Researches on the Distribution of the Mean Motions of the Asteroids. 21 



The limits are simpler than in the case of the first order. The 

 libration of Type 1 is impossible throughout o^nù the librations are 

 restricted to that about n. 



lift 

 26. The point at which -^ changes its sign is given by the 



equations 



=-^«±y(^^;_c 



-±y^-(^T« 



Only one point is possible on the positive side of e. Hence, there 

 is no zero-point in the revolutions and one and only one point, in the 

 librations. 



27. General Case of the Third or Higher Orders^ s—s* >2. We 

 have in this case 



acßM-^y"'p,..'e'-' co^d 



where jo,_,/ is a function of «o with a positive value. Putting 



we have by (14) 



x'^=-C—{ — Vf'''&p,.,.yco^d 



(28) 



1 dd g-a'-a 



— -^=^'x + {-^y~''{s-s'fp,.AJ '-"' cos 6 



The curve A becomes a parabola of various degrees with the axis 

 parallel to the axis of y, and the vertex on the axis of x. The 

 constant C2 becomes 0. The table of the limits is much simplified 

 by this change. The libration of Type 1 is always impossible 

 and the librations are restricted to those about when s—s is odd 

 and those about n when s—s' is even. The libration of Type 3 



