﻿906 Mr. C. Gr. Darwin : A Theory of the 



If the number of atoms in a cubic centimetre is N the 

 mean free path for a fast moving particle is 1/Nttct 2 . If v is 

 the velocity at a distance x from the source we have 



1 dv _ 

 Riro*dx'~~' p9 



so that if V is the initial velocity 



sr » \ dv 



Putting in the value of p found for either volume or surface 

 distribution of electrons, this gives : 



SttA-N 



WA= l 



A W 



4 — 



(I i 



' aw 



•J o-vi/X 2 

 lo 



=1 



"«('+?) 



og(i-ni;)-7(ii0 



e u du 



^) 



For either subscript we shall put 

 C z e u tlu 



M "/i, 3 (^-l) 



= ^M- 



Except for a constant the functions E ]; E 2 are very similar to 

 the integral-exponential function Ez ; indeed, for large values 

 of z they only differ by this constant from e a Ei (z—a) where 

 a 1 = 2j3 and a 2 = 2. 



The velocity curve of the u particles thus is : — - 



S7rlc$na*x = R\og(l+~\-E\og(l+°^\ ... (6) 



It is interesting that the form of the curve should depend on 

 the whole number of electrons in a cubic centimetre, and on 

 the radius of the atom, but not at all on the number of atoms 

 or the number of electrons in each atom. For very high 

 velocities the curve is of the form l-—x=Av i /\ogv. With 

 different substances a will be different and will make the 

 shape of the curve vary. To consider this we write it in the 

 form 



l-.r = E log (1 + 10y)/E log (I + 10") 



