Researches on the Distribution of the Mean Motions of the Asteroids. ^7 



(20) I^s'WE'—Çe'C'-Qp.'EC+^P.' + ^^O 



2. Condition of the contact of the second order of A and Bo or 

 Bi. The equation (20) gives three real roots for ^ when G is 

 great. Let them be denoted by Ei, E^ and E^ in which Ei is the 

 smallest and Ez, the greatest, algebraically. E^ is always real. 

 E2 and Ez are positive when real and may be equal. The condi- 

 tion of the equal roots is 



1/3 ,\i 



(21) (7=3(|p,f = C, ^=10^) 



which is the condition of the contact of the second order, geo- 

 metrically. When Cis smaller than Ci two roots E^ and E^ become 

 imaginary. 



3. Case in which A and ^0 or B^^ intersect on the axis of x. 

 Putting e=0 in the equations (16) and (18), and equating x, we 

 get 



(22) E=±^^C=±F, 



4. Case in which A and Bo or B^ intersect on the axis of e. 



5. Singular case in which A and Bo or B^ intersect on the 

 axis of X and touch each other, simultaneously. Putting the 

 condition (22) in the equation (20) we get easily 



(24) c=|(-|^,.=)*aC, E-Uhi 



22. The following table is constructed to show how different 

 types of the motion are related to the magnitudes of the arbitrary 

 constants C and E. 



