﻿and its Relation to the Quantum Theory. 861 



, s = ? ^[_.001875w-'0O14702^--0011G5w; 3 ~'000944w; 1 



— -000766m? 5 --000607z<; 6 ] cos 3B* 



+ m^[--0000125^ 2 --0000140w 3 - -0000125m; 4 --0000067z<; 5 



-•0000026m; 6 ] cos 5Bt 



+ ^[--0000007w 3 --0000011?6- 4 --0000013z<; 5 



-'0000013z«; 6 ] cos 7B£+ negligible terms in cos 9B£, etc., 



2 



C= u P \ A =-875 - -973125m? + -032072/^ + '021074m; 3 



+ -014987m; 4 + '0ll207iv» + -008694m? 6 + -006522m; 7 

 (17) 



n = A[Vl-ivcos 2 Bt + p]. Z = A[^ 2 cos Bt+s] 9 



t 



*=jS + « (") 







This family of orbits is the most general solution in which 

 R and Z are periodic functions of the time. The power 

 series expansions of the perturbations p and s have been 

 carried far enough to enable one to compute the energy 

 through terms of the seventh order in w *. This solution 

 contains four arbitrary constants (p, w, an epoch angle for q> 

 and one for R and Z) f, while two other arbitrary constants 

 are eliminated by the requirement of periodicity for R and 

 Z. The mean period of <£ differs from that of R and Z J, 

 giving a preces-ion of the line of nodes formed by the inter- 

 section of the orbit with a plane normal to the axis of 

 symmetry. The Cartesian coordinates #=Rcos</> and 

 ^/ = Rsin<£ therefore contain trigonometric terms having 

 two distinct periods, and the motion is conditionally 

 periodic, the orbits not being reentrant (except when pro- 

 jected on the RZ plane). In carrying out the solution we 

 have nowhere assumed the existence of solutions in which 

 R and Z are periodic functions of the time, but wer* 1 

 automatically led to solutions of this character on performing 



* For proof of convergence of power series expansions in w, cf. Moulton, 

 4 Periodic Orbits,' pp. 15-19. 



t In performing the calculations, the epoch angle for R and Z has 

 been so chosen that R = Z= at £ = 0. This involves no loss of generality, 

 as the origin of time is immaterial in computing the energy. 



X The mean period of <p is found by taking 2tt times reciprocal of mean 

 angular velocity which is the constant term in the Fourier expansion of 



P 

 2mR 2 ' 



