326 
being the number of impacts which a molecule (moving at speed 
v) performs in time d&. This quantity is known to be of the form 
( 2 ) 
vB(v)= \n MR 2 a 1 j# e~ x2 -f- (2x 2 -|- 1)J* Q dz e~ s2 
where M represents the number of molecules per unit volume (cf. 
§ I.), R the radius of the molecular "sphere of action” and 
(3) x — vja. 
Now expression (1) as it stands seems to be intractable. If instead 
of B(v) we put its mean value C 
C= -4vr- f°°dvv 9 -£-*°!« 2 B(v) 
a z ]7iJo v ' 
( 4 ) 
— 2 I /27i MR 2 a 
expression (1) reduces to 
( 5 ) d'd'Ce~ c & 
which agrees very approximately with that used by Lorentz. 
With these assumptions it follows that for the mean value of 
the displacement £ of an electron (belonging to a definite class) we 
can write 
( 6 ) 
where 
|T_ e (E x ~ f~ d) P x ) 
m(n 0 2 — n 2 ) 
(21 + *®) 
(?) 
C 2 [ n 0 -\-n i n„ — n \ 
2n 0 [C 2 -\- («o — n) 2 ' C 2 -)- ( n 0 -(- n) 2 \ 
m * = 1 1 
' ' 2n 0 I C 2 -(- (n 0 — rif C 2 -)- ( n 0 -)- n ) 2 
Now if we put 
(9) C 2 -f- n 0 2 — n 2 = B\ 2Cn=S 
we have 
(10) ( C 2 + («„ + n) 2 ) ( c 2 + (no - n) 2 ) = R 2 + S 2 ; 
hence (7) and (8) become 
(11) (E 2 -|-S 2 ) 81 = (V — n 2 )R 
(12) (R 2 +S 2 )% = — (n 0 2 — n 2 )S. 
