and the Spectrum of Helium. 95 



with rings of more ordinary atomic size, for which the only 

 conclusion is that h = c. This signifies that the electron A is 

 always equidistant from the other two which form a ring. 

 It therefore rotates with the same angular velocity, and from 

 this point the reader can easily supply the proof that all 

 three electrons must be in one ring. The conclusion extends 

 not only to the law Xr n for the force between two electrons, 

 but to any law f(r) which can be expressed in a power series, 

 — such a law, for example, as a mixed inverse square and cube, 

 of the form 



We must finally suppose that the law of foroe between 

 electrons bound in an atom which can admit Bohr's lithium 

 model is either — 



(1) No force at all, which evidently constitutes a 

 solution, or 



(2) Any law whatever, provided that the three electrons 

 are all in one ring. 



This last alternative cannot take account of the valency of 

 lithium, as in the last paper, whose investigation can be 

 extended. 



But, in addition, direct distance laws are formally possible 

 for electrons which are actually on the confines of the 

 nucleus. These, however, are not rings of electrons in the 

 sense required by Bohr's theory, but are ^-particles. They 

 would be too close to the nucleus to be capable of behaving 

 with it otherwise than as neutral doublets. We do not need 

 to consider their possibility any further at present. For we 

 cannot effect a compromise, such as would be suggested by 

 putting two electrons close to the nucleus (within a distance 

 10 -13 ) under this law, and the third as an outer valency 

 electron of lithium at a distance 10 ~ 8 , acted on by the other 

 two according to the inverse square law. For such an 

 arrangement would give lithium, on Bohr's theory, a spectrum 

 which could not be distinguished from that of hydrogen. 

 Goplanar rings in Bohr's sense are therefore only possible if 

 bound electrons exert no force on each other, — for the case 

 of lithium at least. We can generalize this result from 

 lithium to any other element, but it is not necessary to give 

 the details of the argument. If the possibility of zero force 

 is admitted, there is no limit to the number of such rings 

 which are possible, or to the number of electrons which each 

 can contain. The atom becomes remarkably indefinite, for 

 any electron is only acted upon by the nucleus, and its 



