﻿340 Prof. P. Lowell on the Asteroids. 



multiplied by m' 2 , and are therefore a thousand times smaller 



than ~. The fourth and fifth are multiplied by m, which 

 dt 



is evanescently small. We thus get 



Now 



d 2 d 



dn 





dt 2 



~di 



' 



dn 



3 



dR 



dt " 



a 2 



de 



whence (2) 

 dt 2 



3*W ff 7l >_#/ C0S ( OT _ w ')} S m0 + A"sin(w--«')cosA 



2 era L 1 • 



For k 2 put its value, n 2 a z , and let 



3n 2 m'tx{Ji—h" cos («r— -cr')}=7i 1 



and 3rcWa /i" sin (-or— «r') = Zt 2 . 



^2$ 



Then -yy- may be put in the form 



where . — A 2 



tan 7 = — r — . 

 hi 



Multiply both sides by 2 - and integrate, 



(JJ= v/(V+I?)[c 0S (^- 7 ) + c] 



= iA[cos(<9- 7 )+c], 



if A = 2 v/(V + V), 



whence (3) ^0 / . , r -^ : =r 



^= v / iA[cos^- 7 j + c]. 



The action depends upon the value of the constant c. If 



A is positive and c is negative and greater than unity -y- 

 becomes imaginary for all values of 6 — <y. in 



(1) If c is positive and greater than one -j- is real and 



never changes sign. Consequently the action is continuously 

 in one direction and increases or decreases indefinitely. 



