1302 



respect to the intermediary orbit with tlie period 2 jt/,^. Each of 

 these oscillations corresponds to a small change of the initial values, 

 i. e. a small deviation of the constants of integration from those 

 of the intermediary orbit. The term corresponding to a root /i =: 

 is not an oscillation, but a constant correction to one of the elements, 

 which does not aiTect the character of the motion. Now there are 

 six possible deviations, i. e. six constants of integration by a change 

 in which the intermediary orbit is not essentially altered. These are: 



1. A change of the zero of the longitudes and the time. This 

 evidently does not affect the motion at all, and since two constants 

 of integration are involved, it corresponds to two roots ^ = 0. 



dr 



2. A change of 7i, — 7i, = — and of x. The first is evidently 



dt 



only a change in the unit of time. The other does affect the motion 

 of the three innei- satellites, but only in so far as the intermediary 

 orbit is replaced by another of entirely the same character. 



3. A change of c,, say to c^ ~\- dc^. We can then call c + dc^ 

 again c% and nothing essential will be altered. 



4. A change of to^. In the intermediary orbit we assumed 

 10^ =z 0. In doing this we neglected a small quantity, and evidently 

 the exact amount of the neglected cpiantity is of no importance. 

 This corresponds to the fact that all coefficients ^,4 and ^^(4 are zero, 

 as is found when they are worked out. 



It must therefore be possible to transform the equation of the 

 16^'' degree in 3 to an equation of the 5^'' degree in /i'. This is 

 effected as follows. 



By differentiating the second and fourth of (23) we find equations 

 of the form : 



d'ki 



-- + ^ Aij kj 4- ^ Bij ioj = 0, 



"^ J j 



(26) 



Hence we find for c'/./, c"/^ and i%j the conditions 

 c'iq ^% - 2: Aij c'j^ — 2 Bij c = 0, 1 



(27) 



The determinant of these equations is 



