with temperature. In this case the solution of the equation reduces to the form

*x*(

*l*−

*x*)C

^{2}R

_{0}/2

*lqk*.(9)

By a property of the parabola, the mean temperature is ⅔rds of the maximum temperature, we have therefore

_{0})/

*a*R

_{0}=

*l*C

^{2}R

_{0}/12

*qk*,(10)

which gives the conductivity directly in terms of the quantities
actually observed. If the dimensions of the bar are suitably chosen,
the distribution of temperature is always very nearly parabolic,
so that it is not necessary to determine the value of the critical
current C^{2}=*hpl*/*a*R_{0} very accurately, as the correction for external
loss is a small percentage in any case. The chief difficulty is that
of measuring the small change of resistance accurately, and of avoiding
errors from accidental thermo-electric effects. In addition to
the simple measurements of the conductivity (M‘Gill College, 1895–1896),
some very elaborate experiments were made by King (*Proc.*
*Amer. Acad.*, June 1898) on the temperature distribution in the case
of long bars with a view to measuring the Thomson effect. Duncan
(*M‘Gill College Reports*, 1899), using the simple method under King’s
supervision, found the conductivity of very pure copper to be 1.007
for a temperature of 33° C.

(*b*) The method of Kohlrausch, as carried out by Jaeger and
Dieselhorst (*Berlin Acad.*, July 1899), consists in observing the difference
of temperature between the centre and the ends of the bar
by means of insulated thermo-couples. Neglecting the external
heat-loss, and the variation of the thermal and electric conductivities
*k* and *k*′, we obtain, as before, for the difference of temperature
between the centre and ends, the equation

_{max}−θ

_{0}= C

^{2}R

*l*/8

*qk*= EC

*l*/8

*qk*= E

^{2}

*k*′/8

*k*,(11)

where E is the difference of electric potential between the ends.
Lorenz, assuming that the ratio *k*/*k*′ =*a*θ, had previously given

^{2}

_{max}−θ

_{0}

^{2}= E

^{2}/4

*a*,(12)

which is practically identical with the preceding for small differences
of temperature. The last expression in terms of *k*/*k*′ is very simple,
but the first is more useful in practice, as the quantities actually
measured are E, C, *l*, *q*, and the difference of temperature. The
current C was measured in the usual way by the difference of
potential on a standard resistance. The external heat-loss was
estimated by varying the temperature of the jacket surrounding
the bar, and applying a suitable correction to the observed difference of
temperature. But the method (*a*) previously described
appears to be preferable in this respect, since it is better to keep
the jacket at the same temperature as the end-blocks. Moreover,
the variation of thermal conductivity with temperature is small
and uncertain, whereas the variation of electrical conductivity is
large and can be accurately determined, and may therefore be
legitimately utilized for eliminating the external heat-loss.

From a comparison of this work with that of Lorenz, it is evident that the values of the conductivity vary widely with the purity of the material, and cannot be safely applied to other specimens than those for which they were found.

