measuring the diffusivity of a metal, since the conditions may be
widely varied and the correction for external loss of heat can be
made comparatively small. Owing, however, to the laborious nature
of the observations and reductions, the method does not appear to
have been seriously applied since its first invention, except in one
solitary instance by the writer to the case of cast-iron (fig. 2). The
equation of the method is the same as that for the linear flow with
the addition of a small term representing the radiation loss.

The heat per second gained by conduction by an element *dx* of the
bar, of conductivity *k* and cross section *q*, at a point where the
gradient is *d*θ/*dx*, may be written *qk*(*d*^{2}θ/*dx*^{2})*dx*. This is equal to
the product of the thermal capacity of the element, *cqdx*, by the
rate of rise of temperature *d*θ/*dt*, together with the heat lost per
second at the external surface, which may be written *hp*θ*dx*, if *p* is
the perimeter of the bar, and 'h* the heat loss per second per degree*
excess of temperature θ above the surrounding medium. We thus
obtain the differential equation

*qk*(

*d*

^{2}θ/

*dx*) =

*cdq*θ/

*dt*+

*hp*θ,

which is satisfied by terms of the type

*e*

^{−ax}sin (2π

*nt*−

*bx*),

where *a*^{2}−*b*^{2} =*hp*/*qk*, and *ab*=π*nc*/*k*.

The rate of diminution of amplitude expressed by the coefficient *a*
in the index of the exponential is here greater than the coefficient
*b* expressing the retardation of phase by a small term depending
on the emissivity *h*. If *h*=0, *a*=*b*= √(π*nc*/*k*), as in the case of
propagation of waves in the soil.

The apparatus of fig. 2 was designed for this method, and may
serve to illustrate it. The steam pressure in the heater may be
periodically varied by the gauge in such a manner as to produce an
approximately simple harmonic oscillation of temperature at the
hot end, while the cool end is kept at a steady temperature. The
amplitudes and phases of the temperature waves at different points
are observed by taking readings of the thermometers at regular
intervals. In using mercury thermometers, it is best, as in the
apparatus figured, to work on a large scale (4-in. bar) with waves
of slow period, about 1 to 2 hours. Ångström endeavoured to find
the variation of conductivity by this method, but he assumed *c* to
be the same for two different bars, and made no allowance for its
variation with temperature. He thus found nearly the same rate
of variation for the thermal as for the electric conductivity. His
final results for copper and iron were as follows:—

*k*=0.982 (1–0.00152 θ) assuming

*c*=.84476.

Iron,

*k*=0.1988 (1–0.00287 θ)”

*c*=.88620.

Ångström’s value for iron, when corrected for obvious numerical
errors, and for the probable variation of *c*, becomes—

*k*=0.164 (1–0.0013 θ),

but this is very doubtful as *c* was not measured.

The experiments on cast-iron with the apparatus of fig. 2 were varied by taking three different periods, 60, 90 and 120 minutes, and two distances, 6 in. and 12 in., between the thermometers compared. In some experiments the bar was lagged with 1 in. of asbestos, but in others it was bare, the heat-loss being thus increased fourfold. In no case did this correction exceed 7 %. The extreme divergence of the resulting values of the diffusivity, including eight independent series of measurements on different days, was less than 1 %. Observations were taken at mean temperatures of 102° C. and 54°C., with the following results:-

*k*/

*c*=.1296,

*c*=.858,

*k*=.1113.

””54°C.,

*k*/

*c*=.1392,

*c*=.823,

*k*=.1144.

The variation of *c* was determined by a special series of experiments.
No allowance was made for the variation of density with temperature,
or for the variation of the distance between the thermometers, owing
to the expansion of the bar. Although this correction should be
made if the definition were strictly followed, it is more convenient
in practice to include the small effect of linear expansion in the
temperature-coefficient in the case of solid bodies.

17. *Lorenz’s Method.*— F. Neumann, H. Weber, L. Lorenz and
others have employed similar methods, depending on the observation
of the rate of change of temperature at certain points of bars, rings,
cylinders, cubes or spheres. Some of these results have been widely
quoted, but they are far from consistent, and it may be doubted
whether the difficulties of observing rapidly varying temperatures
have been duly appreciated in many cases. From an experimental
point of view the most ingenious and complete method was that of
Lorenz (*Wied. Ann.* xiii. p. 422, 1881). He deduced the variations
of the mean temperature of a section of a bar from the sum S of the
E.M.F.’s of a number of couples, inserted at suitable equal intervals
*l* and connected in series. The difference of the temperature
gradients D/*l* at the ends of the section was simultaneously obtained
from the difference D of the readings of a pair of couples at either end
connected in opposition. The external heat-loss was eliminated by
comparing observations taken at the same mean temperatures
during heating and during cooling, assuming that the rate of loss of
heat *f*(S) would be the same in the two cases. Lorenz thus obtained
the equations:—

Heating, *qk* D/*l*=*cql d*S/*dt*+*f*(S).

Cooling, *qk* D′/*l*=*cql d*S′/*dt*′+*f*(S′).

