﻿Electron in the Neighbourhood of an Atom. 781 



We shall write V = M^F(r, 6), where F(r, 0) is a function 



of r and alone, and F(r, &)=■—- , where o> is the solid 



angle subtended by the magnetic wheel at the point (r, 6). 



The Lagrangian function L = T — V, where T and V are 

 the kinetic and potential energies of the system respectively, 

 is given by the equation 



L==iA^ + im(r 2 +r 2 ^ 2 + r 2 sin 2 ^ 2 )--M^F(r, 0). (3) 



The equations of motion of the system are therefore 



At-M.(|£.f + g,*)-0, (4) 



• ?>F 

 ^(f-f^-rsin^^+M^^ssO, .... (5) 



m{r0 + 2rr0 -r 2 sin cos 06 2 ) + m^\~ =0, . . (6) 



m{r 2 sin 2 6ty + 2r sm 2 0r<j) + 2r* sin (9 cos (9 . 0<£) = 0. (7) 



3. If we multiply equations (4), (5), (6), and (7) by i/r, r, 

 0, and </> respectively, add, and integrate we obtain 



i A^ 2 + im(r 2 + r 2 2 + r 2 sin 2 0c£ 2 ) = constant, 



which is the equation of conservation of energy of the 

 system ; if we suppose that initially the wheel is at rest and 

 that the electron is projected from an infinite distance with 

 velocity w , the equation takes the form 



iA^ 2 + imv 2 =imu Q 2 , (8) 



where v and yfr are respectively the velocity of the electron 

 and the angular velocity of the wheel at any moment. 



Moreover the coordinate cf> is ignorable, the equation of 

 motion corresponding to this coordinate being (7), which on 

 integration gives 



?m ,2 sin 2 # . </> = constant ; .... (9) 



this integral may be interpreted as the integral of angular 

 momentum of the electron about the axis of the wheel. 

 Equation (9) shows that if <j> is initially zero, it will remain 

 so always, i. e., <f> will have a constant value throughout the 

 motion, as we should expect from the symmetry of the 

 system. If, however initially, when r is infinite, <£ is not 

 zero, and therefore necessarily sin is not zero, we see that 