19. *Conduction in Gases and Liquids*.—The theory of conduction
of heat by diffusion in gases has a particular interest, since it
is possible to predict the value on certain assumptions, if the
viscosity is known. On the kinetic theory the molecules of a gas
are relatively far apart and there is nothing analogous to friction
between two adjacent layers A and B moving with different
velocities. There is, however, a continual interchange of molecules
between A and B, which produces the same effect as
viscosity in a liquid. Faster-moving particles diffusing from A to
B carry their momentum with them, and tend to accelerate B;
an equal number of slower particles diffusing from B to A act as a
drag on A. This action and reaction between layers in relative
motion is equivalent to a frictional stress tending to equalize the
velocities of adjacent layers. The magnitude of the stress per
unit area parallel to the direction of flow is evidently proportional
to the velocity gradient, or the rate of change of velocity per cm.
in passing from one layer to the next. It must also depend on the
rate of interchange of molecules, that is to say, (1) on the number
passing through each square centimetre per second in either
direction, (2) on the average distance to which each can travel
before collision (*i.e.* on the “mean free path”), and (3) on the
average velocity of translation of the molecules, which varies as
the square root of the temperature. Similarly if A is hotter than
B, or if there is a gradient of temperature between adjacent
layers, the diffusion of molecules from A to B tends to equalize
the temperatures, or to conduct heat through the gas at a rate
proportional to the temperature gradient, and depending also on
the rate of interchange of molecules in the same way as the
viscosity effect. Conductivity and viscosity in a gas should vary
in a similar manner since each depends on diffusion in a similar
way. The mechanism is the same, but in one case we have
diffusion of momentum, in the other case diffusion of heat.
Viscosity in a gas was first studied theoretically from this point of
view by J. Clerk Maxwell, who predicted that the effect should
be independent of the density within wide limits. This, at first
sight, paradoxical result is explained by the fact that the mean
free path of each molecule increases in the same proportion as
the density is diminished, so that as the number of molecules
crossing each square centimetre decreases, the distance to which
each carries its momentum increases, and the total transfer of
momentum is unaffected by variation of density. Maxwell himself
verified this prediction experimentally for viscosity over
a wide range of pressure. By similar reasoning the thermal
conductivity of a gas should be independent of the density.
This was verified by A. Kundt and E. Warburg (*Jour. Phys.* v.
118), who found that the rate of cooling of a thermometer in air
between 150 mm. and 1 mm. pressure remained constant as the
pressure was varied. At higher pressures the effect of conduction
was masked by convection currents. The question of the variation
of conductivity with temperature is more difficult. If the
effects depended merely on the velocity of translation of the
molecules, both conductivity and viscosity should increase
directly as the square root of the absolute temperature; but the
mean free path also varies in a manner which cannot be predicted
by theory and which appears to be different for different gases
(Rayleigh, *Proc. R.S.*, January 1896). Experiments by the
capillary tube method have shown that the viscosity varies more
nearly as θ^{¾}, but indicate that the rate of increase diminishes
at high temperatures. The conductivity probably changes with
temperature in the same way, being proportional to the product
of the viscosity and the specific heat; but the experimental
investigation presents difficulties on account of the necessity
of eliminating the effects of radiation and convection, and the
results of different observers often differ considerably from theory
and from each other. The values found for the conductivity of
air at 0° C. range from .000048 to .000057, and the temperature coefficient
from .0015 to .0028. The results are consistent with
theory within the limits of experimental error, but the experimental
methods certainly appear to admit of improvement.

The conductivity of liquids has been investigated by similar
methods, generally variations of the thin plate or guard-ring
method. A critical account of the subject is contained in a paper
by C. Chree (*Phil. Mag.*, July 1887). Many of the experiments
were made by comparative methods, taking a standard liquid
such as water for reference. A determination of the conductivity
of water by S. R. Milner and A. P. Chattock, employing an
electrical method, deserves mention on account of the careful
elimination of various errors (*Phil. Mag.*, July 1899). Their
final result was *k*=.001433 at 20° C., which may be compared
with the results of other observers, G. Lundquist (1869), .00155
at 40° C.; A. Winkelmann (1874), .001104 at 15° C.; H. F.
Weber (corrected by H. Lorberg), .00138 at 4° C., and .00152 at
23.6° C.; C. H. Lees (*Phil. Trans.*, 1898), .00136 at 25° C., and
.00120 at 47° C.; C. Chree, .00124 at 18° C., and .00136 at 19.5° C.
The variations of these results illustrate the experimental
difficulties. It appears probable that the conductivity of a
liquid increases considerably with rise of temperature, although
the contrary would appear from the work of Lees. A large mass
of material has been collected, but the relations are obscured by
experimental errors.

See also Fourier, *Theory of Heat*; T. Preston, *Theory of Heat*,
cap. vii.; Kelvin, *Collected Papers*; O. E. Meyer, *Die kinetische*
*Theorie der Gase*; A. Winkelmann, *Handbuch der Physik.* (H. L. C.)

**CONE** (Gr.κῶνος), in geometry, a surface generated by a line
(the generator) which always passes through a fixed point
(the vertex) and through the circumference of a fixed curve
(the directrix). The two sheets of the surface, on opposite
sides of the vertex, are called the “nappés” of the cone. The
solid formed between the vertex and a plane cutting the surface
is also called a “cone”; this is contained by a conical surface
and the plane of section. Euclid defines a “right cone” as the