104 Prof. F. Slate or Electronic 



can be traced from the fact that this electron appropriates 



the advantage of the general relation in equations (5), in 



combination with equations (10). 



Since (,a) vanishes with (r), (m") depends upon an excess 



above initial magnitude ; and because the final result involves 



time-rate alone of (m"), any special value (m Q n ) can be used 



in expressions like 



n a n • • ,i dm dm 9 , nn ^ 



m" = m " + ?n 2 ; giving correctly — - — * . (23) 



The algebraic link with Abraham's treatment may now be 

 closed bv simple verifications. First, that the ''longitudinal 

 mass" (m„) and the "transverse mass" (???,.) satisfy the 

 relation 



dv ( d , \dv cl , . m . , n .. 



^rr'sH* s 3 H = r ' • (24) 



and thus stand squarely upon the completed second law of 

 Newton. Secondly, that the separate time-rates of magnetic 

 energy and of electrostatic energy, as Abraham allots them, 

 match exactly the bracketed segregation of activity in equa- 

 tions (18), when (?7i 2 = m r ). And finally, that Abraham's 

 "Lagrange function" ((L) of his equation (113)) belongs 

 to the plan of equations (12), contributing, in fact, the partial 

 derivative there. 



A second auxiliary momentum, also derivable from equa- 

 tion (16), shunts the calculations towards the Lorentz 

 electron. Using (m r ) consistently with equations (5), 

 define 



Q 1 "=[m a {v)] [wyOO] = m'v>= [m l7 »>. . (25) 



In respect to (Qi"), (P) is a partial derivative ; from either 

 factoring it proves that 



= 7 2(r)(2P- m] g). . (26) 



By simple reduction this yields the forms 



/ x w o/-r> m ± dv\ dv m c 2 + v J dv rl , m dv ,__. 



^( P+ T St) =m Tt + 2 J=? It =i + 2 di ■ l27 ) 

 Since it is true that 



dm _ mv 2 dv _m c 2 -\-v 2 dv f m dv\ 

 V dt~ c 2 -v 2 dt ~ 2 ' (F^v 2 dt + \Jdi) 



( dm\ / dm\ . rtOX 