Whence *k* = c*l*^{2}(*d*S/*dt*−*d*S′/*dl*)/(D−D′).

It may be questioned whether this assumption was justifiable,
since the rate of change and the distribution of temperature were
quite different in the two cases, in addition to the sign of the change
itself. The chief difficulty, as usual, was the determination of the
gradient, which depended on a difference of potential of the order
of 20 microvolts between two junctions inserted in small holes 2 cms.
apart in a bar 1.5 cms. in diameter. It was also tacitly assumed
that the thermo-electric power of the couples for the gradient was
the same as that of the couples for the mean temperature, although
the temperatures were different. This might give rise to constant
errors in the results. Owing to the difficulty of measuring the
gradient, the order of divergence of individual observations averaged
2 or 3 %, but occasionally reached 5 or 10 %. The thermal conductivity
was determined in the neighbourhood of 20° C. with a
water jacket, and near 110° C. by the use of a steam jacket. The
conductivity of the same bars was independently determined by the
method of Forbes, employing an ingenious formula for the heat-loss
in place of Newton’s law. The results of this method differ 2 or 3 %
(in one case nearly 15 %) from the preceding, but it is probably less
accurate. The thermal capacity and electrical conductivity were
measured at various temperatures on the same specimens of metal.
Owing to the completeness of the recorded data, and the great experimental
skill with which the research was conducted, the results
are probably among the most valuable hitherto available. One
important result, which might be regarded as established by this
work, was that the ratio *k*/*k*′ of the thermal to the electrical conductivity,
though nearly constant for the good conductors at any
one temperature such as 0° C., increased with rise of temperature
nearly in proportion to the absolute temperature. The value found
for this ratio at 0° C. approximated to 1500 C.G.S. for the best
conductors, but increased to 1800 or 2000 for bad conductors like
German-silver and antimony. It is clear, however, that this relation
cannot be generally true, for the cast-iron mentioned in the last
section had a specific resistance of 112,000 C.G.S. at 100° C., which
would make the ratio *k*/*k*′ = 12,500. The increase of resistance with
temperature was also very small, so that the ratio varied very little
with temperature.

18. *Electrical Methods.*—There are two electrical methods
which have been recently applied to the measurement of the
conductivity of metals, (*a*) the resistance method, devised by
Callendar, and applied by him, and also by R. O. King and J. D.
Duncan, (*b*) the thermo-electric method, devised by Kohlrausch,
and applied by W. Jaeger and H. Dieselhorst. Both methods
depend on the observation of the steady distribution of temperature
in a bar or wire heated by an electric current. The
advantage is that the quantities of heat are measured directly in
absolute measure, in terms of the current, and that the results are
independent of a knowledge of the specific heat. Incidentally it
is possible to regulate the heat supply more perfectly than in
other methods.

(*a*) In the practice of the resistance method, both ends of a short
bar are kept at a steady temperature by means of solid copper
blocks provided with a water circulation, and the whole is surrounded
by a jacket at the same temperature, which is taken as the
zero of reference. The bar is heated by a steady electric current,
which may be adjusted so that the external loss of heat from the
surface of the bar is compensated by the increase of resistance of
the bar with rise of temperature. In this case the curve representing
the distribution of temperature is a parabola, and the conductivity
*k* is deduced from the mean rise of temperature (R−R^{0})/*a*R^{0} by
observing the increase of resistance R−R^{0} of the bar, and the
current C. It is also necessary to measure the cross-section *q*, the
length *l*, and the temperature-coefficient *a* for the range of the
experiment.

In the general case the distribution of temperature is observed
by means of a number of potential leads. The differential equation
for the distribution of temperature in this case includes the majority
of the methods already considered, and may be stated as follows.
The heat generated by the current C at a point *x* where the temperature-excess
is θ is equal per unit length and time (*t*) to that lost by
conduction −d(*qkd*θ/*dx*)/*dx*, and by radiation *hp*θ (emissivity *h*,
perimeter *p*), together with that employed in raising the temperature
*qcd*θ/*dt*, and absorbed by the Thomson effect *s*C*d*θ/*dx*. We thus
obtain the equation—

^{2}R

_{0}(1+

*a*θ)/

*l*=−

*d*(

*qkd*θ/

*dx*)/

*dx*+

*hp*θ+

*qcd*θ/

*dt*+

*s*C

*d*θ/

*dx*. (8)

If C=0, this is the equation of Ångström’s method. If *h* also is
zero, it becomes the equation of variable flow in the soil. If *d*θ/*dt*=0,
the equation represents the corresponding cases of steady flow. In
the electrical method, observations of the variable flow are useful
for finding the value of *c* for the specimen, but are not otherwise
required. The last term, representing the Thomson effect, is eliminated
in the case of a bar cooled at *both* ends, since it is opposite in
the two halves, but may be determined by observing the resistance
of each half separately. If the current C is chosen so that C^{2}R_{0}*a*=*hpl*,
the external heat-loss is compensated by the variation of resistance