming with vectors, in particular, is coming and may be near at
hand. It has, in my opinion, been retarded by the want of special
treatises on vector-analysis adapted for use in mathematical physics,
Professor Tait's well-known profound treatise being, as its name
indicates, a treatise on Quaternions. I have not found the Hamilton-
Tait notation of vector-operations convenient, and have employed, for
some years past, a simpler system. It is not, however, entirely a
question of notation that is concerned. I reject the quaternionic basis
of vector-analysis. The anti-quaternionic argument has been recently
ably stated by Professor Willard Gibbs.* He distinctly separates
this from the question of notation, and this may be considered fortunate,
for whilst I can fully appreciate and (from practical experience) endorse
the anti-quaternionic argument, I am unable to appreciate his notation,
and think that of Hamilton and Tait is, in some respects, preferable,
though very inconvenient in others.
In Hamilton's system the quaternion is the fundamental idea, and
everything revolves round it. This is exceedingly unfortunate, as it
renders the establishment of the algebra of vectors without metaphysics
a very difficult matter, and in its application to mathematical analysis
there is a tendency for the algebra to get more and more complex
as the ideas concerned get simpler, and the quaternionic basis forms
a real difficulty of a substantial kind in attempting to work in harmony
with ordinary Cartesian methods.
Now, I can confidently recommend, as a really practical working
system, the modification I have made. It has many advantages, and
not the least amongst them is the fact that the quaternion does not
appear in it at all (though it may, without much advantage, be brought
* Professor Gibbs's letters will be found in Nature, vol. 43, p. 511, and vol. 44,
p. 79 ; and Professor Tait's in vol. 43, pp. 535, 608. This rather one-sided dis-
cussion arose out of Professor Tait stigmatizing Professor Gibbs as ' ' a retarder of
quaternionic progress." This may be very true ; but Professor Gibbs is anything
but a retarder of progress in vector analysis and its application to physics.
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 529
in sometimes), and also that the notation is arranged so as to harmonize
with Cartesian mathematics. It rests entirely upon a few definitions,
and may be regarded (from one point of view) as a systematically
abbreviated Cartesian method of investigation, and be understood and
practically used by any one accustomed to Cartesians, without any
study of the difficult science of Quaternions. It is simply the elements
of Quaternions without the quaternions, with the notation simplified to
the uttermost, and with the very inconvenient minus sign before scalar
products done away with.*
7. Quantities being divided into scalars and vectors, I denote the
scalars, as usual, by ordinary letters, and put the vectors in the plain
black type, known, I believe, as Clarendon type, rejecting Maxwell's
German letters on account of their being hard to read. A special type
is certainly not essential, but it facilitates the reading of printed com-
plex vector investigations to be able to see at a glance which quantities
are scalars and which are vectors, and eases the strain on the memory.
But in MS. work there is no occasion for specially formed letters.
Thus A stands for a vector. The tensor of a vector may be denoted
by the same letter plain ; thus A is the tensor of A. (In MS. the
tensor is A Q .) Its rectangular scalar components are A lt A 2 , A 3 . A
unit vector parallel to A may be denoted by A 1? so that A = AA l . But
little things of this sort are very much matters of taste. What is
important is to avoid as far as possible the use of letter prefixes, which,
when they come two (or even three) together, as in Quaternions, are
very confusing.
The scalar product of a pair of vectors A and B is denoted by AB,
and is defined to be A
AE = A l B 1 + A 2 B 2 + A 3 B B = ABcosA3 = EA (6)
* 7, 8, 9 contain an introduction to vector-analysis (without the quaternion),
which is sufficient for the purposes of the present paper, and, I may add, for
general use in mathematical physics. It is an expansion of that given in my
paper "On the Electromagnetic Wave Surface," Phil. Mag., June, 1885, (vol. n.,
pp. 4 to 8). The algebra and notation are substantially those employed in all my
papers, especially in " Electromagnetic Induction and its Propagation," The
Electrician, 1885.
Professor Gibbs's vectorial work is scarcely known, and deserves to be well
known. In June, 1888, 1 received from him a little book of 85 pages, bearing the
singular imprint NOT PUBLISHED, Newhaven, 1881-4. It is indeed odd that the
author should not have published what he had been at the trouble of having
of
say
the
subject.
In The Electrician for Nov. 13, 1891, p. 27, I commenced a few articles on
elementary vector-algebra and analysis, specially meant to explain to readers of
my papers how to work vectors. I am given to understand that the earlier ones,
on the algebra, were much appreciated ; the later ones, however, are found diffi-
cult. But the vector-algebra is identically the same in both, and is of quite a
rudimentary kind. The difference is, that the later ones are concerned with
analysis, with varying vectors ; it is the same as the difference between common
algebra and differential calculus. The difficulty, whether real or not, does not
indicate any difficulty in the vector-algebra. I mention this on account of the
great prejudice which exists against vector-algebra.
H.E.P. VOL. II. 2L
530 ELECTRICAL PAPERS.
The addition of vectors being as in the polygon of displacements, or
velocities, or forces; i.e., such that the vector length of any closed
circuit is zero ; either of the vectors A and B may be split into the sum
of any number of others, and the multiplication of the two sums to
form AB is done as in common algebra ; thus
(a + b)(c + d) = ac + ad + be + bd = ca + da + cb + db ....... (7)
If N be a unit vector, NN or N 2 = 1 ; similarly, A? = A 2 for any vector.
The reciprocal of a vector A has the same direction ; its tensor is the
reciprocal of the tensor of A. Thus
AA-^^1; and AB- 1 = B- 1 A = 4 = 4 COS ^ ...... ( 8 )
A 15 x)
The vector product of a pair of vectors is denoted by VAB, and is
A
defined to be the vector whose tensor is ABsin AB, and whose direc-
tion is perpendicular to the plane of A and B, thus
VAB = i(A 2 B 3 - A 3 B 2 ) + j(A 3 B l - A&) + k(^^ - A 2 B,) = - VBA, (9)
where i, j, k, are any three mutually rectangular unit vectors. The
tensor of VAB is V AB ; or
V AB = ^BsinAB ......................... (10)
Its components are iVAB, JVAB, kVAB.
In accordance with the definitions of the scalar and vector products,
we have
i'=l, J 2 =l, k* = l; |
ij = 0, jk = 0, ki = 0; ................... (11)
Vij=k, Vjk = i, Vki=:j;l
and from these we prove at once that
V(a + b)(o + d) = Vac + Vad + Vbc + Vbd,
and so on, for any number of component vectors. The order of the
letters in each product has to be preserved, since Vab= - Vba.
Two very useful formulae of transformation are
1 C 2 -&); ....(12)
and VAVBC = B.CA-C.AB, or =B(CA)-C(AB) ....... (13)
Here the dots, or the brackets in the alternative notation, merely
act as separators, separating the scalar products CA and AB from the
vectors they multiply. A space would be equivalent, but would be
obviously unpractical.
A
As is a scalar product, so in harmony therewith, there is the
B A
vector product V. Since VAB = - VBA, it is now necessary to make
B
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 531
a convention as to whether the denominator comes first or last in
V=. Say therefore, VAB" 1 . Its tensor is
D
(U)
8. Differentiation of vectors, and of scalar and vector functions of
vectors with respect to scalar variables is done as usual. Thus,
.(15)
= AVBC + AVBC + AVBC.
The same applies with complex scalar differentiators, e.g., with the
differentiator
used when a moving particle is followed, q being its velocity. Thus,
?AB = A?? + B~ = AB + BA + A
ct Ct ot
Here qV is a scalar differentiator given by
?AB = A?? + B~ = AB + BA + A.qV.B + B.qV.A .......... (16)
ct Ct ot
so that A.qV.B is the scalar product of A and the vector qV.B; the
dots here again act essentially as separators. Otherwise, we may write
it A(qV)B.
The fictitious vector V given by
k ................... (18)
is very important. Physical mathematics is very largely the mathe-
matics of V. The name Nabla seems, therefore, ludicrously inefficient.
In virtue of i, j, k, the operator V behaves as a vector. It also, of
course, differentiates what follows it.
Acting on a scalar P, the result is the vector
VP = iV 1 P+jV 2 P + kV 3 P, ........................ (19)
the vector rate of increase of P with length.
If it act on a vector A, there is first the scalar product
VA = V 1 ^ 1 + V 2 ^ 2 + V 3 ^3 = divA, .................. (20)
or the divergence of A. Regarding a vector as a flux, the divergence
of a vector is the amount leaving the unit volume.
The vector product WA is
VVA = i(V 2 ^ 3 - V 3 ^ 2 ) + j(V 3 ^! - V^ 3 ) + k(V^ 2 - V^) = curl A. (21)
532 ELECTRICAL PAPERS.
The line-integral of A round a unit area equals the component of the
curl of A perpendicular to the area.
We may also have the scalar and vector products NV and VNV,
where the vector N is not differentiated. These operators, of course,
require a function to follow them on which to operate; the previous
qV. A of (16) illustrates.
The Laplacean operator is the scalar product V 2 or VV ; or
(22)
and an example of (13) is
WWA = V. VA - V 2 A, or curPA = V div A - V 2 A, ..... (23)
which is an important formula.
Other important formulae are the next three.
divPA = PdivA + AV.P, ........................ (24)
P being scalar. Here note that AV.P and AVP (the latter being the
scalar product of A and VP) are identical. This is not true when for P
we substitute a vector. Also
divVAB = BcurlA-AcurlB; .................... (25)
which is an example of (12), noting that both A and B have to be
differentiated. And
curlVAB = BV.A + AdivB-AV.B-BdivA ............ (26)
This is an example of (13).
9. When one vector D is a linear function of another vector E, that
is, connected by equations of the form
A =
(27)
in terms of the rectangular components, we denote this simply by
D = cE, ................................... (28)
where c is the linear operator. The conjugate function is given by
D' = c'E, ................................. (29)
where D' is got from D by exchanging c 12 and c 21 , etc. Should the nine
coefficients reduce to six by C 12 = c 21 , etc., D and D' are identical, or D
is a self-conjugate or symmetrical linear function of E.
But, in general, it is the sum of D and D' which is a symmetrical
function of E, and the difference is a simple vector-product. Thus
where c is a self-conjugate operator, and e is the vector given by
(31)
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 533
The important characteristic of a self-conjugate operator is
B^EA, or E lC() E 2 = E 2 c E l5 ............. (32)
where E x and E 2 are any two E's, and D I} D 2 , the corresponding D's.
But when there is not symmetry, the corresponding property is
E 1 D 2 = E 2 D(, or B 1 cB 2 - BjC^ ............... (33)
Of these operators we have three or four in electromagnetism con-
necting forces and fluxes, and three more connected with the stresses
and strains concerned. As it seems impossible to avoid the considera-
tion of rotational stresses in electromagnetism, and these are not usually
considered in works on elasticity, it will be desirable to briefly note
their peculiarities here, rather than later on.
On Stresses, irrotational and rotational, and their Activities.
10. Let P^ v be the vector stress on the N-plane, or the plane whose
unit normal is N. It is a linear function of N. This will fully specify
the stress on any plane. Thus, if P v P 2 , P 3 are the stresses on the
i, j, k planes, we shall have
[ (34)
Let, also, Q v be the conjugate stress ; then, similarly,
\ (35)
Q 3 = iP 13 +jP 23
Half the sum of the stresses P^ and Q, v is an ordinary irrotational
stress ; so that
P. V = 4> N + V N, Q^=< N-VeN, ............ (36)
where < is self-conjugate, and
2e = i(P 23 -P 32 )+j(P 31 -P 13 ) + k(P 12 -P 21 ) ........... (37)
Here 2 is the torque per unit volume arising from the stress P.
The translational force, F, per unit volume is (by inspection of a
unit cube)
F = V 1 P 1 + V 2 P 2 + V 3 P 3 ........................ (38)
= idivQ 1 +jdivQ 2 + kdivQ 3 ; ............ (39)
or, in terms of the self-conjugate stress and the torque,
F = (i div < i + j div $<$ +k div < k) - curl e, .......... (40)
where -curie is the translational force due to the rotational stress
alone, as in Sir W. Thomson's latest theory of the mechanics of an
"ether."*
* Mathematical and Physical Papers, vol. 3, Art. 99, p. 436.
