504 
5[R. O. W. ETCHARDSON OX THP] ELECTRICAL COXDITCTTVITr 
neighbouring empty space. There is, as we have seen already, a discontinuity in the 
pressure of the corpuscles at the surface of the metal, and by the conservation of 
energy 
■•A 
= .( 11 ), 
• 1 
where w is the work done in taking unit-mass of the corpuscles through the surface 
layer, p is the pressure, and v the volume of unit mass of the corpuscles at any point. 
Similarly, in order to obtain the equations satisfied by the corpuscles outside the 
metal when the equilibrium stage has been reached, we use the principle that the 
work along any path extending from a point a to a point h due to expansion is equal 
to the work done by the electric forces. This gives 
= 0 , 
V ]:)eing the electrostatic potential, the charge on a corpuscle, and n the number of 
corpuscles in unit volume, since everytliing is independent of y and Now nc = N^, 
the number of corpuscles in unit mass, whence 
11^ dv 
V dx 
+ 7 " 
(lx 
= 0. 
In additioii to this the electrostatic potential has to satisfy Poissox’s equation, wliich 
takes the form 
fy; = _ , 
dx~ V 
Cq being the numerical value of the negative charge. 
The equation to be satisfied is therefore 
q_ 4^ N, Spy 
dx^ 
0 
^ /dv\~ . , 47rNy~t',|~ 
or -f C = 0, wliere 
dX' V \dx/ rm 
(LS). 
A first integral of this e([nation is 
d {log v) ^ 2c 
dx \ 
\ i 
\- 
v1 
B beino- an Inteo'ration constant. 
O o 
Now when v is infinite ^^8-- = O, and therefoi'e B = 0, so that 
dx 
\/2c dx = v~- dv. 
