﻿466 



A NEW METHOD FOR CORRECTING A PLANET S ORBIT. 



with rec^ard to the ecliptic) be employed as a plane of reference for the elements ; and 

 again, if / h denote, with respect to the other plane (II.), what geocentric longitude and 

 latitude do to the ecliptic ; if, yet again, R, S, W denote, with Encke, the directions 

 respectively of the radius-vector, the perpendicular to it in the direction of the motion, 

 and the perpendicular to the orbit ; and also F, the direction of the planet as seen from 

 the earth ; E, a direction perpendicular to F in a plane perpendicular to (II.), the 

 « longitude," /, of any line in which is the same as that of F ; and finally, D, a direc- 

 tion perpendicular to both E and F (the positive direction of E being that of 6 = 90^ 

 and that of D, that of I + 90°) ; we shall have (using, with Encke, R S to denote 

 the angle made between R and S, and so with other letters), 



J cos i (J ? = cos E D (5 r + r cos S D (5 t; -f 3 ^o) + 15 Z cos W D, 

 ^8h: 



Erg. Heft, 

 : cos R E 3 r + r cos S E ((J i; + (5 ^o) + "J Z cos "SV E, > ^^^ 



3 J = cos R F <J r + r cos S F (.5 f + 5 ^o) + 5 Z cos W F, J 



In these formulae we have put (see (20) above) our 8 Z for its equivalent, 



r sin (« -j- (u) sin i i — r cos {v -\- w) sin i d SI, 



which is a distance in an absolutely determined direction, and must therefore be the 

 same if we employ cjo, io, •i^o^ instead of o, i, SI. In general, however, the plane 

 (I.), will be the ecliptic. If we transform the above-cited formulee, as Goetze has 

 done, into the following, 



cos R D = sin W D sin M' cos R E = sin W E sin N' cos R F = sin "W F sin P' ; 



cos S D = sin W D cos M' cos S E = sin W E cos N' cos S F = sin "W F cos P' ; 

 and make 



Dz r = D, r = 9° cos Q», 



0° cos Q" = -r e smv 

 q" sin Q" = 



.-^ 



tan qg sin E ; 



_^-Vi', 



whence, . 



rD. (i' + ;f,)=5''sin Q% 



we shall have, by similar equations to his (11), (12), in the article above cited, 



cos 5 D. ? = ^ sin W D sin (M' + Q°), 

 A 



D- 5 = '^ sin W E sin (N' + Q"), 

 A 



D: ^; = 5° sin W F sin (P' + Q°). 

 Again, as D„ r = r, 



D„ V = 0, 



the equations before quoted give us. 



(21) 



