50 Dr. L. Silberstein on the Quantum 



series v= -yA TT, is reduced to quadratures, and ultimately 



to finding ft and W from the two transcendental equations 

 (containing elliptic integrals) by successive approximations.. 

 This gives rise to a variety of intricate spectral types, whose 

 description, however, need not — for reasons stated a moment 

 ago — detain us any further in the present paper. 



It remains only to say a few words about the limits of the 

 integrals to be applied in (8). Since, by (6), £ and rj vanish 

 only together with j>\ anc l Pi respectively, the required limits- 

 will be determined by the roots of the equations 



i'i(f) = o, P2 (v) =■ o, (10) 



the left-hand members of these equations being as on the 

 right hand of (9). The first of (10) is a quartic for 

 coshf=i£, say, and the second a quadratic for cos 2 77. Of 

 the four roots of the former two only, say u ± , u 2 , will be 

 found available *, and if we require the electron not to leave 

 the system (not to escape to "infinity "), both of these will 

 certainly be available. And the roots tj 1 , ?; 2 °f the second 

 equation will be either both complex or both real. Thus, in 

 general, the electron's orbit will be contained between the 

 two ellipsoids (of revolution) f^const., f 2 = cons t-; and it 

 will either pierce incessantly all the hyperboloids 7) = const., 

 or be hedged in between the two hyperboloids ^^eonst.,. 

 77 2 = const.t Thus p l will be integrated from the inner to the 

 outer ellipsoid and back again, and p 2 either over 2tt, if 

 rj increases (or decreases) incessantly or between the two 

 limiting hyperboloids (twice), if rj oscillates. In particular, 

 we may have £= const, throughout, and therefore f x = f 2 ; 

 this for instance is possible for (/> = 0, when the electron 

 describes an ellipse in the meridian plane. And if the two 

 roots rji, rj 2 (are real and) coincide with one another we 

 have 7/ = const. ; a possible motion of this kind occurs, for 

 example, for <fi = 0, when the electron oscillates along an arc 

 of an hyperbola stretching within an ellipse f= const, in the 

 meridian plane. A sub-case of this is the obviously possible 

 to-and-fro motion along a straight perpendicular to the axis 

 and passing through the mid-point of the two electric centres. 

 The formulae for the special types of spectra corresponding to 



* I.e. not only real "but also positive and ^1, so as to yield 

 real £, , | 2 . 



t Needless to say that |= const., ?7 = const. are all confocal ellipsoids 

 and hyperboloids (of two sheets) with the axis of the nucleus as axis of 

 symmetry and the two electric centres as foci. 



