the present article is to describe more recent work, and to discuss
experimental difficulties and methods of measurement.

1. *Mechanism of Conduction.*—Conduction of heat implies
transmission by contact from one body to another or between
contiguous particles of the same body, but does not include
transference of heat by the motion of masses or streams of matter
from one place to another. This is termed *convection*, and is most
important in the case of liquids and gases owing to their mobility.
Conduction, however, is generally understood to include diffusion
of heat in fluids due to the agitation of the ultimate molecules,
which is really molecular convection. It also includes diffusion of
heat by internal radiation, which must occur in transparent
substances. In measuring conduction of heat in fluids, it is
possible to some extent to eliminate the effects of molar convection
or mixing, but it would not be possible to distinguish
between diffusion, or internal radiation, and conduction. Some
writers have supposed that the ultimate atoms are conductors,
and that heat is transferred through them when they are in
contact. This, however, is merely transferring the properties of
matter in bulk to its molecules. It is much more probable that
heat is really the kinetic energy of motion of the molecules, and
is passed on from one to another by collisions. Further, if
we adopt W. Weber’s hypothesis of electric atoms, capable of
diffusing through metallic bodies and conductors of electricity,
but capable of vibration only in non-conductors, it is possible
that the ultimate mechanism of conduction may be reduced in all
cases to that of diffusion in metallic bodies or internal radiation
in dielectrics. The high conductivity of metals is then explained
by the small mass and high velocity of diffusion of these electric
atoms. Assuming the kinetic energy of an electric atom at any
temperature to be equal to that of a gaseous molecule, its
velocity, on Sir J. J. Thomson’s estimate of the mass, must
be upwards of forty times that of the hydrogen molecule.

2. *Law of Conduction.*—The experimental law of conduction,
which forms the basis of the mathematical theory, was established
in a qualitative manner by Fourier and the early experimentalists.
Although it is seldom explicitly stated as an experimental law,
it should really be regarded in this light, and may be briefly
worded as follows: “*The rate of transmission of heat by conduction*
*is proportional to the temperature gradient.*”

The “rate of transmission of heat” is here understood to
mean the quantity of heat transferred in unit time through unit
area of cross-section of the substance, the unit area being taken
perpendicular to the lines of flow. It is clear that the quantity
transferred in any case must be jointly proportional to the area
and the time. The “gradient of temperature” is the fall of
temperature in degrees per unit length along the lines of flow. The
*thermal conductivity* of the substance is the constant ratio of the
rate of transmission to the temperature gradient. To take the
simple case of the “wall” or flat plate considered by Fourier for
the definition of thermal conductivity, suppose that a quantity of
heat Q passes in the time T through an area A of a plate of
conductivity *k* and thickness *x*, the sides of which are constantly
maintained at temperatures θ′ and θ″. The rate of transmission
of heat is Q/AT, and the temperature gradient, supposed uniform,
is (θ′−θ″)/*x*, so that the law of conduction leads at once to the
equation

*k*(θ′ −θ″)/

*x*=

*kd*θ/

*dx*.(1)

This relation applies accurately to the case of the steady flow
of heat in parallel straight lines through a homogeneous and
isotropic solid, the isothermal surfaces, or surfaces of equal
temperature, being planes perpendicular to the lines of flow.
If the flow is steady, and the temperature of each point of the
body invariable, the rate of transmission must be everywhere the
same. If the gradient is not uniform, its value may be denoted by
*d*θ/*dx*. In the steady state, the product *kd*θ/*dx* must be constant,
or the gradient must vary inversely as the conductivity, if the
latter is a function of θ or *x*. One of the simplest illustrations of
the rectilinear flow of heat is the steady outflow through the upper
strata of the earth’s crust, which may be considered practically
plane in this connexion. This outflow of heat necessitates a
rise of temperature with increase of depth. The corresponding
gradient is of the order of 1° C. in 100 ft., but varies inversely with
the conductivity of the strata at different depths.

3. *Variable State.*—A different type of problem is presented
in those cases in which the temperature at each point varies
with the time, as is the case near the surface of the soil with
variations in the external conditions between day and night or
summer and winter. The flow of heat may still be linear if the
horizontal layers of the soil are of uniform composition, but the
quantity flowing through each layer is no longer the same. Part
of the heat is used up in changing the temperature of the successive
layers. In this case it is generally more convenient to
consider as unit of heat the thermal capacity *c* of unit volume,
or that quantity which would produce a rise of one degree of
temperature in unit volume of the soil or substance considered.
If Q is expressed in terms of this unit in equation (1), it is necessary
to divide by *c*, or to replace *k* on the right-hand side by the
ratio *k*/*c*. This ratio determines the rate of diffusion of temperature,
and is called the *thermometric conductivity* or, more shortly,
the *diffusivity*. The velocity of propagation of temperature
waves will be the same under similar conditions in two substances
which possess the same diffusivity, although they may differ
in conductivity.

4. *Emissivity.*—Fourier defined another constant expressing
the rate of loss of heat at a bounding surface per degree of difference
of temperature between the surface of the body and its
surroundings. This he called the *external conductivity*, but the
term *emissivity* is more convenient. Taking Newton’s law of
cooling that the rate of loss of heat is simply proportional to
the excess of temperature, the emissivity would be independent
of the temperature. This is generally assumed to be the case
in mathematical problems, but the assumption is admissible
only in rough work, or if the temperature difference is small.
The emissivity really depends on every variety of condition,
such as the size, shape and position of the surface, as well as on
its nature; it varies with the rate of cooling, as well as with
the temperature excess, and it is generally so difficult to calculate,
or to treat in any simple manner, that it forms the greatest
source of uncertainty in all experimental investigations in which
it occurs.

5. *Experimental Methods.*—Measurements of thermal conductivity
present peculiar difficulties on account of the variety
of quantities to be observed, the slowness of the process of
conduction, the impossibility of isolating a quantity of heat,
and the difficulty of exactly realizing the theoretical conditions
of the problem. The most important methods may be classified
roughly under three heads—(1) Steady Flow, (2) Variable Flow,
(3) Electrical. The methods of the first class may be further
subdivided according to the form of apparatus employed. The
following are some of the special cases which have been utilized
experimentally:—

6. *The* “*Wall*” *or Plate Method.*—This method endeavours to
realize the conditions of equation (1), namely, uniform rectilinear
flow. Theoretically this requires an infinite plate, or a perfect
heat insulator, so that the lateral flow can be prevented or rendered
negligible. This condition can generally be satisfied with sufficient
approximation with plates of reasonable dimensions. To find the
conductivity, it is necessary to measure all the quantities which
occur in equation (1) to a similar order of accuracy. The area A
from which the heat is collected need not be the whole surface of the
plate, but a measured central area where the flow is most nearly
uniform. This variety is known as the “Guard-Ring” method, but
it is generally rather difficult to determine the effective area of the
ring. There is little difficulty in measuring the time of flow, provided
that it is not too short. The measurement of the temperature
gradient in the plate generally presents the greatest difficulties. If
the plate is thin, it is necessary to measure the thickness with great
care, and it is necessary to assume that the temperatures of the
surfaces are the same as those of the media with which they are in
contact, since there is no room to insert thermometers in the plate
itself. This assumption does not present serious errors in the case of
bad conductors, such as glass or wood, but has given rise to large
mistakes in the case of metals. The conductivities of thin slices
of crystals have been measured by C. H. Lees (*Phil. Trans.*, 1892)
by pressing them between plane amalgamated surfaces of metal.
This gives very good contact, and the conductivity of the metal
being more than 100 times that of the crystal, the temperature of
the surface is determinate.