534 ELECTRICAL PAPERS.
Next, let N be the unit-normal drawn outward from any closed
surface. Then
SP^SF, (41)
where the left summation extends over the surface and the right sum-
mation throughout the enclosed region. For
P J , = ^ 1 P 1 + JV 2 P 2 + JV 3 P 3 = i.NQ 1 +j.NQ 2 + k.NQ 3 ; (42)
so the well-known theorem of divergence gives immediately, by (39),
2P J = 2(idivQ 1 +jdivQ, + kdivQ s )= s 2F (43)
Next, as regards the equivalence of rotational effect of the surface-
stress to that of the internal forces and torques. Let r be the vector
distance from any fixed origin. Then VrF is the vector moment of a
force, F, at the end of the arm r. Another (not so immediate) appli-
cation of the divergence theorem gives
2VrP. v = 2VrF + 22e (44)
Thus, any distribution of stress, whether rotational or irrotational, may
be regarded as in equilibrium. Given any stress in a body, terminating
at its boundary, the body will be in equilibrium both as regards trans-
lation and rotation. Of course, the boundary discontinuity in the stress
has to be reckoned as the equivalent of internal divergence in the
appropriate manner. Or, more simply, let the stress fall off continuously
from the finite internal stress to zero through a thin surface-layer. We
then have a distribution of forces and torques in the surface-layer which
equilibrate the internal forces and torques.
To illustrate; we know that Maxwell arrived at a peculiar stress,
compounded of a tension parallel to a certain direction, and an equal
lateral pressure, which would account for the mechanical actions apparent
between electrified bodies ; and endeavoured similarly to determine the
stress in the interior of a magnetized body to harmonize with the similar
external magnetic stress of the simple type mentioned. This stress in
a magnetized body I believe to be thoroughly erroneous ; nevertheless,
so far as accounting for the forcive on a magnetized body is concerned,
it will, when properly carried out with due attention to surface-discon-
tinuity, answer perfectly well, not because it is the stress, but because
any stress would do the same, the only essential feature concerned being
the external stress in the air.
Here we may also note the very powerful nature of the stress-function,
considered merely as a mathematical engine, apart from physical reality.
For example, we may account for the forcive on a magnet in many
ways, of which the two most prominent are by means of forces on
imaginary magnetic matter, and by forces on imaginary electric currents,
in the magnet and on its surface. To prove the equivalence of these
two methods (and the many others) involves very complex surface-
and volume-integrations and transformations in the general case,
which may be all avoided by the use of the stress-function instead
of the forces.
11. Next as regards the activity of the stress P A and the equivalent
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 535
translational, distortional, and rotational activities. The activity of P^
is P v q per unit area, if q be the velocity. Here
P. v q = ft .NQ 1 + ?2 .NQ 2 + ?3 .NQ 3) .................. (45)
by (42) ; or, re- arranging,
P.vq = N( ?1 Q 1 -f ?2 Q 2 + (?3Q 3 ) = N2 ? Q = N ? Q 9 , ............. (46)
where Q 7 is the conjugate stress on the q-plane. That is, qty q or 2 Qg
is the negative of the vector flux of energy expressing the stress-activity.
For we choose P yx . so as to mean a pull when it is positive, and when
the stress P v works in the same sense with q, energy is transferred
against the motion, to the matter which is pulled.
The convergence of the energy-flux, or the divergence of ,
and here h will be the intrinsic force of magnetization, such that /xh
is the intensity of intrinsic magnetization. But I have shown that
when there is motion, another impressed term is required, viz., the
motional magnetic force
h = VDq, .................................. (88)
making the first circuital law become
curl(H-h -h) = J = C + D + q/> .................. (89)
Maxwell's other connection to form the equations of propagation is
made through his vector-potential A and scalar potential Mf. Finding
this method not practically workable, and also not sufficiently general,
I have introduced instead a companion equation to (89) in the form
-curl(E-e -e) = G = K + B + qo-, ............... (90)
where e expresses intrinsic force, and e is the motional electric force
given by
e = VqB, ................................ (91)
which is one of the terms in Maxwell's equation of electromotive force.
As for e , it includes not merely the force of intrinsic electrization,
the analogue of intrinsic magnetization, but also the sources of energy,
voltaic force, thermoelectric force, etc.
(89) and (90) are thus the working equations, with (88) and (91) in
case the medium moves; along with the linear relations before
mentioned, and the definitions of energy and waste of energy per unit
volume. The fictitious K and o- are useful in symmetrizing the equa-
tions, if for no other purpose.
Another way of writing the two equations of curl is by removing the
e and h terms to the right side. Let
curlh=j,
-curie =g, G + g = G .
Then (89) and (90) may be written
curl(H-h )= J = C
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 541
So far as circuitality of the current goes, the change is needless, and
still further complicates the make-up of the true current, supposed now
to be J . On the other hand, it is a simplification on the left side,
deriving the current from the force of the flux or of the field more
simply.
A question to be settled is whether J or J should be the true
current. There seems only one crucial test, viz., to find whether e J
or e J is the rate of supply of energy to the electromagnetic system by
an intrinsic force e . This requires, however, a full and rigorous
examination of all the fluxes of energy concerned.
The Electromagnetic Flux of Energy in a stationary Medium.
1 6. First let the medium be at rest, giving us the equations
curl(H-h ) = J = C + D, (94)
-curl(E-eo)=G = K + B (95)
Multiply (94) by (E - e ), and (95) by (H - h ), and add the results.
Thus,
(E - e ) J + (H - h c )G = (E - e ) curl (H - h ) - (H - h ) curl (E - e ),
which, by the formula (25), becomes
e J + h G = EJ + HG + div V(E - e )(H - h ) ;
or, by the use of (82), (83),
e J-fh G = +7+ J+divW, (96)
where the new vector W is given by
W=V(E-e )(H-h ) (97)
The form of (96) is quite explicit, and the interpretation sufficiently
clear. The left side indicates the rate of supply of energy from
intrinsic sources. Thus, (Q+ U+f) shows the rate of waste and of
storage of energy in the unit volume. The remainder, therefore,
indicates the rate at which energy is passed out from the unit volume ;
and the flux W represents the flux of energy necessitated by the
postulated localization of energy and its waste, when E and H are
connected in the manner shown by (94) and (95).
There might also be an independent circuital flux of energy, but,
being useless, it would be superfluous to bring it in.
The very important formula (97) was first discovered and interpreted
by Professor Poynting, and independently discovered and interpreted
a little later by myself in an extended form. It will be observed that
in my mode of proof above there is no limitation as to homogeneity or
isotropy as regards the permittivity, inductivity, and conductivity.
But c and //. should be symmetrical. On the other hand, k and g do
not require this limitation in deducing (97).*
* The method of treating Maxwell's electromagnetic scheme employed in the
text (first introduced in " Electromagnetic Induction and its Propagation," The.
Electrician, January 3, 1885, and later) may, perhaps, be appropriately termed the
542 ELECTRICAL PAPERS.
It is important to recognize that this flux of energy is not dependent
upon the translational motion of the medium, for it is assumed explicitly
to be at rest. The vector W cannot, therefore, be a flux of the kind
Q 9 before discussed, unless possibly it be merely a rotating stress that
is concerned.
The only dynamical analogy with which I am acquainted which
seems at all satisfactory is that furnished by Sir W. Thomson's theory
of a rotational ether. Take the case of e = 0, h = 0, k = 0, g = 0, and
c and //. constants, that is, pure ether uncontaminated by ordinary
matter. Then
curlH = cE, ................................ (98)
-curlE = yaH ................................. (99)
Now, let H be velocity, /A density; then, by (99), -curlE is the
translational force due to the stress, which is, therefore, a rotating
stress; thus,
P^ = VEN, Q^ V = VNE; .................. (100)
and 2E is the torque. The coefficient c represents the compliancy or
reciprocal of the quasi-rigidity. The kinetic energy |/*H 2 represents
the magnetic energy, and the potential energy of the rotation represents
the electric energy ; whilst the flux of energy is VEH. For the activity
of the torque is
and the translational activity is
-HcurlE.
Their sum is - div VEH,
making VEH the flux of energy.*
All attempts to construct an elastic-solid analogy with a distortional
stress fail to give satisfactory results, because the energy is wrongly
localized, and the flux of energy incorrect. Bearing this in mind, the
above analogy is at first sight very enticing. But when we come to
Duplex method, since its characteristics are the exhibition of the electric,
magnetic, and electromagnetic relations in a duplex form, symmetrical with
respect to the electric and magnetic sides. But it is not merely a method of
exhibiting the relations in a manner suitable to the subject, bringing to light
useful relations which were formerly hidden from view by the intervention of the
vector-potential and its parasites, but constitutes a method of working as well.
There are considerable difficulties in the way of the practical employment of
Maxwell's equations of propagation, even as they stand in his treatise. These
difficulties are greatly magnified when we proceed to more general cases, involving
heterogeneity and eolotropy and motion of the medium supporting the fluxes.
The duplex method supplies what is wanted. Potentials do not appear, at least
initially. They are regarded strictly as auxiliary functions which do not represent
any physical state of the medium. In special problems they may be of great
service for calculating purposes ; but in general investigations their avoidance
simplifies matters greatly. The state of the field is settled by and H, and these
are the primary objects of attention in the duplex system.
*This form of application of the rotating ether I gave in The Electrician,
January 23, 1891, p. 360.
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 543
remember that the d/dt in (98) and (99) should be d/cfy and find extra-
ordinary difficulty in extending the analogy to include the conduction
current, and also remember that the electromagnetic stress has to be
accounted for (in other words, the known mechanical forces), the per-
fection of the analogy, as far as it goes, becomes disheartening. It
would further seem, from the explicit assumption that q = in obtaining
W above, that no analogy of this kind can be sufficiently comprehensive
to form the basis of a physical theory. We must go altogether beyond
the elastic solid with the additional property of rotational elasticity. I
should mention, to avoid misconception, that Sir W. Thomson does not
push the analogy even so far as is done above, or give to //, and c the
same interpretation. The particular meaning here given to /A is that
assumed by Professor Lodge in his " Modern Views of Electricity," on
the ordinary elastic-solid theory, however. I have found it very con-
venient from its making the curl of the electric force be a Newtonian
force (per unit volume). When impressed electric force e produces
disturbances, their real source is, as I have shown, not the seat of e ,
but of curl e . So we may with facility translate problems in electro-
magnetic waves into elastic-solid problems by taking the electromagnetic
source to represent the mechanical source of motion, impressed New-
tonian force.
Examination of the Flux of Energy in a Moving Medium*, and Establishment
of the Measure of " True " Current.
17. Now pass to the more general case of a moving medium with
the equations
curlH^ curl(H-h -h) = J = C + D + q/3, (101)
- curlE 1 = -curl(E-e -e) = G = K + B + qo-, (102)
where Ej is, for brevity, what the force E of the flux becomes after
deducting the intrinsic and motional forces ; and similarly for H r
From these, in the same way as before, we deduce
(e + e)J + (h + h)G = EJ + HG + divVE 1 H 1 ; (103)
and it would seem at first sight to be the same case again, but with
impressed forces (e + e ) and (h + h ) instead of e and h , whilst the
Poyntirig flux requires us to reckon only Ej and Hj as the effective
electric and magnetic forces concerned in it.*
*It will be observed that the constant 4?r, which usually appears in the
electrical equations, is absent from the above investigations. This demands a
few words of explanation. The units employed in the text are rational units,
founded upon the principle of continuity in space of vector functions, and the
corresponding appropriate measure of discontinuity, viz. , by the amount of diver-
gence. In popular language, the unit pole sends out one line of force, in the
rational system, instead of 4?r lines, as in the irrational system. The effect of the
rationalization is to introduce 4?r into the formulae of central forces and potentials,
and to abolish the swarm of 47r's that appears in the practical formulas of the
practice of theory on Faraday- Max well lines, which receives its fullest and most
appropriate expression in the rational method. The rational system was explained
by me in The Electrician in 1882, and applied to the general theory of potentials
544 ELECTRICAL PAPERS.
But we must develop (Q+ U+f) plainly first. We have, by (86),
(87), used in (103),
e J + h G = E(C + D + q/)) + H(K + B + qo-)-(eJ + hG)+divVE 1 H 1 . (104)
Now here we have
(IDo)
Comparison of the third with the second form of (105) defines the
generalized meaning of c when c is not a mere scalar. Or thus,
= JM? + JcJStf + faJE* + 6.AE, + c, s E,E s + CaEiE* ...... ( 1 06)
representing the time-variation of U due to variation in the c's only.
