542 Mr. 0. G. Darwin on the 



This expression is valid for any fields, including explicit 

 dependence of <f> and A on the time. In the case of a con- 

 stant magnetic field it is a matter of indifference what 

 particular integral is taken in finding A from H. For the 

 general value of A is given by the addition of a term All to 

 any particular value, where II is a function of x, y, z. This 



adds on to L a term (r l5 AO) = -j- , and if fl is any function 



of cV, y, z and t whatever 



~ cm _ d an __ ^_ <m 



~? dt dt %g ~dq dt " ' ' * * ' l } 



so that the extra terms will be without effect on the equations 

 of motion. 



4. We next find the Lagrangian for a number of freely 

 moving interacting particles. Suppose there is a second one 

 of charge e 2 and mass m, 2 at r 2 . This particle is in motion, 

 but at first we imagine it outside the dynamicnl system, that 

 is we suppose r 2 to be known in terms of the time. Then the 

 motion of e\ is governed by (5), where <£ and A are to be 

 calculated from the position and motion of e 2 . The'potentials 

 are given by 



d> = !? I A = e ^ 5 I ( 7 ) 



y r + (f 2 ,r 2 -r0/O| ret .' A Cr+Cr^rj-r^/Clret.- l ; 



In these expressions r 2 =(r 2 — r x ) 2 and (r 2 , r 2 — rj/r is the 

 component of velocity of e 2 away from <? x . The quantities 

 are all to have retarded values. If the effect reaching e x at 

 time t, left e 2 at time t — r, we have 



OV = ( — t x + r 3 — r 2 T + J r 2 r 2 — . . . ) 2 



= r 2 -2r(r 2 , r 2 -r0 + r 2 {r 2 2 + (r 2 , r 2 - n)} - . . . . 

 Solving by approximation we find 

 t = r _fer^- ri ) + ^_ { ^ + ( ^ r> _ ri) + (i|> r2 _ r [y /r ^ 



Substituting in (7) 



, e'2\ ?2 frV + (r 2 , r 3 — r,) (r 2 , v 2 ~*i) 2 l e 2 r 2 



. - • (9) 



