322 
and 
(3) m= (. i — f 7i ay + (f^) 2 
from which ( 1 — tc€1) and can be found. We obtain 
( 4 ) 
( 5 ) 
(V 2 
^ a + ^)(« ,a — — + , g 
(v 2 — x 2 -j- -2) 2 -\-4v i u 2 3 
2 vu , _ 
(V 2 — x 2 -{- 2) 2 -f- i % 2 9 
Formula (4) will be seen immediately to belong to a class of equa¬ 
tions of which the best known is that discovered by H. A. Lo- 
rentz in 1879. In order to verify this let us admit k = 0 for all 
classes of electrons present; then $ = 0, u = 0 and formula (4) 
reduces to 
( 6 ) 
V 2 — 1 
^+2 
where €C is given by (see (4), § 5) 
i n® 
( 7 ) 
a 
e 2 N 
m n n a — n- 
Equations (6) and (7) constitute Lorentz’ celebrated theorem. They 
can of course be supposed to hold approximately in the case 
when k is very small for all classes of electrons and n 0 2 — n 2 is 
comparatively large. 
§ 7 . As another example let us suppose that g is different from 
zero and that k = 0. Reverting to the general equations (V) and 
(6) of § 5. we conclude that in this case 
a ) 
v i —l 
4n€t 
1 — a) €t ' 
Substituting for m its value from (3), § 4., we obtain 
( 2 ) 
V 2 — 1 
v 2 2 -\- g (v 2 - 1) 
Here €t must be taken from (7), § 6. This again, in the case when 
(7 = 0, reduces to Lorentz’ theorem. 
§ 8. The assumption that 
( 1 ) 
(5 = 0 
