Some Orbits of an Electron. 203 



Since these are connected by the relation 



dm t 



it is easy to show that the equations of motion of the electrons 

 are 



2(«^) = -«E^ 



Let the particle approach with initial velocity V along the 

 negative direction of the axis of x and let the initial value 

 of y be p. Take V = c sin ft. Then the integral of angular 

 momentum is 



*#-% P~V _ , Q 



To find the energy integral we take 



. d x .d y eYi xx + yy 



Now 



d x . d y f v 2 \V 2 dl x 2 + f 



x dtt, A 1 ' 2 V dt/ H u 2 Y /2 V c 2 ) dt2 t? 



c 2 



-w('-a 



so that the integral is 



(>-» 



1 o.f^l 



^2x1/2 SeC P~, ;iC 2 



wc r 



Write — 9 =a an absolute constant not depending on. the 

 mcr L ** 



particular orbit. In polar coordinates we thus have- 

 rs 



^jj2=pc tan j3 



('-» 



P2 



