( 446 ) 
dh 
— are very small in comparison with A e—*, provided only the state 
av 
of the metal differ very little in two points whose mutual distance 
is /. This is seen by remarking that the ratios of the terms in 
question to Ae—”* are of the order of magnitude 
dA 
l 
da =e dh 
atl fp 
4 at 
= 1 
and that, in the second of these expressions, 7? is of the same order as re: 
If the term in (20) which contains X, is likewise divided by 
Ae-*”, we get 
2hl X. 
Now, 2/ X is the square of the velocity an electron would acquire 
if, without having an initial motion, it were acted on by the external 
foree m X over a distance /. If this veiocity is very small as com- 
pared with that of the heat-motion, the term in Y in our equation 
may also be taken to be much smaller than the term A e#. 
It appears in this way that there are many cases in which, as 
we have done, the function p (S, 1,5) may be neglected in the second 
member of the equation (7). 
The above reasoning would not hold however, if, in the case of 
two metals im contact with one another, there were a real discon- 
tinuity at the surface of separation. In order to avoid this difficulty, 
I shall suppose the bodies to be separated by a layer in which the 
properties gradually change. I shall further assume that the thick- 
ness of this layer is many times larger than the length /, and that 
the forces existing in the layer can give to an electron that is initially 
at rest, a velocity comparable with that of the heat-motion, only 
if they act over a distance of the same order of magnitude as the 
thickness. Then, the last terms in (20) are again very small in com- 
parison with the first. 
As yet, a theory of the kind here developed cannot show that 
the values we shall find for certain quantities relating to the contact 
of two metals (difference of potential and Prxtimre-effect) would still 
hold in the limit, if the thickness of the layer of transition were 
indefinitely diminished. This may, however, be inferred from thermo- 
dynamical considerations. 
§ 7. Having found in (20) the law of distribution of the veloci- 
