502 
MR. 0. W. RICHARDSON ON THE ELECTRICAL CONDUCTIVITY 
set free at tlie surface of the metal, as in the case of the experiments to ])e described 
later, the metal must he continually losing energy owing to the emission of the 
orpuscles. This energy is composed of two parts : the first being represented hv 
the work done liy the c(.)rpuscles in passing through the surface laver, while the 
second is ecpial to tlie energy of translation which they possess when they have 
readied the outside of tlie metal. The sum of tlie two is easily calculated, since it is 
etjua] to the energy of translatioji wliich the corpuscles that have passed througli tlie 
surface layer possessed wliile tliey were inside the metal. We have therefore merely 
to multiply tlie nunilier of corpuscles Avhich strike tlie surface by the energy each 
jxissesses and integrate between limits Avhich embrace all values that pass through 
the surface layer. The total loss of energy per second is therefore 
ui (a'" + v~ d- w~) ue J,; (Jir 
( 8 ). 
It (1 + /.:<l>) e--‘* 
2 Trhii^k^ 
ml V 
2we'’’ 
- e 
■nm 
—$ Re 
(9) 
Now the work done in a second by the corpuscles passing through the surface 
layer is obviously = Nff, so that the part oi' the energy lost by the hot metal per 
second which appears in the form of the translational energy of the corpuscles, is 
given by 
Ndi = 
n e 
—2i-$ 
2 
= n 
2we^\- _ 
-e 
mv 
■}> Re 
(10). 
This calculation of the inte of emission of energy only applies, of course, to the 
case where the ions are removed by an external field as fast as they are formed. If 
there is no external field and the ions are allowed to remain, we soon arrive at a 
steady state when as jnany corpuscles possessing a given amount of energy enter the 
surface of the metal in a given time as leave it ; so that in this case there is no loss 
of energy due to this cause. 
The following proof of formula (h), which is due to Professor J. J. Thomson, is 
interesting, since it does not involve the methods of the kinetic tlieory of gases. 
Suppose we have a closed space wliich is bounded liy a surtace of hot metal, then the 
corpuscles will be given oft' from the metal until a stead}- state is reached. In this 
steady state as many corpuscles will pass through the bounding surface from the 
vacuum to the metal as from the metal to the vacuum, liut the pressure will not lie 
the same on both sides of the surface owing to the forces which tend to retain the 
corpuscles in the metal. There will thus be a discontinuity in the pressure at the 
surface of separation, and fii being the work done on an ion when taken through the 
surfa.ce, we have 
••n 
p (Iv = <l>. 
