33fi LE VERRIER ON THE PERTURBATIONS OF PLANETS. 



I') J 



(1.) 



3. Let us designate by R the disturbing function ; by I and l' 

 the mean longitudes of the disturbed planet and of the disturb- 

 ing planet, and let us put 



R = C-\-X{i,i')sin {il + i' l')' 

 + '^li,i']cos{il + i't 



C being a constant. The indices i and i' may have all entire 

 positive and negative values from zero to infinity. But it is suf- 

 ficient also to give to one of them, i for example, positive values 

 only. This we shall suppose. 



Let us first leave the longitude /' constant, and give succes- 

 sively to the longitude I the equidistant values 0, «, 2 a, 3 «, . . ., 

 p ct, a. being an arc which does not exactly divide the circumfe- 

 rence. If, for one of these arcs l=p a, we attribute successively 

 to i the values 0, 1, 2, 3, . . ., i, the corresponding numerical 

 value Rp of the disturbing function may be written, — 



R^ = C + S{ (0, i') sin i' I' + [0, i'] cos i' I'} 



+ "^{{1,1') sin {pu + i' I') + [1,1'] cos {pa + i' I'} 



+ 2{ (i, i') sin {ipu-\-i' I') + [i, i'] cos {ip a + i' I') }, 



the sign S having now only relation to i'. And in developing 

 the sines and cosines, we shall have, 



R^ = C + 2{ (0, i') sin i' I' + [0, i'] cos i' I'} 



+ S-{(1, i') cos i'/'— [1, i'] sin i' I'} sinjoa 

 + X{{l,i') sini'Z'+ [Iji'j cos i' I'} cos pa. 



(2.) 



+ 2{(i, i') cos i' I'— [i, i'l sin i' I'} sin ip u 

 + 2{ (ij i') sin i' I' + [i, i'] cos i' /'} cos ip a. 



By only changing^ in this expression, the quantities contained 

 under the signs S remain constant. We shall designate them 

 more simply by putting 



C + 2-{(0, i') sin i' /' -I- [0, i'] cos i /'} = Bo,-| 



%{\i, i') cos i' l<- [i, i'J sin i' I'} = A,-, I . . . (3.) 

 S{(i, i') sin i' /'+ [i, i] cos i' I') = B„J 



and by giving successively to p the values 0, 1, 2, . . . up to 2 i, 

 the expression (2.) will furnish the following relations : — 



