33 



of the /i's can vary by more than a single unit, in analogy with the considerations 

 in the former section about a linear harmonic vibrator. 



Another example which has more direct physical importance, since il includes 

 all the special applications of the quantum theory to spectral problems mentioned 

 in the introduction, is formed by a conditionally periodic system possessing an axis 

 of symmetry. In all these applications a separation of variables is obtained in a 

 set of three coordinates Qj, q, and q^, of which the first two serve to lix the posi- 

 tion of the particle in a plane through the axis of the system, while the last is 

 equal to the angular distance between this plane and a fixed plane through the 

 same axis. Due to the symmetry, the expression for the total energy in Hamilton's 

 equations will not contain the angular distance q^ but only the angular momentum 

 Pg round the axis. The latter quantity will consequently remain constant during 

 the motion, and the variations of q^ and q„ will be exactly the same as in a condi- 

 tionally, periodic system of two degrees of freedom only. If the position of the 

 particle is described in a set of cylindrical coordinates z, p, a, where z is the dis- 

 placement in the direction of the axis, p the distance of the particle from this axis 

 and & is equal to the angular distance ^3, we have therefore 



z = 2'Cri,T, cos 2 TT {(r^Wj + î'') ^2)^+ ^^1. '"2} 

 and " (32) 



P = -Cri,r, cos 2;r ( {r^to^ -f r^w^) t + c.^^^J ^ 



where the summation is to be extended over all positive and negative entire values 

 of Tj and Tg, and where cu^ and cu^ are the mean frequencies of oscillation of the 

 coordinates Çj and q.^. For the rate of variation of i^ with the time we have further 



-^ = 93 = ^ = fi(li,<li,Pi,Pi,P3) = ±-C"^,r, cos 2 ?r{(î-iWi-f r, «,,)/+ c"^,^J, 



where the two signs correspond to a rotation of the particle in the direction of in- 

 creasing and decreasing ^3 respectiveljt, and are introduced to separate the two 

 types of symmetrical motions corresponding to these directions. This gives 



±ä = 2 7ra.3f+2'C;';,^^ cosln {{r^co^ + r,aj^)t-{- c!;^^,), (33) 



where the positive constant 01,=^,— C'J.o is the mean frequency of rotation round the 



axis of symmetry of the system. Considering now the displacement of the particle 

 in rectangular coordinates x, y and z, and taking as above the axis of sj'mmetry 

 as z-axis, we get from (32) and (33) after a simple contraction of terms 



X = pCOsê == 2'Dri, Tj cos 2 7r((TiWj + TjO), -f- 0^3) /-|- rfr„T, } 



and " (34) 



y = ^ sin é» = ± 2'Z)ri,r2 sin 2n-{ (ri^i -f- Tju^j + «3) f + rfr^r^}, 



where the D's and d's are new constants, and the summation is again to be ex- 

 tended over all positive and negative values of Tj and Tj. 



D, K. D. Vldensk. Selsk. Skr., n.aturvidensk. ogmathem. Afd., 8. Rrekke, IV. 1- 5 



