Dynamical Motions of Chary ed Particles. 547 



For considering quasi-elliptic orbits we naturally take 

 j) x =p !/ —.0 for the integration constants. If this is not 

 done it will be found that the modified Lagrangian 

 deduced from (15), if expressed in polar coordinates, con- 

 tains 6 explicitly. Under these conditions the ordinary 

 integral oi: angular momentum does not exist. But any 

 such case could be worked out easily by taking p x -=zp y =0 

 and when the complete solution has been found, applying 

 a linear relativity transformation to give the system the 

 proper motion of translation. Such a transformation would 

 be expected to introduce the time explicitly into the formula?. 

 So it appears that the angular momentum integral would be 

 replaced by a complicated integral involving both 6 and t. 

 The study of such an integral might have an analytical 

 interest, but it would appear that in any specified case where 

 n x does not vanish (and the same applies to motions of the 

 particles which are not in a plane) the required results could 

 be quickest attained by relativity transformations. Thus, in 

 studying the collision of a moving electron with a stationary, 

 we should worly out the orbit with both moving in such a 

 way that p x =p I/ r=0, and afterwards apply the transformation 

 which would reduce one of them initially to rest. 



Taking p x ^=p y = in (16) we have equations of which the 

 solution is 



20 2 (M + m) 3 *U * V} 



+ 2CV (M + m) 2 [^'+?(^ + ^)]- ' ' ( 17 ) 



Next form L' = L— p x ~X — PyY. This is given by simply 



omitting X, Y from (15), since X, itself of the order C -2 , 

 occurs everywhere either squared or else multiplied by 

 0~ 2 . In polar coordinates we then have 



, 1 Jim 2 J_ Mm(.M? - Mm + m 2 ) 4 



1 2M + m tC + *(' 2 (M + m) 3 



Ee Ee Mm 2r 2 + r 2 <9 2 , A Q , 



+ ^ + 20 2 7M^p"^ ' " (lb) 



where w 2 =r 2 + r 2 2 . The integral of angular momentum is 



Mm .ill, J M 2 -Mm + m 2 , Be 1\ _ 



M + m' \ + 2C 2 ~ (M-fmf : W + 2 (M + m) rj ~ P 



. . . (19) 