Similarly f = HB - JH/iH = HB - f^ ................... (107)
with the equivalent meaning for p. generalized.
Using these in (104), we have the result
e J + h G = (Q + U+ T} + q(E/) + Ho-) + (JBcB + JH/1H)
-(eJ + hGJ + divVEjHj. (108)
Here we have, besides (Q+ U+T\ terms indicating the activity of a
and connected functions in 1883. (Reprint, vol. 1, p. 199, and later, especially
p. 262. ) I then returned to irrational formulas because I did not think, then, that
a reform of the units was practicable, partly on account of the labours of the B. A.
Committee on Electrical Units, and partly on account of the ignorance of, and
indifference to, theoretical matters which prevailed at that time. But the circum-
stances have greatly changed, and I do think a change is now practicable. There
has been great advance in the knowledge of the meaning of Maxwell's theory, and
a diffusion of this knowledge, not merely amongst scientific men, but amongst a
large body of practicians called into existence by the extension of the practical
applications of electricity. Electricity is becoming, not only a master science, but
also a very practical one. It is fitting, therefore, that learned traditions should
not be allowed to control matters too greatly, and that the units should be ration-
alized. To make a beginning. I am employing rational units throughout in my
work on " Electromagnetic Theory," commenced in The Electrician in January,
1891, and continued as fast as circumstances will permit; to be republished in
book form. In Section XVII. (October 16, 1891, p. 655) will be found stated
more fully the nature of the change proposed, and the reasons for it. I point out,
in conclusion, that as regards theoretical treatises and investigations, there is no
difficulty in the way, since the connection of the rational and irrational units may
be explained separately ; and I express the belief that when the merits of the
rational system are fully recognised, there will arise a demand for the rationaliza-
tion of the practical units. We are, in the opinion of men qualified to judge,
within a measurable distance of adopting the metric system in England. Surely
the smaller reform I advocate should precede this. To put the matter plainly, the
present system of units contains an absurdity running all through it of the same
nature as would exist in the metric system of common units were we to define the
unit area to be the area of a circle of unit diameter. The absurdity is only
different in being less obvious in the electrical case. It would not matter much if
it were not that electricity is a practical science.
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 545
translational force. Thus, E/> is the force on electrification p, and Eqp
its activity. Again,
so that we have c = -^ - qV . c,
ot
.(109)
and, similarly, (L = J- - qV . /*,
the generalized meaning of which is indicated by
^2 + JEcE = -iE(qV.c)E= -qVZ7 c ; (110)
where, in terms of scalar products involving E and D,
-qV/" c = - J(E.qV.D-D.qV.E) (Ill)
This is also the activity of a translational force. Similarly,
'dT
(112)
is the activity of a translational force. Then again,
-(eJ + hG) = - JVqB-GVDq = q(VJB + VDG) ....... (113)
expresses a translational activity. Using them all in (108), it becomes
. (114)
It is clear that we should make the factor of q be the complete trans-
lational force. But that has to be found ; and it is equally clear that,
although we appear to have exhausted all the terms at disposal, the
factor of q in (114) is not the complete force, because there is no term
by which the force on intrinsically magnetized or electrized matter
could be exhibited. These involve e and h . But as we have
.................. (115)
a possible way of bringing them in is to add the left member and
subtract the right member of (115) from the right member of (114);
bringing the translational force to f, say, where
f=E/> + Ho--VZ7 c -V^ + V(J+j )B + VD(G + g ) ...... (116)
But there is still the right number of (115) to be accounted for. We
have
-div(Veh + Ve h) = ej + hg + e j+h g, .......... (117)
and, by using this in (114), through (115), (116), (117), we bring it to
e J + h G = (Q + U+ T) + fq - (e j + h g) + div (VE^ - Veh - Ve h)
+ |(^ c + ^); (US)
H.E.P. VOL. II. 2M
546 ELECTRICAL PAPERS.
or, transferring the e , h terms from the right to the left side,
= Q+i/ r +r+fq+div(VE 1 H 1 -Veh -Ve h)+^(C7 c +r,). (119)
Here we see that we have a correct form of activity equation, though it
may not be the correct form. Another form, equally probable, is to be
obtained by bringing in Yeh ; thus
div Veh = h curl e - e curl h = - (ej + hg) = q( VjB + VDg), (120)
which converts (119) to
e Jo+h A = Q+ ^7+r+Fq+div(VE 1 H l -Veh-Veh -Ve li)+|( U e +T fl \ (121)
where F is the translational force
P = B/) + H + V curl E.D-V*7 C + etc.], ............................ (138)
where the unwritten terms are the similar magnetic terms. This being
the N-component of F, the force itself is given by (122), as is necessary.
It is Vcurlh .B that expresses the translational force on intrinsically
magnetized matter, and this harmonizes with the fact that the flux B
due to any impressed force h depends solely upon curl h .
Also, it is - V^ that explains the forcive on elastically magnetized
matter, e.g., Faraday's motion of matter to or away from the places of
greatest intensity of the field, independent of its direction.
If S be the torque, it is given by
therefore S = VDE+ VBH ........................... (139)
But the matter is put more plainly by considering the convergence
of the stress flux of energy and dividing it into translational and other
parts. Thus
), ...(140)
where the terms following Fq express the sum of the distortional and
rotational activities.
Shorter Way of going from the Circuital Equations to the Fliu 1 of
Energy, Stresses, and Forces.
21. I have given the investigation in ^ 17 to 19 in the form in
which it occurred to me before I knew the precise nature of the results,
being uncertain as regards the true measure of current, the proper form
of the Poynting flux, and how it worked in harmony with the stress
flux of energy. -But knowing the results, a short demonstration may
be easily drawn up, though the former course is the most instructive.
Thus, start now from
on the assumptioual understanding that J and G are the currents which
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 551
make e J and h G the activities of e and h the intrinsic forces. Then
e J + h G = EJ + HG + divW, .................. (142)
where W - V(E - e )(H - h ) ; ...................... (143)
and we now assume this to be the proper form of the Poynting flux.
Now develop EJ and HG thus :
EJ + HG = E(C + D + q/> + curl h) + H(K + B + qo- - curl e), by (93) ;
= Q l + U +U, + Eqp + E curl VDq
+ Q 3 + T+ T u + Hqo- + H curl VBq, by (88) and (91) ;
= C>i + U+ U c + Eq/> + E(D div q + qy. D - q div D - Dv.q)
= Q l + U+ U c + 2 U div q + E. qV. D - E . Dv. q
+ magnetic terms,
+ magnetic terms
Nowhere qy.Z7= JE.qV.D + JD.qV.E,
so that the terms in the third pair of brackets in (144) represent
with the generalized meaning before explained. So finally
EJ + HG = g+6* + r+divq(Z7+r) + ^(f7 c + r /i )
+ (^divq-E.DV.q) + (rdivq-H.BV.q), ...(145)
which brings (142) to
ivq-H.BV.q), (146)
which has to be interpreted in accordance with the principle of con-
tinuity of energy.
Use the form (127), first, however, eliminating Fq by means of
which brings (127) to
e J + h l) G = ( t )+^+r+div{W + q(/: T + r)}-i:Qv^ + Sa; (147)
and now, by comparison of (147) with (146) we see that
-Sa + 2QV( 7 = (E.DV.q-^divq)- 3 5' : + (H.BV.q-rdivq)-^; i ; (148)
ot ot
from which, when /a and c do not change intrinsically, we conclude that
552 . ELECTRICAL PAPERS.
as before. In this method we lose sight altogether of the translational
force which formed so prominent an object in the former method as a
guide.
Some JRemarks on Hertz's Investigation relating to the Stresses.
22. Variations of c and /*, in the same portion of matter may occur
in different ways, and altogether independently of the strain-variations.
Equation (146) shows how their influence affects the energy transforma-
tions ; but if we consider only such changes as depend on the strain,
i.e., the small changes of value which /x and c undergo as the strain
changes, we may express them by thirty -six new coefficients each (there
being six distortion elements, and six elements in //,, and six in c), and
so reduce the expressions for 'dUJ'dt and 'dT^/'dt in (148) to the form
suitable for exhibiting the corresponding change in Q iV and in the stress
function P^. As is usual in such cases of secondary corrections, the
magnitude of the resulting formula is out of all proportion to the
importance of the correction-terms in relation to the primary formula
to which they are added.
Professor H. Hertz* has considered this question, and also refers to
von Helmholtz's previous investigation relating to a fluid. The c and /*
can then only depend on the density, or on the compression, so that a
single coefficient takes the place of the thirty -six. But I cannot quite
follow Hertz's stress investigation. First, I would remark that in
developing the expression for the distortional (plus rotational) activity,
he assumes that all the coefficients of the spin vanish identically ; this
is done in order to make the stress be of the irrotational type. But it
may easily be seen that the assumption is inadmissible by examining
its consequence, for which we need only take the case of c and //. intrin-
sically constant. By (139) we see that it makes S = 0, and therefore
(since the electric and magnetic stresses are separable), VHB = 0, and
VED = ; that is, it produces directional identity of the force E and
the flux D, and of the force H and the flux B. This means isotropy,
and, therefore, breaks down the investigation so far as the eolotropic
application, with six /* and six c coefficients, goes. Abolish the assump-
tion made, and the stress will become that used by me above.
Another point deserving of close attention in Hertz's investigation,
relates to the principle to be followed in deducing the stress from the
electromagnetic equations. Translating into my notation it would
appear to amount to this, the a priori assumption that the quantity
-, l( r "> < 150 >
where v indicates the volume of a moving unit element undergoing
distortion, may be taken to represent the distortional (plus rotational)
activity of the magnetic stress. Similarly as regards the electric stress.
Expanding (150) we obtain
* Wiedemann's Annalen, v. 41, p. 369.
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 553
Now the second circuital law (90) may be written
-e ) = K + + (Bdivq-BV.q) ........... (152)
Here ignore e , K, and ignore the curl of the electric force, and we obtain,
by using (152) in (151),
H.BV.q-HBdivq + rdivq-^ = H.BV.q-rdivq-?^, (153)
ot ot
which represents the distortional activity (my form, not equating to
zero the coefficients of curl q in its development). We can, therefore,
derive the magnetic stress in the manner indicated, that is, from (150),
with the special meaning of 3B/3J later stated, and the ignorations or
nullifications.
In a similar manner, from the first circuital law (89), which may be
written
-DV.q), ........... (154)
we can, by ignoring the conduction-current and the curl of the mag-
netic force, obtain
, .............. (155)
which represents the distortional activity of the electric stress.
The difficulty here seems to me to make it evident a priori that (150),
with the special reckoning of 3B/d, should represent the distortional
activity (plus rotational understood) ; this interesting property should,
perhaps, rather be derived from the magnetic stress when obtained by
a safe method. The same remark applies to the electric stress. Also,
in (150) to (155) we overlook the Poynting flux. I am not sure how
far this is intentional on Professor Hertz's part, but its neglect does
not seem to give a sufficiently comprehensive view of the subject.
The complete expansion of the magnetic distortional activity is, in
fact,
r-HG ; ...... (156)
and similarly, that of the electric stress is
tf-EJc, ...... (157)
It is the last term of (156) and the last term of (157), together,
which bring in the Poynting flux. Thus, adding these equations,
(158)
where (E J + HG ) = (e J + h G ) - div W ; .............. (159)
and so we come round to the equation of activity again, in the form
(146), by using (159) in (158).
554 ELECTRICAL PAPERS.
Modified Form of Stress-Vector, and Application to the Surface separating
two Regions.
23. The electromagnetic stress, P v of (149) and (136) may be put
into another interesting form. We may write it
NH.B) ...... (160)
Now, ND is the surface equivalent of div D and NB of div B ; whilst
VNE and VNH are the surface equivalents of curl E and curl H. We
may, therefore, write
P. v = (Ep' + VDG') + (Hcr / + VJ / B), ............. (161)
and this is the force, reckoned as a pull, on unit area of the surface
whose normal is N. Here the accented letters are the surface equiva-
lents of the same quantities unaccented, which have reference to unit
volume.
Comparing with (122) we see that the type is preserved, except as
regards the terms in F due to variation of c and ^ in space. That is,
the stress is represented in (161) as the translational force, due to E
and H, on the fictitious electrification, magnetification, electric current,
and magnetic current produced by imagining E and H to terminate at
the surface across which' P v is the stress.
