( 440 ) 
the time f, this element will contain a certain number (in fact, a 
very large number) of electrons moving in different ways. 
Now, we can always imagine a piece of metal of finite dimensions, 
say of unit volume, in which the ‘concentration’, as we may 
call it, of the electrons and the distribution of the different velocities 
among them are exactly the same as in the element /S. In studying 
the said distribution for the N electrons, with which we are then 
concerned, we shall find a diagram representing their velocities to be 
very useful. This is got by drawing, from a fixed point O, N vectors, 
agreeing in direction and magnitude with the velocities of the electrons. 
The ends of these vectors may be called the velocity-points of the 
electrons and if, through the point O of the diagram, we draw axes 
parallel to those used in the metal itself, the coordinates of a velocity- 
point will be equal to the components &, 7, § of the velocity of the 
corresponding electron. 
Writing now 
HS n, 5) da 
for the number of velocity-points within the element dé at the 
point (§, 9,6), we make the exact solution of all problems relating 
to the system of electrons depend on the determination of the 
function f(S, 1, 5). 
We may also say that 
PSs NUSA rn A) 
is the number of electrons in the element dS, whose velocity-points 
lie in dà; in particular 
FE n, 8) dSd and. 3 2 ae} 
is the number of electrons for which the values of the components 
of velocity are included between § and §+ds5, 4 and 4 + dy, 
5 and 6+ do. The expression (2) is got from (1) by a proper choice 
of the element dà. 
If the function in (1) were known, we could deduce from it the 
total number of electrons and the quantities » and WW mentioned 
in § 1. Integrating over the full extent of the diagram of velocities, 
we have 
N= f G5) a Bs. fs) te ote (5) 
v= f EF Enda, | ERE 
and if, in treating of the flux of energy, we confine ourselves to 
the kinetic energy of the particles, 
—_—— = ay oe 
—) 
_ 
