48 Dr. L. Silberstein on the Quantum 



the spectrum is always a (generalized) Balmer series of 

 simple, that is, of ideally sharp lines, showing no fine 

 structure. It will be kept in mind that this Balmerian 

 type of the series and the singleness (and ideal thinness) 

 of its members or lines theoretically persists, no matter 

 how small or how large the dimensions of the stationary 

 orbits. Both are general features, being consequences of 



the -=-law of the field and of the irrelevance of the diameter 

 r 2 



of the simple nucleus. 



Such being the well-known result for a point-nucleus, 

 our problem will be to investigate the complications of 

 the series, and more especially of the fine structure of its 

 members, due to the complexity of the nucleus. 



In what follows the meaning of the above symbols will be 

 retained throughout. 



3. Nucleus consisting of tivo fixed centres (point-charges). 

 This is the only case of a complex nucleus which can, for any 

 dimensions of the orbits, be rigorously solved, i. e. reduced to 

 quadratures, and ultimately to known elliptic integrals. This 

 is the reason why it is here mentioned at all. But the general 

 and complete solution of this problem, famous since the times 

 of Jacobi*, has in our connexion but a purely theoretical 

 interest. (Since those orbits only are spectroscopically 

 relevant whose dimensions are very large as compared with 

 those of the nucleus.) It will, therefore, be enough to give 

 here but a general outline of the theory of such a system, 

 omitting all the particular results arrived at by the writer f. 



Let 2a be the mutual distance of the two centres, and let 

 each carry one-half of the total charge of the nucleus, i. e. 

 \ice. (The case of unequal charges is not essentially more 

 complicated.) Take the axis of symmetry (join of centres) 

 as the .r-axis, with mid-point as origin, and denote by y the 

 distance from this axis. Then, with the well-known trans- 

 formation of Jacobi, 



x = a cosh f . cos 77, y = a sinh f . sin 77, 



the potential energy will be 



V — /* cosh g _ tee* , 



~ cosh 2 f- cos V / *~ a' .' ' ' ' W 



* Its integrability was already discovered "by Euler. 



t Some of these results with the corresponding type of spectra were 

 described at the Bournemouth meeting, without being recorded in the 

 much abbreviated Report. 



