46 



\vhere the differential symbols on the lefl sides are written to indicate mean values 

 of the rates of variation of the orbital constants during an approximate period of 

 the perturbed motion. From the definition of '/' it follows that this quantity in 

 general will depend on «, as well as on o.„, ... «s, ^-^j,, ... ?,s, but that it will not 

 depend upon ,îj. From the above considerations it follows further thai, with the 

 approximation in question. «^ may be considered as a constant in the expressions 

 on the right sides of (46 1, while for «., . . . . d^, .j',, . . . ,9s we may take a set of values 

 corresponding to some moment within the period to which the mean values on the 

 left sides refer. 



It Vvill be seen that the equations (46) allow to follow the secular perturbations 

 during a time interval sufficiently long for the external forces to produce a con- 

 siderable change in the shape and position of the original orbit, if in the total 

 variations of the orbital constants «,,... «s, ß^ ■ ■ ■ ßs ^^'e look apart from small 

 quantities of the same order as the small oscillations of these constants within a 

 single period. As a consequence of the secular variations, the orbit will pass through 

 a cycle of shapes and positions, which -^ill depend on its original shape and posi- 

 tion and on the character of the perturbing field, but not on the intensity of this 

 field. In fact, as seen from (46), the variations in the shape and position of the orbit 

 ■v^ill remain the same if ?' is multiplied by a constant factor, which will only in- 

 fluence the rate at which these variations are performed. It will further be observed 

 that the problem of determining the secular perturbations hy means of (46) con- 

 sists in solving a set of equations of the same tj'pe as the Hamiltonian equations 

 of motion for a system of s — 1 degrees of freedom. In these equations the quantity 

 I- plays formally the same part as the total energy in the usual mechanical problem, 

 and in analogy with the principle of conservation of energ}' it follows directly from 

 (46) that, with neglect of small quantities proportional to /-, the value of '/'will 

 remain constant during the perturbations, even if the external forces act 

 through a time interval of the same order as «^ ;.. In fact, with neglect of small 

 quantities proportional to /.", we have 



RI ^^ i^P^^EJlPÆ] = V'/'— — ~- ' — ^\ = 

 Dt ~^ [dak Dt ■ dßu Dt) ~^ \ da.kdßu'^ dßkdaul ~ ' 



Since at any moment '/' will ditïer only by small quantities proportional to /^ 

 from the mean value of the potential of the external forces taken over an approx- 

 imate period of the perturbed motion, it follows from the above that, with neglect 

 of small quantities of this order, also the mean value of the inner energy a^ of the 

 perturbed system, taken over an approximate period, will remain constant during 

 the perturbations, even if the perturbing forces act through a time interval long 

 enough to produce a considerable change in the shape and position of the orbit. 

 In the special case, where the perturbed system allows of separation of variables, 

 this last result may be shown to follow directly from formula (28j in Part I. 

 Taking for the time interval îf in this formula the period a of the undisturbed 



