﻿Prof. P. Lowell on the Asteroids. 341 



(2) I£ c, irrespective of sign, is less than unity and A is 



negative, -j- becomes imaginary for values of $ — y around 0°. 



— y is consequently oscillatory about 180°; unevenly so in 

 consequence of 7. 



(3) If c, irrespective of sign, is less than unity and A 



positive, -j~ becomes imaginary for values of 6—<y near 180°. 



6 — y is consequently similarly oscillatory around 0°. 



Case (1) is the usual one in celestial mechanics. 



Case (2) finds interesting exemplification in Jupiter's 

 satellites I., II., and III. and in Hyperion perturbed by 

 Titan. 



Case (3) has apparently been exemplified in the cases of 

 Enceladus and Dione and in the inclinations by Mimas 

 perturbed by Tethys. 



5. Let us now consider this equation : 



^= x /iA(cos6> + c). 



To begin with, the right side of it demands attention. In 

 it n and n' may without loss of accuracy be taken as constant. 

 And here a simplification may be introduced. 

 Since 



n 2 a 3 = P and n t2 a ;B =k 2 (1 + m'), 



if w r e denote the mass of the Sun by unity, we may for a 

 first criterion take 



n 2 a z =n' 2 a' 3 inasmuch as m'= 1A ,_ „ =- , 



lU4roo 



and furthermore calculate 



a 

 u= — 



a 



with the values of a and a' corresponding to 



9 1 !"• 



n = Zn' or a= -=—. - 



This will usually suffice to determine the character of the 

 action. Where the result of the two sides is close we must 

 use the true values of n, which are found from 



'aJ- ^ V 1+W 



