Nc.u/; coxrr.Mi't^iy.iKY .un\i.\ci.s ix I'/ivsics-ix fi<o 



riiis is to 111' iiiu-i;r.ilc(l in llu- ii>ti,il \\.i\ . li\ iiiuliipK iiii; r,i( li icim 

 \\ill« •-'((//• (//); llii' result is 



{,.] = - 1>' iii>-+'2<-li i>ir — C. 



lii.".) 



iIr- last symbol st.iiuiing for a constant of intCKralioii. I'in.illy 

 (dr di>)- = ((lr (it)- {(l(t>;dl)- = {dr/dty-(m-r*/p-) 



= - Cinr' /)' + 2eEmr\p- - r. (3(5) 



\\c nro^iiizi- at oiuc the iilentical form of this equation for ihe path 

 in which the attracted particle moves and the equation (31) for the 

 ellipse drawn about the centre of attraction as focus. 



It remains only to identify the constants. Equating the co- 

 etTicients of /■' in the two equations, we have 



p-^eEnia (l-t-). (37) 



This is the equation giving the angular momentum of the electron 

 in terms of the major axis and the eccentricity- of the orbit. Fviiiating 

 the coefficients of r* in (31) and (3G) we have 



C = {y'j>na-il-t-)=eE,a (38) 



to determine the constant of integration in (35). If now the reader 

 will take the expression for the energy of the system 



n'= \tmr-e'/r = \m(Sdr/dt)- + r(d<i>/dt)y-e';r (39) 



and substitute for {d<t>/dt) according to (33) and for {dr/dt) according 

 to (35) and (38), he should arrive at 



W= -e-,2a. (40) 



This is the equation giving the energy of the system in terms of the 

 constants of the ellipse; we see that the energy depends only on the 

 major axis, not on the eccentricity, of the ellipse. 



The period of revolution 7" is a little more difficult to calculate. 

 The most logical procedure would be to take the reciprocal of the 

 expression (35) for dr dt, and integrate 



/= )"(-/>= m-r- + 2eE, mr-eE, a)~'''dr (41) 



around a complete revolution. The derivative dr/dt passes twice 

 through zero in the course of the revolution, once at the point of the 

 orbit nearest to the nucleus (perihelion) and once at the point farthest 

 away. At these points r = a{\^t), as can be seen from the geometry 

 of the ellipse or by inserting these values into the expression for dr/dt. 



