SOMF. a)A'7/:".l//'()A'./A')' .IKr.l.WIS l\ rilVSICS l\ (V^t 



anil tn 2(- lor llio ioni/cd-liclinin .iinm-inodtl. no oihrr ciscs in.illcr 

 fnr tlir liiiu' iK-inR." 



./I. Motion of an Electron in tin hnrrsc-Sijiiure Central Field 



Most people rccognizi' tlif i'(|ualit)n ol llu- ellipse must i-.i>il\- in ilif 

 form 



.v'-, a' + V", 6-= 1 



iti >i i'oor(liiiate-s\stiMn ol wliicli tlie oriv;iii is .it the centre of tlie 

 ellipse, the .v-axis and the v-axis par.illel respeeti\el\' to the major 

 and the minor axes of the ellipse. 



The symbol a and b denote the semi-major and semi-minor axes 

 of the ellipse; they are related b\- 



h- = a'{l-f-) (30) 



in which « stands for the "eccentricity" of the ellipse. The foci 

 of the ellipse lie on the major axis at distances at to either side of its 

 centre. Transferring the origin to one focus, say the focus at x= -\-at, 

 and using coordinate-axes parallel to the former ones, we have 



(j' + ae)- «--|- v'-, b'-= 1 



Transforming coordinates again, this time into polar coordinates r 

 and <t> with the origin at the focus of the ellipse and the direction 4> = 

 pointing along the .v-axis, by means of the substitutions 



j' = r cos ((> y = r sin 



we arri\e after somewhat tedious but not difficult algebra ■* at the 

 e(|uation for the ellipse in the form in which we shall use it 



^^ a(l-^') 

 l+« cos <i> 



and at the derivative thereof 



/ dry _ r*t^ ^m"" 4, ^ r* 2r^ 



W«y a'ii-i'y- a'il-i-yaiX-t-) ^ ' 



' The allowance to be made for the motion of the nucleus never differs perce|)til)ly 

 from that already made by introducing n into ef|uation (16), and the magnetic 

 fields arising from the motions of the electron and of the nucleus are without per- 

 ceptible effect (C. (J. Darwin. Phil. Mag. 30, pp. 537-551; 1920). The correction 

 which would be required if the nucleus or the electron were o<ldly shaped, if the 

 nucleus were a magnet, or if there were entrainnient of the (jotential energy of the 

 svstem bv the moving electron, have been evaluated bv various people; consult 

 .■\. K. Kuark, .\stroph. Jl.. 58, pp. 46-58 (1923). 



' ["he ambiguitv of sign which arises in the course of the development may be 

 res lived by thinkmg of the limiting case of the circle f« = 0). 



