Dynamical Motions of Charged Particles. 543 



The solution of A is only carried to this degree, because of 

 the further factor C" 1 in L which multiplies it. Substi- 

 tuting in (5) we have 



r_ 02/0 e ^ e -2 ^2 f r 2 2 + (r 2 , r 2 -r 1 )-2(r 1 , r 2 ) 



_ (^2, r 2 - ri) 2 ) 



A 1 ] l ji • ,i • r-2/3 i ^ e l e 2 ( r 25 r 2 — r l) 



Add to this the expression — m 2 L< p 2 + t; ?r7T9 • 



1 ar ZO ?• 



The first term is without effect because it is a pure function 

 of' the time, the second by (6). The result is 



(f 1? r,— rOfrj, r 2 -r0 



+ 



}• (-to) 



From its complete symmetry (10) will also give the motion 

 of e 2 when r x is regarded as known in terms of the time. 

 Thus the equations of motion of e 1 are 2) ri L = and of 

 e 2 are ® r .,L = 0. If q be any generalized coordinate 

 involving both r x and r 2 we have 



as may be seen by writing out the values of 2) ri and 2> re 

 or directly from the co-variance of the operator ® for point 

 transformations. Thus (10) is the Lagrangian for the 

 simultaneous motion of the two particles, which can now 

 be regarded as both belonging to the dynamical system. 



The last term in (10) is only accurate to the terms in G~ 2 , 

 so for the sake of consistency the first two should only be 

 expanded to this degree. They are then of the form 



— m^C 2 + i^f f + g^ wii'i 4 - 



Thus we have the complete Lagrangian for any number of 

 charged particles in any field in the form : 



■ ,„ :. 4 V. jl , V^l/:. »n v<^l g 2 



?'12 



(r 1? r 2 ) , (r 1 ,r 2 -r 1 )(r 2 , r 2 -r L ) 



f^S^V+S^-^V^^+t^c^'^" 22 * 



• VK e i p 2 I i?i, r 2 ) , (r 1; r 2 -r 1 )(r 2 , r 2 — r L ) ) 



• ■ • (U) 



