Rosearches on the Distril>utioii of the Meau Motious of the Asteroids. 47 



and 



Ç_i=no— -p-(n-i + ?i-2+ + »-9 — n_B— ^î-u — w.is) 



may represent the widtli in the positive and negative directions of 

 X respectively. The sum of these two quantities becomes zero 

 when the distribution is uniform, and becomes negative when 

 there is some condensation near no. The number of the mean 

 motions in each direction may be taken more or less. But if this 

 number be too small the result will be inaccurate, while if too 

 many it may be affected by other irregularities of the distribution. 



57. The width and position of the gaps thus determined, 

 according to the Berliner Jahrbuch for 1917, are as follows: — 



The existence of the gaps ?//> to the ffth order seems established beyond 

 doubt. It may also be observed that the width becomes narrower 

 as the order advances and also as the denominator or the 

 numerator of the ratio increases. 



58. In the portion with smaller mean motions than 500", 

 though the number of the asteroids is very small we see at once 

 that the character is reversed (PLATE). No asteroid can be found 