The coefficient J which occurs in (161) is understandable by sup-
posing the fictitious quantities ("matter" and "current") to be distri-
buted uniformly within a very thin layer, so that the forces E and H
which act upon them do not then terminate quite abruptly, but fall off
gradually through the layer from their full values on one side to zero
on the other. The mean values of E and H through the layer, that is,
JE and ^H, are thus the effective electric and magnetic forces on the
layer as a whole, per unit volume-density of matter or current ; or JE
and JH per unit surface-density when the layer is indefinitely reduced
in thickness.
Considering the electric field only, the quantities concerned are
electrification and magnetic current. In the magnetic field only they
are magnetification and electric current. Imagine the medium divided
into two regions A and B, of which A is internal, B external, and let N
be the unit normal from
the surface into the
external region. The
mechanical action be-
tween the two regions
is fully represented by
the stress P^ v over their
interface, and the for-
cive of B upon A is
fully represented by the
E and H in B acting upon
the fictitious matter and
current produced on the boundary of B, on the assumption that E and
H terminate there. If the normal and P A - be drawn the other way,
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 555
thus negativing them both, as well as the fictitious matter and current
on the interface, then it is the forcive of A on B that is represented by
the action of E and H in A on the new interfacial matter and current.
That is, the E and H in the region A may be done away with alto-
gether, because their abolition will immediately introduce the fictitious
matter and current equivalent, so far as B is concerned, to the influence
of the region A. Similarly E and H in B may be abolished without
altering them in A. And, generally, any portion of the medium may
be taken by itself and regarded as being subjected to an equilibrating
system of forces, when treated as a rigid body.
24. When c and /x do not vary in space, we do away with the forces
- %E' 2 Vc and - J-fl^V/*, and make the form of the surface and volume
translational forces agree. We may then regard every element of p or
of o- as a source sending out from itself displacement and induction
isotropically, and every element of J or Gr as causing induction or
displacement according to Ampere's rule for electric current and its
analogue for magnetic current. Thus
(163)
where r x is a unit vector drawn from the infinitesimal unit volume in
the summation to the point at distance r where E or H is reckoned.
Or, introducing potentials,
, .................. (165)
These apply to the whole medium, or to any portion of the same,
with, in the latter case, the surface matter and current included, there
being no E or H outside the region, whilst within it E and H are the
same as due to the matter and current in the whole region (" matter,"
p and cr; "current," J and G-). But there is no known general method
of finding the potentials when c and p vary.
We may also divide E and H into two parts each, say Ej and Hj due
to matter and current in the region A, and E. 7 , H. 7 due to matter and
current in the region B surrounding it, determinable in the isotropic
homogeneous case by the above formulas. Then we may ignore Ej
and Hj in estimating the forcive on the matter and current in the region
A; thus,
^(H^ + VJ^ + ^E^ + VDA), .............. (166)
where o- 1 = div B x = div B, and J l = curl Hj = curl H in region A, is the
resultant force on the region A, and
-(H^ + VJ.BJ + SCE^ + VDjG,,) ................ (167)
is the resultant force on the region B ; the resultant force on A due to
556 ELECTRICAL PAPERS.
its own E and H being zero, and similarly for B. These resultant forces
are equal and opposite, and so are the equivalent surface-integrals
2(HX+VJ{B 2 ) + 2(E^ + VD 2 GO, ................ (168)
and SfHjoJ + VJ^B 1 ) + 2(E^ + VD 1 GO, ................ (169)
taken over the interface. The quantity summed is that part of the
stress-vector, P v , which depends upon products of the H of one region
and the B of the other, etc. Thus, for the magnetic stress only,
+ (H 2 .B 2 N - N. JH 2 B 2 ) + (H^N - N.JH^), (170)
and it is the terms in the second and fourth brackets (which, be it
observed, are not equal) which together make up the magnetic part of
(168) and (169) or their negatives, according to the direction taken for
the normal; that is, since H 1 B 2 = H 2 B 1 ,
O = 2(HX + VJ{B 2 ) = 2(H ^ making n negative. When
ja 2 = 0, the repulsion is
m
when /A 2 = oo, it is turned into an attraction of equal amount.
Similarly, if we consider the attraction to be the resultant force
between m and the interfacial matter cr, we shall get the same result by
the quantity summed (over the interface) being o- x normal component
of magnetic force due to matter m in a medium of unit inductivity, or
the normal component of induction due to m in its own medium.
For this is
ma - m
(203 ) again.
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 567
Another way is to calculate the variation of energy made by displac-
ing either the pole m or the /x 2 mass. The potential energy is expressed
by
J(P+^)m = JPm + |ZPo-/>t, ................. (205)
where P = m/lir^r and p = 2 o-/4?rr, the potentials of matter m/ft, and cr,
where r is the distance from m or from o- to the point where P and p
are reckoned.
The value of the second part in (205), depending upon o-, comes to
- m*
'
and its rate of decrease with respect to a expresses the repulsion
between the pole and the /x 2 region. This gives (203) again.
A fourth way is by means of the gwwi-electromagnetic force on
fictitious interfacial electric current, instead of matter, the current
being circular about the axis of symmetry AB. The formula for the
attraction is
2 V curl B. B , .......................... (207)
if R be the radial magnetic force from m in its own medium, tensor
m/47r/i 1 r 2 . Here the curl of B is represented by the interfacial discon-
tinuity in the tangential induction, or
zm
Also the tangential component of R is mx/fa^r 3 . Therefore the
repulsion is
(208)
as before, equation (203). This method (207) is analogous to (204).
37. There are several other ways of representing the attraction,
employing fictitious matter and current; but now let us change the
method, and observe how the attraction between the magnetic pole and
the iron mass is accounted for by a stress-distribution, and its space-
variation. The best stress is the third, equation (188), 31. Applying
this, we have simply a tension of magnitude J/*jA f "^i m tne nrst
medium and \p. 2 R% = T z in the second, parallel to B x and B 2 respec-
tively, each combined with an equal lateral pressure, so that the tensor
of the stress-vector is constant.
But, so far as the attraction is concerned, we may ignore the stress
in the second medium altogether, and consider it as the 2 P^ of the
stress-vector in the first medium over the surface of the second medium.
The tangential component summed has zero resultant; the attraction
is therefore the sum of the normal components, or ST^cos 20 P where
6 l is the angle between R x and the normal. This is the same as
568 ELECTRICAL PAPERS.
2 %p^(R$ - R%), if R x and R T are the normal and tangential components
of R x j or
which on evaluation gives the required result (203).
But this method does not give the true distribution of translational
force due to the stresses. In the first medium there is no translational
force, except on the magnet. Nor is there any translational force in
the second /* 2 medium. But at the interface, where /x changes, there is
the force - J72 2 V/* per unit volume, and this is represented by the
stress-difference at the interface. It is easily seen that the tangential
stress-difference is zero, because
(210)
and both the normal induction and the tangential magnetic force are
continuous. The real force is, therefore, the difference of the normal
components of the stress-vectors, and is, therefore, normal to the inter-
face. This we could conclude from the expression - ^R^Vp. But
since the resultant of the interfacial stress in the second medium is
zero, we need not reckon it, so far as the attraction of the pole is con-
cerned. The normal traction on the interface, due to both stresses, is
of amount
................. (211 >
per unit area. Summed up, it gives (203) again.
That (211) properly represents the force - J72 2 V/* when fi is discon-
tinuous, we may also verify by supposing ft to vary continuously in a
very thin layer, and then proceed to the limit.
The change from an attraction to a repulsion as p 2 changes from
being greater to being less than /*j, depends upon the relative import-
ance of the tensions parallel to the magnetic force and the lateral
pressures operative at different parts of the interface. In the extreme
case of /x 2 = 0, we have Rj tangential, with, therefore, a pressure every-
where. For the other extreme, R T is normal, and there is a pull on
the second medium everywhere. When /* 2 is finite there is a certain
circular area on the interface within which the translational force due
to the stress in the medium containing the pole m is towards that
medium, whilst outside it the force is the other way. But when both
stresses are allowed for, we see that when /A 2 >/>t 1 the pull is towards
the first medium in all parts of the interface, and that this becomes a
push in all parts when ^ > p 2 .
A Definite Stress only obtainable by Kinetic Consideration of the Circuital
Equations and Storage and Flux of Energy.
38. We see that the stress considered in the last paragraph gives a
rationally intelligible interpretation of the attraction or repulsion. The
same may be said of other stresses than that chosen. But the use of
ON THE; FORCES IN THE ELECTROMAGNETIC FIELD. 569
Maxwell's stress, or any stress leading to a force on inductively mag-
netized matter as this stress does, leads us into great difficulties. By
(198) we see that there is first a bodily force on the whole of the /u, 9
medium, because it is magnetized, unless ^ 2 = ^ When summed up,
the resultant does not give the required attraction. For, secondly, the
/*! medium is also magnetized, unless /^ = 1, and there is a bodily force
throughout the whole of it. When this is summed up (not counting
the force on the magnet), its resultant added on to the former resultant
still does not make up the attraction (i.e., equivalently, the force on the
magnet). For, thirdly, the stress is discontinuous at the interface
(though not in the same manner as in the last paragraph). The
resultant of this stress-discontinuity, added on to the former resultants,
makes up the required attraction. It is unnecessary to give the details
relating to so improbable a system of force.
Our preference must naturally be for a more simple system, such as
the previously considered stress. But there is really no decisive settle-
ment possible from the theoretical statical standpoint, and nothing
short of actual experimental determination of the strains produced and
their exhaustive analysis would be sufficient to determine the proper
stress-function. But when the subject is attacked from the dynamical
standpoint, the indeterminateness disappears. From the two circuital
laws of variable states of electric and magnetic force in a moving
medium, combined with certain distributions of stored energy, we are
led to just one stress-vector, viz. (136). It is, in the magnetic case, the
same as (188): that is, it reduces to the latter when the medium is
kept at rest, so that J and G- become J and G.
It is of the simple type in case of isotropy (constant tensor), but is a
rotational stress in general, as indeed are all the statically probable
stresses that suggest themselves. The translational force due to it
being divisible conveniently into (a), the electromagnetic force on
electric current, (6), the ditto on the fictitious electric current taking
the place of intrinsic magnetization, (c), force depending upon space-
variation of p ; we see that the really striking part is (b). Of all the
various ways of representing the forcive on an intrinsic magnet it is
the most extreme. The magnetic " matter " does not enter into it, nor
does the distribution of magnetization ; it is where the intrinsic force
h has curl that the translational force operates, usually on the sides of
a magnet. From actual experiments with bar-magnets, needles, etc.,
one would naturally prefer to regard the polar regions as the seat of
translational force. But the equivalent forcive 2j B has one striking
recommendation (apart from the dynamical method of deducing it),
viz., that the induction of an intrinsic magnet is determined by curl h ,
not by h itself; and this, I have shown, is true when h is imagined to
vary, the whole varying states of the fluxes B, D, C due to impressed
force being determined by the curls of e and h , which are the sources
of the disturbances (though not of the energy).
The rotational peculiarity in eolotropic substances does not seem to
be a very formidable objection. Are they not solid ?
As regards the assumed constancy of p, a more complete theory
570 ELECTRICAL PAPERS.
must, to be correct, reduce to one assuming constancy of /*, because, as
Lord Rayleigh* has shown, the assumed law has a limited range of
validity, and is therefore justifiable as a preparation for more complete
views. Theoretical requirements are not identical with those of the
practical engineer.
But, for quite other reasons, the dynamically determined stress might
be entirely wrong. Electric and magnetic "force" and their energies
are facts. But it is the total of the energies in concrete cases that
should be regarded as the facts, rather than their distribution ; for
example, that, as Sir W. Thomson proved, the " mechanical value " of
a simple closed current C is %LC' 2 , where L is the inductance of the
circuit (coefficient of electromagnetic capacity), rather than that its
distribution in space is given by JHB per unit volume. Other distri-
butions may give the same total amount of energy. For example, the
energy of distortion of an elastic solid may be expressed in terms of the
square of the rotation and the square of the expansion, if its boundary
be held at rest; but this does not correctly localize the energy. If,
then, we choose some other distribution of the energy for the same dis-
placement and induction, -we should find quite a different flux of energy.
