﻿Prof. P. Lowell on the Asteroids. 313 



may substitute —j- got from the longer expression for that 



de d 2 



obtained previously for the shorter in -x-j and integrate term 



by term. After which we evaluate each term separately and 

 extract the square root of the result. 



7. The effect of these terms depends upon the values 

 of 0, 'st— 's/, and e. If = 180° approximately and also 

 st — «r' = 180° approximately, the aggregate of the terms 

 depending on becomes as large as possible ; the terms in 

 20 add to it whatever the value of e, while the terms in 30 

 do likewise. They are, however, smaller unless e be very 

 large. 



If <9=zl80° approximately and sr — ot' = approximately 

 the aggregate of the terms in is smaller than it otherwise 

 would be ; the terms in 20 add to it if ehe large, but subtract 

 from it if e be small. Those in 30 do the same. 



8. Turning now to the other side of the inequality 



d^i ^ or, de t d'STi 



dt * " i —'" + St + w> 



the subscripts being introduced to denote that the elements 

 are taken at the epoch of osculation, the several terms are 



where n = 299"'12S a day, and 7i l varies from the double of this 

 by a few seconds for the asteroids considered. Twenty-five 

 seconds is as far as it is necessary to go. The second term is 



de^_2_ dU Jl=? 'T~7\ — ™- — 



dt na' da na 2 e ■ ■* ' da na 2 \/l — e 2% di' 



Now 



da da da* 



which since dot _ 1 



da ~a" 



**l(5W> + bP)< 



whence 



dll ira ,] ni f" /t , .. . . . ^ ., ."] 



da=- iW [WW>)cos*-V°>J- 



