1889 - 90 .] Sir William Thomson on Ohmic Resistance. 
167 
no difficulty in understanding that there should have been amply 
sufficient current through an exceedingly thin shell of the bar to 
produce very suddenly the high temperature of the surface which 
Lord Armstrong perceived, and yet that the total amount of heat 
generated was insufficient to heat the bar to any sensible degree 
after the second or two required for the thermal diffusion (diffusivity 
•225 of a sq. cm. per sec.), to spread it nearly uniformly through the 
body of the bar. The heat lost outside the bar by surface emissivity 
(which is about 1/4000 of a gramme water thermal unit per second 
per sq. cm. of surface per degree of excess) would be quite ineffective 
to considerably diminish the whole quantity in the time required for 
diffusion to nearly equal temperature throughout the bar. If the 
dynamo had been doing no work externally at the time of the 
accident, the time required to get up a strong enough current out 
of the dynamo to produce much heating effect would have been very 
much longer than it was. The result to Lord Armstrong might 
not have been very noticeably different from what it was, but the 
attendant’s fingers would have been burned also. 
where Jds denotes integration all round the border of the cross-section. 
Hence the condition for constant total amount of current is simply 
For the case of circular cross-section with uniform electric conductivity in all 
parts of it ; and with the circuit- completing conductor either a coaxial, 
cylindric sheath, or a conductor of any form whatever, provided only that no 
part of it is near enough to the considered part of the given conductor to 
sensibly disturb the distribution, if the current, through its circular cross- 
section, from being of equal current- density at equal distances from the axis, 
the condition for constancy of total amount becomes simply 
at the boundary of the conductor, where r denotes distances from the axis. 
The full numerical solution of this problem, from the instantaneous commence- 
ment of a current of given total strength (which must necessarily be in the 
very outer skin, and must require an infinite current for the first instant) 
through the whole time until the current becomes as nearly as may be 
uniform throughout the cross-section, is particularly easy, but must be re- 
served for a future communication. It is identical with the following par- 
ticular case of Fourier’s thermal problem : — Let a given quantity of heat be 
initially distributed uniformly through an infinitely thin surface-layer of a 
solid cylinder coated with an impermeable varnish ; it is required to find, for 
any subsequent time, the temperature at any distance inwards from the 
surface of the cylinder. 