But I have not succeeded in making any other arrangement than Max-
well's work practically, or without an immediate introduction of great
obscurities. Perhaps the least certain part of Maxwell's scheme, as
modified by myself, is the estimation of magnetic energy as ^HB in
intrinsic magnets, as well as outside them, that is, by JB/^~ 1 B, however
B may be caused. Yet, only in this way are thoroughly consistent
results apparently obtainable when the electromagnetic field is con-
sidered comprehensively and dynamically.
APPENDIX.
Received June 27, 1891.
Extension of the Kinetic Method of arriving at the Stresses to cases of Non-
linear Connection between the Electric and Magnetic Forces and the
Fluxes. Preservation of Type of the Flux of Energy Formula.
39. It may be worth while to give the results to which we
are led regarding the stress and flux of energy when the restriction
of simple proportionality between "forces" and "fluxes," electric
and magnetic respectively, is removed. The course to be followed,
to obtain an interpretable form of the equation of activity, is
sufficiently clear in the light of the experience gained in the case
of proportionality.
First assume that the two circuital laws (89) and (90), or the two in
(93), hold good generally, without any initially stated relation between
the electric force E and its associated fluxes C and D, or between the
* Phil. Mag., January, 1887.
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 571
magnetic force H and its associated fluxes K and B. When written in
the form most convenient for the present application, these laws are
~ + (Ddivq-DV.q), ......... (212)
q-Bv.q) .......... (213)
Now derive the equation of activity in the manner previously followed,
and arrange it in the particular form />^ ^>
e J + h G + conv Y(E - e )(H - h ) = (EC + HK) + ( E?? + H^) '
\ (jv (Jv J \ (~\
+ (E.DV.q-EDdivq) + (H.BV.q-HBdivq), (214)
which will best facilitate interpretation.
Although independent of the relation between E and D. etc., of course
the dimensions must be suitably chosen so that this equation may really
represent activity per unit volume in every term.
Now, guided by the previous investigation, we can assume that
(e J + h G ) represents the rate of supply of energy from intrinsic
sources, and also that V(E - e )(H - h ? ), which is a flux of energy
independent of q, is the correct form in general. Also, if there be
no other intrinsic sources of energy than e , h , and no other fluxes of
energy besides that just mentioned except the convective flux and that
due to the stress, the equation of activity should be representable by
(e J + h G ) + conv [V(E - e )(H - h ) + q(Z7+ T)]
U+ ) + Fq +
U+T) + 2Qvq, ........................... ............. (215)
where Q is the conjugate of the stress-vector, F the translational force,
and Q, U 1 and T the rate of waste and the stored energies, whatever
they may be.
Comparing with the preceding equation (214), we see that we require
+ [E.DV.q-(ED- tf)divq] + [H.Bv.q-(HB -T)divq]. (216)
Now assume that there is no waste of energy except by conduction; then
(217a)
Also assume that =E, =H .................. .(217ft)
at ot ot ot
These imply that the relation between E and D is, for the same particle
of matter, an invariable one, and that the stored electric energy is
(218)
'0
where E is a function of D. Similarly,
(219)
= T
572 ELECTRICAL PAPERS.
expresses the stored magnetic energy, and H must be a definite function
of B.
On these assumptions, (216) reduces to
^QV^ = [E.DV.q-(ED- Z7)div q] + [H.BV.q- (HB - J) div q], (220)
from which the stress-vector follows, namely,
P^=[E.DN-N(ED- CT)] + [H.BN-N(HB-r)] (221)
Or, P, v =(VDVEN + NZ7) + (VBVHN + Nr) (222)
Thus, in case of isotropy, the stress is a tension U parallel to E com-
bined with a lateral pressure (ED - U] ; and a tension T parallel to H
combined with a lateral pressure (HB - T).
The corresponding translational force is
F = EdivD + DV.E-V(ED- /) + HdivB + BV.H - V(HB-7), (223)
which it is unnecessary to put in terms of the currents.
Exchange E and D, and H and B, in (221) or (222) to obtain the
conjugate vector Q v ; from which we obtain the flux of energy due to
the stress,
- $Q g = D.Eq - q(ED - U) +B.Hq - q(HB - T)
= VEVDq + VHVBq + q(CT+T), (224)
or -^Q^VeH + VEh + q^+T 7 ), (225)
where e and h, are the motional electric and magnetic forces, of the same
form as before, (88) and (91) ; so that the complete form of the equation
of activity, showing the fluxes of energy and their convergence, is
e J + h G + conv [ V(B - e )(H - h ) + q( 17+ T)]
- conv [VeH + VEh + q( U+ T)] = Fq + (Q + U+ f), (226)
where F has the above meaning.
There is thus a remarkable preservation of form as compared with
the corresponding formulae when there is proportionality between force
and flux. For we produce harmony by means of a Poynting flux of
identical expression, and a flux due to the stress which is also of
identical expression, although U and T now have a more general
meaning, of course.*
* As the investigation in this Appendix has some pretensions to generality, we
should try to settle the amount it is fairly entitled to. No objection is likely to
be raised to the use of the circuital equations (212), (213), with the restriction of
strict proportionality between and H and the fluxes D and B, or C and K entirely
removed ; nor to the estimation of J and G as the " true " currents ; nor to the
use of the same form of flux of electromagnetic energy when the medium is
stationary. For these things are obviously suggested by the preceding investi-
gations, and their justification is in their being found to continue to work, which
is the case. But the use in the text of language appropriate to linear functions,
which arose from the notation, etc., being the same as before, is unjustifiable.
We may, however, remove this misuse of language, and make the equation (226),
showing the flux of energy, rest entirely upon the two circuital equations. In
fact, if we substitute in (226) the relations (217a), (2176), it becomes merely a
f writing (214).
to (21 la), (217&) that we should look for limitations. As regards
particular way of writing (214).
It is, therefore,
ON THE FORCES IN THE ELECTROMAGNETIC FIELD. 573
Example of the above, and Remarks on Intrinsic, Magnetization when
there is Hysteresis.
40. In the stress-vector itself (for either the electric or the magnetic
stress) the relative magnitude of the tension and the lateral pressure
varies unless the curve connecting the force and the induction be a
straight line. Thus, if the curve be of the type shown in the first
figure, the shaded area will
represent the stored energy
and the tension, and the
remainder of the rectangle
will represent the lateral
pressure. They are equal
when H is small ; later on
the pressure preponderates,
and more and more so the
bigger H becomes.
But if the curve be of the
type shown in the second
figure, then, after initial
equality, the tension pre-
ponderates ; though, later
on, when H is very big, the
pressure preponderates.
To obtain an idea of the
effect, take the concrete
example of an infinitely
long rod, uniformly axially
inductized by a steady
current in an overlapping
solenoid, and consider the
forcive on the rod. Here
both H and B are axial or longitudinal ; and so, by equation (223), the
translational force would be a normal force on the surface of the rod,
acting outwards, of amount
per unit area ; this being the excess of the lateral pressure in the rod
over |#o^>o. the lateral pressure just outside it.
In case of proportionality of force to flux, the first pressure is \RB,
and, if there is no intrinsic magnetization, H and H are equal. The
(217a), there does not seem to be any limitation necessary. That is, there is no
kind of relation imposed between E and C, and H and K. This seems to arise
merely from Q meaning energy wasted for good, and having no further entry into
the system. But as regards (2176), the case is different. For it seems necessary,
in order to exclude terms corresponding to E(dcfdt)E and H(3/i/30H in the linear
theory, when there is rotation, that E and D should be parallel, and likewise H and
B. At any rate, if such terms be allowed, some modification may be required in
the subsequent reckoning of the mechanical force. In other respects, it is merely
implied by (2176) that E and D are definitely connected, likewise H and B, so that
there is no waste of energy other than that expressed by Q.
574 ELECTRICAL PAPERS.
outward force is therefore positive for paramagnetic, and negative for
diamagnetic substances, and the result would be lateral expansion or
contraction, since the infinite length would prevent elongation.
But if the curve in the rod be of the type of the first figure, and the
straight line ac be the air-curve to correspond, it is the area abc that now
represents the outward force per unit area when the magnetic force has the
value ad. If the straight line can cross the curve ab, we see that by suffi-
ciently increasing H we can make the external air-pressure preponderate,
so that the rod, after initially expanding, would end by contracting.
If the rod be a ring of large diameter compared with its thickness, the
forcive would be approximately the same, viz., an outward surface-force
equal to the difference of the lateral pressures in the rod and air. The
result would then be elongation, with final retraction when the external
pressure came to exceed the internal.
Bid well found a phenomenon of this kind in iron, but it does not seem
possible that the above supposititious case is capable of explaining it,
though of course the true explanation may be in some respects of a
similar nature. But the circumstances are not the same as those
supposed. The assumption of a definite connection between H and B,
and elastic storage of the energy T, is very inadequate to represent the
facts of magnetization of iron, save within a small range.
Magneticians usually plot the curve connecting H-h and B } not
between H and B, or which would be the same, between H-h and
B - B , where B is the intrinsic magnetization. Now when an iron
ring is subjected to a given gaussage (or magnetomotive force), going
through a sequence of values, there is no definite curve connecting
Hh and J5, on account of the intrinsic magnetization. But, with
proper allowance for A , it might be that the resulting curve connecting
H and B in a given specimen would be approximately definite, at any
rate, far more so than that connecting H-h Q and B. Granting perfect
definiteness, however, there is still insufficient information to make a
theory. The energy put into iron is not wholly stored ; that is, in
increasing the coil-current we increase B as well as B, and in doing so
dissipate energy ; but although we know, by Ewing's experiments, the
amount of waste in cyclical changes, it is not so clear what the rate of
waste is at a given moment. There is also the further peculiarity that
the energy of the intrinsic magnetization at a given moment, though
apparently locked up, and really locked up temporarily, however loosely
it may be secured, is not wholly irrecoverable, but comes into play
again when H is reversed. Now it may be that the energy of the
intrinsic magnetization plays, in relation to the stress, an entirely
different part from that of the elastic magnetization. It is easy to make
up formulae to express special phenomena, but very difficult to make a
comprehensive theory.
But in any case, apart from the obscurities connected with iron, it is
desirable to be apologetic in making any application of Maxwell's
stresses or similar ones to practice when the actual strains produced are
in question, bearing in mind the difficulty of interpreting and harmonizing
with Maxwell's theory the results of Kerr, Quincke, and others.
THE POSITION OF 4* IN ELECTROMAGNETIC UNITS. 575
LIII. THE POSITION OF 4?r IN ELECTROMAGNETIC UNITS.
[Mature, July 28, 1892, p. 292.]
THERE is, I believe, a growing body of opinion that the present system
of electric and magnetic units is inconvenient in practice, by reason of
the occurrence of 4?r as a factor in the specification of quantities which
have no obvious relation with circles or spheres.
It is felt that the number of lines from a pole should be ra rather
than the present 47rra, that "ampere turns" is better than 471-71(7, that
the electromotive intensity outside a charged body might be a- instead
of 4;r(r, and similar changes of that sort; see, for instance, Mr. Williams's
recent paper to the Physical Society.
Mr. Heaviside, in his articles in The Electrician and elsewhere, has
strongly emphasized the importance of the change and the simplifi-
cation that can thereby be made.
In theoretical investigations there seems some probability that the
simplified formulae may come to be adopted
/A being written instead of 4^, and k instead of -^ ;
but the question is whether it is or is not too late to incorporate the
practical outcome of such a change into the units employed by electrical
engineers.
For myself I am impressed with the extreme difficulty of now
making any change in the ohm, the volt, etc., even though it be only a
numerical change ; but in order to find out what practical proposal the
supporters of the redistribution of 4?r had in their mind, I wrote to
Mr. Heaviside to inquire. His reply I enclose ; and would merely say
further that in all probability the general question of units will come
up at Edinburgh for discussion.
OLIVER J. LODGE.
My dear Lodge, I am glad to hear that the question of rational
electrical units will be noticed at Edinburgh if not thoroughly dis-
cussed. It is, in my opinion, a very important question, which must,
sooner or later, come to a head and lead to a thoroughgoing reform.
Electricity is becoming not only a master science, but also a very
practical science. Its units should therefore be settled upon a sound
and philosophical basis. I do not refer to practical details, which may
be varied from time to time (Acts of Parliament notwithstanding), but
to the fundamental principles concerned.
