680 BELL SYSTEM TECHXICAL JOURNAL 



netic field made in a snmewli.n difk-rint manner, I reserve the details 

 for the following section. 



Vet it cannot be said that equation (71) is the utterance of the 

 much-desired General Principle, of the distinctive feature par excel- 

 lence which sets all permissible orbits apart from all non-permissible 

 orbits in every case. The most that can be said is this, that equation 

 (71), if properly interpreted, is the widest partial principle that has 

 yet been discovered. But it suffers limitations. I do not mean, as 

 might be thought, that cases have been discovered in wliich ilie per- 

 missible orbits determined by such equations as (71) have energy- 

 values not agreeing with those of the observed stationary states. 

 The difficulty is, that equations such as (71) cannot e\en be formu- 

 lated in many cases, because the necessary mechanical conditions 

 do not exist. 



This matter is a hard one to make clear; but the limitation can 

 be at least partialh' expressed in the following way. Re\ert to the 

 equations (70) which were applied to the rosette orbits. The first 

 of the integrals in (70) is to be taken over an entire cycle of the \ari- 

 able r. Now it was said in section J2 that the periods of the two 

 \arial)les r and are not equal, and in general they are incommensur- 

 al>le. W'lien the variable r describes a complete cycle, r and drdt 

 l)oth return to their initial values; but <t> and d(t>/dt do not have, at 

 tlie end of the cycle of r, the same values as they had at its begin- 

 ning. It follows that if pr depends on (j> or on d<i>/dt, the first of the 

 two integrals in equation (70) will have different values for differ- 

 ent cycles of r. If so, the conditions imposed upon the permissible 

 orbits by (70) would have no meaning. The conditions ha\e a 

 meaning, only if each of the integrals in (70) has the same value 

 for e\ery cycle of its variable — therefore, only if pr depends on r 

 only, and p^ depends on <^ only. And in general, such a set of equa- 

 tions as (71) has a meaning, only if it is possible to find a set of vari- 

 ables such that the momentum corresponding to each of them dejiends 

 on and only on the \ariable to which it corresponds; or, in technical 

 language, only if it is possible to effect separation of variables. 



Separation of variables is po.ssible in some cases, and in others it 

 is not. When the periods of all the variables are equal, as the\' are 

 when we imagine an electron of changeless mass revoking in an 

 inverse-square field, it is clearly always possible; the difficulty de- 

 scribed in the foregoing paragraph does not occur. In the other 

 cases which I ha\e outlined — when the electron is imagined to move 

 in an inverse-scjuare field according to the laws of relativistic me- 

 chanics, and when it is imagined to mo\c in a field compounded of 



