87 



the absence of the magnetic field, only by the addition of a term which is linear 



g 



in the momenta and equal to (t>3(). In fact, denoting E expressed as a function 



of the q's and p's by ^ (p, q), we get from (83) together with (4), with the approx- 

 imation under consideration, 



3 3 3 



From this it follows that, with neglect of small quantities proportional to the 

 square of the magnetic forces, the perturbations of the orbit of the electron in a 

 hydrogen atom, which besides to a small external electric field of potential ø is 

 exposed to a small external magnetic field of vector potential 3t, are given by a 

 set of equations of the same form as (44) in § 2, but where the a's and ß's are 

 replaced by a set of quantities a[, a'^, u'^, ß[, ß[, ß'^, which are related to the q's 

 and p"s and the time in the same way as the orbital constants a^, a^, us, ß^, ß2, ßs 

 for the undisturbed atom are related to the g-'s and p's and the time through the 



equations (18), and where ß is replaced by the expression — e(/>-|- -(ti2(), considered 



as a function of the a"s and ^"s and the time. Since now, at any moment, the 

 quantities a[, a^, a'^, ß[, ß'^, ß\ differ from the corresponding orbital constants a^, 

 «2, «3, /9j, ß^, /Î3 only by small terms proportional to the intensity of the magnetic 

 field, we see therefore that, with neglect of small quantities of the same order as 

 the variation in the orbital constants within a single period, the secular per- 

 turbations of the shape and position of the orbit of the electron will 

 again be given by the equations (46), if in the present case ?' is taken equal 

 to the sum of the mean value '/'£ of the potential energy — e(P of the electron 

 relative to the external electric forces and the mean value ¥m of the quantity 



-(ö9l), both taken over an osculating orbit corresponding to some moment during 



the revolution and expressed as functions of a^, a,, a^, ß^, /?,.') The latter mean 

 value, however, is easily seen to allow of a simple interpretation. In fact, we have 



^î< 



ta 



'PM-ll\i^^i)dt = -'-^B, (84) 



where cu is the frequency of revolution of the electron in the osculating orbit, and 

 where B represents the total flux of magnetic force through this orbit, taken in 



') If the relativity modifications are taken into account, the orbit of the electron in the undis- 

 turbed atom is not strictly periodic, but it will be seen that the secular variations of this orbit are 

 still obtained from the equations (46), if only, to the expression for Ï'' as defined in the text, a term is 

 added which is equal to the expression for 'J' given by formula (70) in § 3. 



12* 