If we were to define the unit area to be the area of a circle of unit
diameter, or the unit volume to be the volume of a sphere of unit
diameter, we could, on such a basis, construct a consistent system of
units. But the area of a rectangle or the volume of a parallelepiped
would involve the quantity TT, and various derived formulae would
576 ELECTRICAL PAPERS.
possess the same peculiarity. No one would deny that such a system
was an absurdly irrational one.
I maintain that the system of electrical units in present use is founded
upon a similar irrationality, which pervades it from top to bottom.
How this has happened, and how to cure the evil, I have considered in
my papers first in 1882-83, when, however, I thought it was hopeless
to expect a thorough reform; and again in 1891, when I have, in my
" Electromagnetic Theory," adopted rational units from the beginning,
pointing out their connection with the common irrational units sepa-
rately, after giving a general outline of electrical theory in terms of the
rational.
Now, presuming provisionally that the first and second stages to
Salvation (the Awakening and Repentance) have been safely passed
through, which is, however, not at all certain at the present time, the
question arises, How proceed to the third stage, Reformation 1 Theo-
retically this is quite easy, as it merely means working with rational
formulae instead of irrational ; and theoretical papers and treatises may,
with great advantage, be done in rational formulae at once, and irre-
spective of the reform of the practical units. But taking a far-sighted
view of the matter, it is, I think, very desirable that the practical units
themselves should be rationalized as speedily as may be. This must
involve some temporary inconvenience, the prospect of which, unfortu-
nately, is an encouragement to shirk a duty; as is, likewise, the
common feeling of respect for the labours of our predecessors. But
the duty we owe to our followers, to lighten their labours permanently,
should be paramount. This is the main reason why I attach so much
importance to the matter; it is not merely one of abstract scientific
interest, but of practical and enduring significance ; for the evils of the
present system will, if it continue, go on multiplying with every
advance in the science and its applications.
Apart from the size of the units of length, mass, and time, and of
the dimensions of the electrical quantities, we have the following
relations between the rational and irrati Dnal units of voltage V, electric
current (7, resistance R, inductance L, permittance S, electric charge Q,
electric force E, magnetic force JJ, induction B. Let # 2 stand for 47r,
and let the suffixes r and t mean rational and irrational (or ordinary).
Also let the presence of square brackets signify that the "absolute"
unit is referred to. Then we have
[ft]
The next question is, what multiples of these units we should take to
make the practical units. In accordance with your request I give my
ideas on the subject, premising, however, that I think there is no
finality in things of this sort.
First, if we let the rational practical units be the same multiples
THE POSITION OF 4ir IN ELECTROMAGNETIC UNITS. 577
of the "absolute" rational units as the present practical units are
of their absolute progenitors, then we would have (if we adopt the
centimetre, gramme, and second, and the convention that />t = l in
ether)
[Jt r ] x 10 9 = new ohm =x 2 times old.
[L r ~\ x 10 = new mac =x 2
[S r ] x 10~ 9 = new farad = ar 2
[C r ] x 10~ 1 = new amp =or 1
[KJxlO 8 = new volt =x
10 7 ergs = new joule =old joule.
10" ergs per sec. = new watt =old watt.
I do not, however, think it at all desirable that the new units should
follow on the same rules as the old, and consider that the following
system is preferable :
y.2
[ r ] x 10 8 = new ohm = ~~- x old ohm.
x 2
[L r ] x 10 8 = new mac =-xoldmac.
I '] x 10- s = new farad = 15 x old farad.
x 2
[6V] x 1 = new amp = x old amp.
[T r ] x 10 s =new volt = x x old volt.
10 8 ergs = new joule = 10 x old joule.
10 8 ergs per sec. = new watt = 10 x old watt.
It will be observed that this set of practical units makes the ohm, mac,
amp, volt, and the unit of elastance, or reciprocal of permittance, all
larger than the old ones, but not greatly larger, the multiplier varying
roughly from 1J to 3J.
What, however, I attach particular importance to is the use of one
power of 10 only, viz., 10 8 , in passing from the absolute to the practical
units instead of, as in the common system, no less than four powers,
10 1 , 10 7 , 10 8 , and 10 9 . I regard this peculiarity of the common system
as a needless and (in my experience) very vexatious complication. In
the 10 8 system I have described, this is done away with, and still the
practical electrical units keep pace fairly with the old ones. The
multiplication of the old joule and watt by 10 is, of course, a necessary
accompaniment. I do not see any objection to the change. Though
not important, it seems rather an improvement. (But transformations
of units are so treacherous, that I should wish the whole of the above
to be narrowly scrutinized.)
H.E.P. VOL. ii. 2o
578 ELECTRICAL PAPERS.
It is suggested to make 10 9 the multiplier throughout, and the
results are :
x 10 9 = new ohm =x- x old ohm.
[L r ] x 10 = new mac =#- x old mac.
[S r ] x 10- 9 = new farad = ar 2 x old farad.
[C r ] x 1 = new amp = x old amp.
[V r ] x 10 9 = new volt = IQx x old volt.
10 9 ergs = new joule = 10 2 x old joule.
10-' ergs p. sec. =new watt = 10' 2 x old watt.
But I think this system makes the ohm inconveniently big, and has
some other objections. But I do not want to dogmatize in these
matters of detail. Two things I would emphasize : First, rationalize
the units. Next, employ a single multiplier, as, for example, 10 s .
OLIVER HEAVISIDE.
PAIGNTON, DEVON, July 18, 1892.
CORRECTIONS. VOL. II.
p. 69, equation (516), change sign of last term from - to +, as in (73), p. 192.
p. 69, equation (526), change sign of last term from + to - , as in (72), p. 192,
and for )[ read )-[, to agree with (72), p. 192.
p. 316, equation (400), the lower limit should be 2 .
p. 355, last line, for 361 read 301.
p. 387, seventh line, for 153 read 393.
p. 400, second line, for fraction to read fraction of to.
INDEX.
Absorption, (1) 428, 432, 479, 480
Action at a distance, (2) 490
Activity, equations of, (1) 450, 521 ;
(2) 174, 535, 547, 572
mutual, (1) 522
Admittance, (2) 357
Ampere, theory of magnetism, (1) 181
electrodynamics, (1) 238, 282, 482,
559
Analogies, conduction, induction, and
displacement, (1) 472
magnetization and electrization, (1)
489
electric and magnetic (various), (1)
509-15
moving isotropic and stationary eolo-
tropic medium, (2) 499
induction in core and current in wire,
(2) 30, 57
waves along circuit and waves along
cord, (2) 349, 401
hydraulic, (1) 96
telegraph cable and inductized core,
(1)399
liquid in pipe and current in wire,
(2) 60, 182
Anglo-danish cable, unilateral effect,
(1), 61
speeds of working on, (1) 62
Arrival-curves on cables, (1) 50-1, 68,
72-4
calculation of, (1) 78-95, 125
in cores and wires, (1) 398 ; (2) 58
Atomic currents, (1) 490
Attenuation, (2) 120, 129, 166
tables of, (2) 346, 350
Ayrton and Perry, (1) 39, 337 ; (2) 245,
367, 486
Axioms of thermodynamics, (1) 487
Bain, (1) 138
Balances, true and false, (2) 100, 115
periodic, (2) 106
iron against copper, (2) 115
with the Christie, (2) 33-38, 256-292,
366
H.E.P. VOL. II.
2o2
Berliner, (1) 183
Bessel functions, (1) 173, 360, 387 ; (2)
48, 176, 445
different forms of, (2) 445
of any order, (2) 467
complementary, (2) 445, 467
in plane waves, (2) 477
in spherical waves, (2) 428
Bidwell, retraction of iron, (2) 574
Blaserna, oscillations, (1) 61
Blyth, arc microphone, (1) 182
Bosscha's corollaries, (1) 21
Bottomley, (2) 42, 113
Boundary functions, connection with
electrical distributions, (1) 552-6
Bridge (see Christie)
system of telephony, (2) 251
across circuit, effect of, (2) 123
Budde, (1) 328
Capacity (see Permittance)
Cardinal formula, (2) 501
Carnot, (1) 316, 486
Cartesian expansions, (2) 16
Cayley, A., (2)389
Characteristic function, (1)412-15; (2)
261, 371
degree of, (1) 540
Chemical contact force, (1) 337-42,
472
Christie balance, (2) 102, 256
of exact copies, (2) 257
of reduced copies, (2) 104, 258
conjugate conditions of, (2) 263
of self-induction, (2) 263
practical use of, (2) 265
peculiarities of, (2) 270
simple-periodic, (2) 106, 270
disturbance of, by metal, (2) 273
of resistance, permittance, and induct-
ance, (2) 280
of self and mutual inductance, (2) 107,
291
miscellaneous arrangements of, (2) 286
of thick wires, (2) 116
applied to telegraph circuit, (2) 105
580
ELECTRICAL PAPERS.
Circuital, (1) 279, 344, 435, etc.
distributions, (2) 470
law, first, (1) 443
law, second, (1) 447
equations, (1) 449; (2) 8, 174, 468,
497, 540, 541, 543, 571.
Clark, Latimer, (1)2
Clausius, (1) 179, 296, 316, 327, 487 ;
(2) 501
Closure of electric current, (1) 559
Coils with cores, combinations of, (1)
402-416
in parallel, equivalent to one, (2) 292
combinations of, with S.H. voltage,
(1)114
Compliancy, (2) 542
Condensers, in sequence, (1) 425
theory of signalling with, (1) 47, 53
theory of combination of shunted, (1)
536-42
Condenser, electromagnetic wave on dis-
charge of, (2) 465
Conductance, (1) 399; (2) 24, 125
Conduction and displacement (simul-
taneous), (1) 494-509
Conductors, diffusion of current in
(nature of), (2) 385
Conjugacy of conductors (conditions of),
(2) 259
Conjugate property, of normal systems,
(1) 81, 128, 390, 396, 401 ; (2) 53,
178, 202
general, (1) 143, 523
in electrical arrangements, (2) 205
Conjugate vector functions, (2) 19
Conservation of energy, (1) 291-303
Contact layers, (1) 342, 350
Convection-current, (2) 490-518
produces plane wave, (2) 493, 511
equatorial concentration, (2) 493, 496,
511
energy of, (2) 493, 505, 514
mutual energy of two point-charges,
(2) 507
general theory of, (2) 508
at speed greater than light, (2) 494,
496, 515
at speed less than light, (2) 495
equilibrium surfaces, (2) 514
charged straight line, (2) 516
charged plane, (2) 517
bidimensional solution, (2) 517
Convergence, (1) 210, 215
Coulomb, (1)278
Culley, R. S., (1) 62
Cumming, (1) 311
Curl (of a vector), (1) 199, 443
at a surface, (1) 200
inverted, (1) 220
of impressed forcive (source of dis-
turbances), (2) 60, 361
Current, a function of magnetic force,
(1) 198
straight, magnetic force of, (1) 198
true (Maxwell's), (1) 433
sheet, (1) 205, 227
elements, (2) 310, 501
in wires, magnetic theory, (2) 58, 181
Cycles in a mesh of conductors, (2) 108
Daniell'o cell, (1) 2
Davies, (2) 41
Deflection of wave, (2) 519
Deprez, Marcel, (1)238
Determinantal equation, (1) 415
and differential equations, (2) 261
Determinateness of distributions, (1)
497-506
Determination of potential from surface
value, (1) 553
Dielectric, (1) 433
moving, (2) 492
Diffusion of current in wires, (2) 44-61
Diffusion effect, (2) 274
nature of, (2) 385
conductive, (1) 384
Differentiation of vectors, (2) 531
Displacement, (1) 432, 475
circuital, (1)466
instantaneous vanishing of, (1) 534
persistence of, (2) 481
Dissipativity, (1), 431
Distortion, (2) 120, 166
of plane waves, (2) 482
abolition of, (2), 512
in telephony, causes of, (2) 347
Distortionless circuit, (2) 123-155
short theory of, (2) 307
with terminal short-circuit, (2) 131,
312
with terminal resistance, (2) 130
with terminal complete absorption,
(2)127,311
with terminal partial absorption, (2)
133-5, 312
best arrangement of, (2) 136, 323
in parallel, (2) 137
with intermediate resistance, (2), 138,
315
of different types, (2) 152
with variable speed of current, (2)
153, 316
with intermediate bridges, (2) 315
approximate, (2) 345
establishment of current in, (2) 313
Divergence of a vector, (1) 209, 444
of coefficients in normal systems, (1)
90, 530
Divided core, (1) 374
Divided iron equivalent to self-induction,
(2) 275
INDEX.
