( 685 ) 
The mean motions in longitude are the same as in the solutions of 
the first kind, viz. : 
Dy == CV — x, 
The mean motions in anomaly remain rigorously commensurable.') 
I will now restrict the discussion to the special case represented 
in the system of Jupiter, viz. : 
ty a i a ek 
For the general case similar results will be found, which I do not 
however at present propose to investigate. 
Moreover I will limit myself to the consideration of small excen- 
tricities, which is the only case that is of immediate practical value. 
Whether the conditions (1) do also admit solutions with large excen- 
tricities, is a question which can only be answered by a special 
investigation. 
Under these restrictions we find that out of the 16 combinations 
satisfying the conditions of symmetry, there are only 4 which also 
satisfy the conditions (1. For two of these x is positive, and for 
the two others it is negative. Further, if the quantity 
4,-—84,+24,=—K 
is formed for each of these solutions, it will be found that one of 
the solutions with a positive x has K = 0° and the other has K = 180°, 
and similarly for the solutions with a negative x. Of these four 
solutions that with A —= 180° and x positive (the case of nature) is 
the only stable one. 
These solutions of the second kind thus appear, on both sides of 
the exceptional point x = 0, as the natural continuations of the two 
possible solutions of the first kind (A = 0° and A= 180°). In the 
solutions of the first kind the unperturbed orbit is circular, the 
perturbed orbit is affected by a ‘great inequality”, with the argument 
ct. In the solutions of the second kind this inequality appears as 
an equation of the centre’). In the solutions of the first kind we 
have the condition that the unperturbed excentricity must be zero; 
corresponding to this the excentricities in the solutions of the second 
1)’ These solutions are based on the same principle as those investigated by 
Scuwarzscuitp (Astr. Nachr. 3506). Scuwarzscuitp, however, only considers the 
case of two planets, one of which has an excentricity, and at the same time an 
infinitely small mass. Consequently the orbit of the other planet, which is acircle, 
is not perturbed. 
2) In the integration by the usual method, this inequality presents itself as a 
perturbation of the excentricities and pericentres. 
Besides this “great” inequality there are, of course, a number of others, whose 
arguments are multiples of >, which are the same in the two solutions. 
