Eesearctes on the Distribution of the Mean Motions of the Asteroids. 19 



Since e must be positive, two points, real or imaginary, will be 



determined by these equations. Now evidently only an odd 



number of zero-points in the librations and an even number in the 



revolutions, are possible. Accordingly, since there cannot be 



/Id 

 more than two, one and only one zero-point of-rr will exist in the 



case of the librations. 



The conditions necessary in order that two zero-points may be 

 real, are 



3s'^ 9/ 



E>0, ^^E'<c^^G 



by (25) and (26). Hence C must be positive and therefore both 

 values of x must be negative. Accordingly there exists no zero- 

 point in the revolution of Type 3. For the revolution of Type 1, 

 we have the condition 



■(=1^-^«) 



E^-i-F, 



which contradicts with one of the above conditions. Hence no 

 zero-point exists in the revolution of Type 1, and therefore the 



■fJi 



only type of the motion, in which two zero-points of -^ may exist, is 

 the revolution of Type 2. 



rift 



Evidently the sign of -rr is negative in the revolution of Type 1 

 and positive in the revolution of Type 3. 



24. General Case of the Second Order, s-s'=2. We have in 

 this case 



a(ß,c=p^ß^ co^ 

 where p,=i>,<'>=^ [^(4s^_55)6^^'4-(4s-2)ao^+ao='^] 

 Putting ^=^y 