581
Division of discharge, (1) 106
Duplex method (electromagnetic), (1)
449, 542 ; (2) 172
Duplex telegraphy, Gintl's method, (1)
18; Frischen's, (1) 19; Eden's, (1)
21; Stearns', (1)21
by balancing batteries, (1) 22
by Bridge system, theory of sensitive-
ness, (1) 24
by differential system, theory of
sensitiveness, (1) 30
variations of balance in, (1) 33
with balanced capacity, (1) 25
Earth, as a return conductor, (1) 190
magnetic force of current in, (1) 224
currents, (1) 389
Edison, T. A., problem, (1) 34
etheric force, (1) 61 ; (2) 85
Effective resistance and inductance, or
conductance and permittance, (2)
357
Elastance, (1)512; (2) 125
Elastivity, (2) 125
Elastic solid (generalized), (2) 535-9
Electret, (2) 488
Electric energy, (1) 432, 466
various expressions for, (1) 506
Electrification in a conductor, (1) 476
Electric impulse, (1) 517
Electric connexions (summary), (2) 375
Electrization, (1) 488
Electromagnets, (1) 95
Electromagnetic force, from stress, (1)
545
Electromotive forces, method of com-
paring, (1)1
Electromagnetic field, (2) 251
flux of energy in, (2) 525, 541-3
equations of the, (2) 539
stress in the, (2) 548
force in the, (2) 546, 558
Electrostatic time-constant of circuit,
(2) 128
induction, (1) 117
Energy, electric, (1) 432
magnetic, (2) 434
mutual, of magnetic shells, (1) 234
of linear currents, (1) 235
of current systems, (1) 240
self, of current system, (1) 248
magnetic, localization of, (1) 248
minimum property of, (1) 251
transfer of, (1) 282, 434-41, 450; (2)
541-3, 571
Equal roots (in normal systems), (1)
529
Equilibrium surf aces in movingmedium,
(2) 514
Eolotropic potential function, (2) 499
Eolotropy in Ohm's law, (I) 280-90, 430
Equilibrium of stressed medium, (1)
547
of stress, (2) 534
Ether, (1) 420, 430, 433 ; (2) 525
gravitational function of, (2) 528
force in free, (2) 557
Euler, (1) 381
Evaluation of constants in normal sys-
tems, (1) 523-5, 529
Everett, (1) 179, 327
Ewing, (1)365; (2)275, 574
Extra-current, (1) 53-61
integral, (1) 121
False electrification, (1) 506
electric current, (1) 506, 512
magnetic current, (1) 509, 512
Faraday, (1) 195, 298, 447, etc.
Faults (leakage), theory of effect on
signalling, (1) 71-95.
Felici's balance, (2) 110
disturbed, theory, (2) 112
Fictitious matter and current on bound-
aries, (1) 549 ; (2) 554
Fitzgerald, G. F., (1) 467 ; (2) 394, 489,
492, 508, 524
Fleming, J. A., (2) 108, 488.
Fluids (electric), (2) 80, 486.
Forbes, (2) 403
Flux of energy (see Transfer)
Flux (initial) due to impressed force,
cancelled later, (2) 412
Force, electromagnetic, (1) 545; (2) 560
magneto-electric, (1) 545
on intrinsic magnets, (2) 550, 559
on inductively magnetized matter, (2)
550
(general) in electromagnetic field, (2)
546, 550, 569, 572
other forms of, got statically, (2)
561-3
between two regions, (2) 554
Forced vibrations of electromagnetic
systems (examples), (2) 233
Foucault currents, (2) 111, 113
Fourier, (1) 201, 333 ; (2) 387
series, to suit terminal conditions, ( 1 )
92, 123, 151 ; (2) 391
integrals, (2) 474 ; evaluation of, (2)
478
Fourier's theorem, extension of, (1) 154
Freedom, degrees of, in electrical com-
binations, (1) 540
Fresnel, (2) 1, 2, 3, 11, 12, 392, .VJI
Friction and electrification, (1) 475
Functions, Fourier's, (1) 151
Bessel's, (1) 173
Murphy's, (1) 176
Legendre's, (1) 177
582
ELECTRICAL PAPERS.
Functions
spherical zonal harmonic, (1) 229;
(2) 405
expansion in series, (1) 142-150; (2)
201, 233
Function of wires, (2) 486
of self-induction, (2) 489
Galvanometer, resistance of, for maxi-
mum magnetic force, (1) 12, 38
differential, for measuring small re-
sistances, (1) 13
differential, resistance of coils for
maximum effect, (1) 16
Generalization of resistance to pass from
characteristic function to differ-
ential equation, (1) 415
Gibbs, Willard, (1) 272 ; (2) 20, 528-9
Giltay, (2) 348
Glaisher, J. W. L., (2) 389
Glazebrook, (2) 521
Goethe, (1) 335
Granville, W. P., (2)496
Grassmann, (1) 272
Gravitation, (2) 527
Gray, Elisha, (2) 156
Green, (1) 555
Hamilton, Sir W. R., (1) 207; (2) 5,
528, 557
Hamilton's cubic, (2) 19
Hall effect, (1) 290
Heat, Joule's law, (1)490
developed in core, (1) 364
Heaviside, A. W., (2) 83, 145, 185, 251,
323
Hertz, H., (2) 444, 489, 490, 503, 523-4,
552-3
Helmholtz, von, (1) 282, 342, 344, 381 ;
(2) 552
Henry, Joseph, (1) 61
Hindrance, (2) 488
Hockin, C., (2) 246
Hughes, D. E., (1) 365-6 ; (2) 28-30, 35,
38, 101, 111, 169
Hydrokinetic analogy, (1) 275
Hysteresis, in telephone, (2) 158
outside mathematical theory, (2) 574
Identities, transcendental, (1) 88; (2)
245, 389, 445-6
Impedance, (1) 371 ; (2) 64, 125, 185
equality rule, (1) 99 ; (2) 143, 354
of a wire, (2) 165
of circuits, (2) 64
equivalent, of telegraph circuit, (2)
72, 341
reduced by inertia, (2) 65
Impedance-
reduced by compliancy, (2) 71
magnetic, of short lines, (2) 67
influence of displacement on, (2) 71-6
fluctuations with frequency, (2) 73,
345
ultimate form with great frequency,
(2)76
extended meaning of, (2) 371
Impressed forces, effect of, (1) 164; (2)
473
in dielectrics, (1) 471
Impulsive inductance and permittance,
(2) 359
inductance of telegraph circuit, (2)
368
E.M.F. generating spherical wave,
(2) 417
Inanity of ^, (2) 511
Index-surface, (2) 9
Inductance, (1) 354 ; (2) 28, 125
generalized, (2) 357
vanishing of, (2) 358
of straight wires, (1) 101 ; (2) 47
of cylinders, (2) 355
coils, (2) 37
of solenoid. (2) 277
(effective) of wires, (2) 64
(effective) of tubes, (2) 69, 192
ultimate form at great frequency, (2)
71
of iron and copper wires, (2) 261
of prisms, (2) 243
and permittance of lines, (2) 303
beneficial effect of, (2) 380, 393
increases amplitude, lessens distor-
tion, (2) 164, 308, 350
effect of increasing, (2) 121-3
of unclosed conductive circuit, (2) 502
of Hertz oscillator, (2) 503
Inductivity, (2) 28, 125
a constant with small forces, (2) 158
Induction, between parallel wires, (1)
116-141
in cores, (1) 353-416
balances with the Christie, (2) 33-38,
366
Inductize, (2) 40
Inductometer, (2) 110, 112, 167, 267
calibration of, (2) 110, 267
with equal coils, (2) 268
Inequalities between wires, (2) 305,
337
Inertia (magnetic), (1) 96; (2) 60
Influence between distant circuits,
telephony by, (2) 237
Intermitter, (2) 272
Intermittences, not S.H. variations,
(2) 270
Iron, divided, (2) 111, 113, 158
Ironic insulators, (2) 123
INDEX.
583
Intrinsic magnetic force, (1) 454
magnetization, (1) 451
electric force, (1)489
electrization, (1) 489
Inversion of vector operators, (2) 22
Irrational units, origin of, (1) 199
Jenkin, Fleeming, (1) 46, 125, 417
Joubert, (1) 116
Joule, (1) 283, 294
Joule's law, (1) 301
Kerr, (2) 574
Kirchhoff, laws, (1)4
theory of telegraph, (2) 81, 191, 395
Kohlrausch, (2) 271
Kundt, (2) 486-7
Lacoine, Emile, (I) 2, 23
Lamb, (1) 382
Leakage, effect on propagation, (1) 53,
71, 138, 535; (2) 71, 122
quickening effect of, (2) 252
Lenz, (1) 281, 482
Leroux, (1) 325
Light, (2)311
electromagnetic theory of, (2) 392
Lightning discharges, (2) 486
Limiting distance of telephony, (2) 121,
347
Linear network, property of, (2) 294
Lodge, 0. J., (1) 416-24; (2) 41, 438,
444, 483, 486, 503, 524, 527, 575
Long-distance telephony, (2) 119, 147,
349
Loop circuits, (2) 303
as induction balances, (2) 334
Mac, (2) 167
Magnetic induction, Faraday's idea of,
(1) 279
conductivity, (1) 441 ; effect of, (2)
480, 483
current, (1) 441, 442, 520
energy, (1) 445-8 ; due to current, (1)
517-19
impulse, (1) 504
retentiveness, (2) 41
force, example of independence of
permeability, (1) 517
disturbances from Sun, (2) 122
energy of moving charges, (1) 446
Magnetization, molecular, (2) 39
Magnetoelectric force, (1) 545 ; (2) 498
Magnus, (1) 313
Mance, (2) 294
Manganese steel, (2) 113
Maximum heat, (1) 499
energy, (1) 499
Maxwell, ;>aWw,
gravitational stress, (1) 544
magnetic stress, (2) 563
naturalness of his views, (1) 478
sketch of his theory, (1) 429-451
Mayer, (1) 294
Mechanical forces on magnets, (1) 457
action between two regions, (1) 548-
558
force between magnets and currents,
(1)556
Michelson, (2) 520
Microphone, theory of, (1) 181
Minimum heat, (1) 303-9, 497
Momentum, magnetic, (1) 59, 120, 480
persistence of, (2) 142, 145, 320, 481
Morse instrument, (1) 20, 23, 33
Motion of sphere through liquid, (1)
276
Motional electric force, (1) 448, 497
magnetic force, (1) 446, 497
Motion of medium, effect of, (2) 497
Mutual inductance, decrease by in-
creasing inductivity, (2) 112, 288
Neumann, J., formula, (1) 236, 281 ;
(2) 501, 503
Newton, (1) 291, 335, etc.
Nomenclature, (2)23-28, 165-8, 302, 327
Normal systems, size of, (2) 206
cylindrical, (1) 385, 393
in heterogeneous telegraph circuits,
(2) 223
general electromagnetic, (1) 521-531
of displacement in conductors, (1)
533
in shunted condensers, (1) 539
of current in wires, (2) 46, 51, 54
Oersted, (1)282
Ohm's law, (1) 282-6, 429
theory of propagation in wires, (1)
286 ; (2) 77, 191
O'Kinealy, (1) 94
Orthogonality of electric and magnetic
forces, (2) 221
Oscillations, condenser and coil, (1) 106 ;
(2)84
on long circuits, (I) 57, 132; (2) 85
got by reducing inductance, (1) 536
Oscillator, permittance and inductance
of, (2) 503
Oscillatory E.M.F. on a telegraph line,
(2) 61-76
subsidence of charge of condenser,
(1)532
subsidence in normal systems, (1) 526
584
ELECTRICAL PAPERS.
