﻿Prof. P. Lowell on the Asteroids. 33'J 



shall neglect the inclinations of the asteroids' orbits. Their 

 inclusion would introduce terms only of the second order in 

 e or less, assuming r\ to be of the order of e. The perturbing 

 function is thus composed of terms as follows : — 



R = secular terms + A cos + A x cos 20 -f A 2 cos 30 &c. 



+ B cos tf' + Bi cos 20' + B 2 cos 30' &c. 



+ terms with other combinations of 

 ot — gt with or 0', 



in which — j ndt + e~ \ 2a dt — 2e' + sj, 



and & = §ndt + e— \2n ' dt -2e ' + ot'. 



We will consider first the two most important terms, 

 A cos and B cos 0', The second term introduces the action 

 due to Jupiter's orbital eccentricity. 



Using the usual symbols of Laplace and Leverrier we 

 have then 



(1) R=- ^^(4// 2 > + /y 2 >) cos0+ ^^(3W + bjP)cos 0', 



ACL — CL 



db {i) 

 in which the subscripts in tlve fr's denote « — = — . 



da 



Now 0' may be written 



ff =s j* n<fr + 6 — §2ndt — 2e / + ^t — oj + sr' 



whence 



R = - ^ { j>(4^ + V 2) ) - e' (3^ l) + bp) cos (isr - *r') ] cos 



-{/ (SbW + b^y) sin {k ^ v')} sin 0\. 



Let the quantity in the first bracket be h' — h" cos (-ar — nr) ; 

 that in the second, A" sin (or— ■*/). 



Since 



0= J n<ft + e + * — J 2 >< Vf- 2e\ 

 <M dn ^ 2 e d?v M /V 



W <^ 2 " dt + <fa 2 + </r 2 w 7/7 "* - 7^ ' 



The second and third members of the right side are 



2 A 2 



