for the Distribution of Electric Energy. 191 
From this we see that the conductor will only rise 50° 0. 
in temperature when the current is of such strength as to 
require a conductor eight and a half centimetres in diameter. 
The question of safety, then, may, at least in the case of un- 
covered copper rod, be left out of consideration when the cur- 
rent is less than 5000 amperes. 
Next take the case of a conductor covered with a substance 
whose thermal conductivity is k. In this case r depends both 
on the rate of cooling from the surface, or the emissivity, and 
on the conductivity k. 
Let e = emissivity of the surface, 
T= difference of temperature between the two surfaces 
of the covering, 
T' = the difference of temperature between the external 
surface of the covering and the surrounding atmo- 
sphere, 
r 2 = the external diameter of the covering, 
q = the quantity of heat conducted through the cover- 
ing per unit of time. 
Then, since for constant temperature we must have the 
quantity conducted through the covering equal to the quantity 
radiated from the surface, we have 
2irr&T = q (8) 
But 
q= ~ 27rk d*> < 9 > 
where x is any radius. Hence 
2irr. 2 eT=-2'jrkv~ , 
#=-pT' (10) 
ax k x 
But the total difference of temperature between the two 
sides of the covering is 
-Hi 
r »dx 
x 
Now, by the conditions of the question, 
T + T x =50; 