Peltier effect. (1) 310
Penetration of current into wires, ('2)
30,32
Permanent magnetic field of telephone,
(2) 156
Permeability, (1) 434
Permeance, (1) 512; (2) 24
Permittance of wires overground, (1)
42-46 ; (2) 159
of wires in loop, (2) 329
Poggendorff, (1)2, 23
Poisson, (1) 279
Pole, dimensions of magnetic, (1) 179
Polar distributions, subsidence of, (2)
469
Potential, of scalars, (1) 202
of vectors, (1) 203
characteristic equation of, (1) 218
in relation to curl, (1) 219
in relation to impressed force, (1) 349
not physical, (1) 502
metaphysical nature of propagation
of, (2) 483, 490
of circular magnetic shell, (1) 229
energy of magnets, (1), 457
Poynting, (2) 93-96, 172, 489, 490, 521,
522, 525, 527, 541
Preece, (2) 119, 160, 165, 305, 367, 380,
488-9
Pressural wave, (2) 485
Prescott, (2) 156
Prisms, magnetic induction in, (2) 240
Propagation along a wire, (2) 62, 82
general equations of, (2) 87-91
along a wire with variable constants,
(1)142; (2)222
along parallel wires, (1) 130, 136, 140
Pyroelectricity, (1) 493
Quaternions, (1) 207, 271 ; (2) 3, 376,
528, 556
Quincke, (2) 574
Rational units, (1) 199, 263 ; (2) 543
Rational current elements, (2) 500, 508
mutual energy of, (2) 501, 507
Rayleigh, Lord, (1) 299, 333, 365 ; (2)
63, 101, 274, 277, 367, 405, 445, 570
Ray, in direction of flux of energy,
(2) 16
Reaction of core currents on coil, (1) 370
Reciprocity, (1) 62, 128
Received current on telegraph cii*cuit,
(2)62
Reis, (1) 181
Reluctance, (2) 125, 168
Reluctivity, (2) 125, 168
Reciprocal relation of permittance and
inductance, (2) 221
Resistance of telegraphic lines, (1) 42
insulation, (1) 42
of carbon contacts, (1) 181
of earth, (1) 193
balances, true and false, (2) 37
increased, of wires, (2) 30, 37
effective, of wires, (2) 64
at great frequency, (2) 71
terminal, (1) 67, 155
negative (equivalent to), (1) 91, 167
of tubes. (2) 69, 192
at great frequency, (2) 71
and inductance of wires, general
formulae, (2) 97, 278-9
ditto, induction longitudinal, (2) 99
table of increased, (2) 98
observation of increased, (2) 100
effective, of wires, balance, (2) 115
at front of a wave along a wire, (2) 436
Resistance operators, general, (2) 205,
355
elementary form of, (2) 356
S.H. form of, (2) 357
of telegraphic circuit, (2) 105
ditto, properties of, (2) 368
of infinitely long circuit, (2) 369
of distortionless circuit, (2), 370
in normal solutions, (2) 371
irrational, (2) 427
theorem relating to, (2) 373
spherical, (2) 439
cylindrical, (2) 447
Resistivity, (2) 24, 125
Resonance on telephone circuits, (2) 71,
73-76
Retardation, electrostatic, (1) 63
and permittance of looped wires, (2)
323
Roots, imaginary, (1) 89, 153, 159
Rotational property, (1) 289, 431, 451
Rowland, H. A., (1) 434; (2) 405
St. Venant, (2) 240
Salvation, (2) 576
Scalar product, (1) 431
Schwendler, (1)4
Seat of E.M.F., (1) 421
Seebeck, (1) 311, 314
Self-contained forced vibrations,
Plane, (2) 377
Spherical, (2) 365, 408, 419, 442
Cylindrical, (2) 365, 450, 454, 455,
458, 467
Self-induction, function of, (2) 396
Sensitiveness of Wheatstone's Bridge,
(1)4
table of, (1)11
Shunt, to differential galvanometer, (1)
17
to electromagnet, (1) 111
INDEX.
585
Siemens-Halske, duplex, (1) 19
Similar electrical systems, (2) 290
Slope of a scalar, (1) 212
Smith, Willoughby, (1) 47 ; (2) 28
Solutions, of electromagnetic equations,
(2) 469
distortionless, (2) 470
for plane waves in conducting di-
electric, (2) 473
Source of magnetic disturbances, (1)
425
Specific heat of electricity, (1) 313
Speculations, (1) 331-7
Specific capacity of conductors, (1) 495
Speed of current, (2) 121, 129
Spherical functions in plane waves, (2)
475
Stationary wave, (1) 548
Stokes, (2) 405, 538
formula for /,, (2) 467
Stresses (1) 542-558 ; (2) 533-574.
Stress vector, (1) 543 ; (2) 533, 572
force due to, (1) 544
torque due to, (1) 544 ; (2) 533
electric, (1) 545
Maxwellian, (1) 546; (2) 563
in plane waves, (1) 547
over surface, (1) 551, 554
rotational and irrotational, (2) 523
activity of, (2) 535
electromagnetic, (2) 549, 551
various kinds of, (2) 561-3
distortional and rotational activity,
(2) 535
statical indeterminateness of, (2) 558
Submarine cables, signalling on, (1) 47,
61, 71
Sumpner, (2) 367
Sun, long waves from, (2) 122, 392
Subsidence of induction in a core, (1)
398
of displacement in a conductor, (1) 533
of current in wires, (2) 49
of current in rectangular rods, (2) 243
Surface condition, (2) 170, 487
Surface conduction, (1) 440
Surface divergence, (1) 216
Sylvester, (2) 201
Tait, P. G., (1) 271, 324-5; (2) 3, 12,
91, 528
Tail of wave, (2) 124
growth of, (2) 318
positive, due to resistance, (2) 141,
318
negative, due to leakage, (2) 145, 320
general, due to both, (2) 150
Tangential continuity, (1) 505
Telegraphy, duplex, (1) 18-34
multiplex, (1)24
Telegraph lines, test for, (1) 41
circuits, classification of, (2) 340, 402
of low resistance, simplified theory,
(2) 343
nearly distortionless, (2) 345
periodic impressed force on, (2) 245
amplitude of received current on,
(2) 249, 400
with terminal apparatus, (2) 250,
401
Telephone, theory of, (2) 155
in induction balances, (2) 33
differential, (2) 33, 43
Telephony, conditions of good, (2) 121
improvement of, (2) 322
Temperature, absolute, (1) 317
Terminal conditions, theory of, (1), 144
conditions, treatment of, (2) 297
conditions, transcendental, (1) 169-72
arbitrary functions, (2) 208, 300
apparatus, effect of, (2) 353, 390, 400
condenser, (1) 85, 156
condenser and coil, (1) 157
induction coil and condenser, (1) 161
Thermodynamics, (1) 315-318, 481-488
Thermoelectric force, (1) 305-331, 441,
484
inversion, (1) 314
diagram, (1) 321
Theorem of divergence, (1) 209
of version, (1)211
of slope, (1)212
of normal systems, (2) 226
of electric and magnetic energy, (2)
360
of dependence of disturbances on
rotation, (2) 3(>1
Time-constants, (1) 57
Thompson, S. P., (1) 181 ; (2) 348, 492
Thomson, J. J., (2) 93, 396, 403, 405,
434, 443, 493, 495, 497, 505-7, 524,
558
Thomson, Sir W. (Lord Kelvin), passim
theory of telegraph, (1) 48, 74, 122,
286, 439 ; (2) 78, 191
thermodynamics, (1) 487
thermoelectricity, (1) 312, 319
magnetic energy, (1) 238
rotational effect, (1) 290
Volta force, (1) 417
sparking distance, (1) 298
Thomson effect, (1) 314
Transferability of impressed forces, (2)61
Transfer of energy, (1) 282, 378, 420;
(2) 174
in general, (2) 525-7
in stationary medium, (2) 541-2
in moving medium, (2) 546-7, 551, 572
along wires, (2) 95
circuital indeterminateness of, (2) 93
auxiliary inactive, (2) 94
586
ELECTRICAL PAPERS.
Transformer with conducting core, (2)
118
Transformation from ascending to
descending series, (2) 446
True current, Maxwell's, (1) 433
extended form, (2) 492, 497
criterion of, (2) 541, 547
expression for, in moving medium,
(2)561 m
Tube and wire coaxial, current longi-
tudinal, (2) 50-55
Tubes, coaxial, theory of, (2) 186, 208-15
Tumlirz, (2) 41
Tyndall, (1) 435
Units, rational and irrational, (1) 199,
262, 432 ; (2) 543, 576
names of, (2) 26
practical, multiplier for, (2) 577
Van Rysselberghe, (2) 250
Varley, C. F. , condenser patent, (1) 47
wave-bisector, (1) 63
gas resistance, (1) 286
Vectors, type for, (1) 199
scalar product of, (1) 431 ; (2) 5
circuital and polar, (1) 520
Vector, curl of a, (1) 199
potential of a, (1) 203
divergence of a, (1) 209, 215 ; (2) 5
function, division into circuital and
divergent parts, (1) 253
product, (1) 431 ; (2) 4
potential of magnetic current, (1)
467
Vector algebra, outline, (2) 4-8
fuller outline, (2) 528-33
to harmonize with Cartesian, (2) 3
Vector operators, (1) 430; (2) 6, 19,
532
conjugate property, (2) 533
differentiation of, (2) 544, 547-9, 562
Vector and scalar potential, insufficient
to specify state of field, (2) 173
Version, theorem of, (1) 211, 444 ; (2) 5
Velocity of electricity, (1) 435, 439 ; (2)
310, 393
of propagation of potential, (2) 484
of plane waves in eolo tropic medium,
(2) 1, 2, 3
Viscous fluid motion and conductive
diffusion, (1) 384
dissipation, (1) 382
Volta-force, (1) 337-42, 416-28
Voltage, transverse, (2) 189
Vortices (Maxwell's), (1) 333
Vorticity, (2) 363
Vortex line, circular, source of waves,
(2) 415
Waves of magnetic induction into cores,
(1) 361, 384
propagation of along wires, (1) 439 ;
(2)62
Wave-surface, duplex electromagnetic,
(2)15
features of, (2) 2
ellipsoidal, (2) 3
Fresnel's, (2) 1, 2
Waves, electromagnetic, (2) 375-520
generat-'on and propagation, (2) 377,
385
in conductors, with distortion re-
moved, (2) 378
in the P.O., (2)489
spherical, from moving charge, (2)
49
convective deflection of, (2) 519
infinite concentration of, (2) 465
reflected (solutions), (2) 387
Waves, plane, distorted, in conducting
medium, (2) 381
with distortion removed, (2) 379
general solution for, (2) 474
Fourier integrals for, (2) 474, 478
integration of differential equations
for, (2) 476
resulting from arbitrary initial states,
(2) 477
interpretation of distorted waves, (2)
479
Waves, spherical, in dielectric, (2) 402-
443
general, (2) 403
condensational, (2) 403
simplest type of, (2) 404
with conical boundaries, (2) 404-5
zonal harmonic, (2) 406
differential equation of, (2) 407
of first order ; generation of shell
wave, (2) 409
reflection at centre, (2) 410
magnetic energy constant, (2) 412
second order, (2) 413
from spherical sheet of radial force,
(2) 414
simply periodic, (2) 418, 443
Waves, spherical, in conductors, (2)
421
in conducting dielectrics, (2) 422
undistorted, (2) 425
general case, (2) 426
special solutions, (2) 427-436
effect of metal screens, (2) 440
effect of reflecting barriers, (2) 438
Waves, cylindrical, (2) 443-67
due to longitudinal impressed force
in thin tube, (2) 447
with two coaxial conducting tubes, (2)
449
effect of barrier on, (2) 451
INDEX.
587
Waves
separate action of two surface sources
of, (2) 453
from a vortex filament, (2) 456
from a filament of impressed force
(2) 460
from a finite cylinder of impressed
force, (2) 461
Webb, F. H., (2) 83, 329
Weber's hypothesis, (1)296, 435; (2) 191
Weber, H., (2) 28
Wheatstone's bridge, (1)3; (2) 256
automatic, (1) 52, 62, 63
velocity of electricity, (2) 395
alphabetical indicator (oscillations),
(1)59
Williams, W., (2) 575
Winter, G. K., (1) 53
Wires, propagation along, (2), 190
approximate equations, (2), 333
Wire*
S.H. waves along, (2) 195
resonance on, (2) 195
impedance fluctuations, (2) 196
practical working system of treating
propagation in terms of transverse
voltage and current, (2) 119
parallel, (2) 220
of varying resistance, etc., (2) 229
homogeneous, (2) 231
Wires and tubes, general equations, (2)
176
differential equations, (2) 179
normal systems, (2) 178, 180
magnetic theory of, (2) 181
S.H. voltage, solution, (2) 183
resistance operators of, (2) 188
Work done by impressed forces, (1)
462-5, 474
(double) of impressed force, (1) 456
END OF VOL. II.
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