﻿A NEW METHOD FOR CORRECTING A PLANET's ORBIT. 465 



Andhence, by(18), (18"), 



"-N 



cosqpcos (yl— x) . esm(A — x) , 



°^ 1 4" e cos (yl — x) 1 -)- e cos (.1 — x) 



, , r cos 01 cos V ^ e >■ sin u . 



= S\ogp—- — 5<f ^ — Sx; 



T COS V 



= S\ogp — -^^-^^^5cp — ta.nq,smY.8xi (19) 



or, =-|^-(2tan,p + ^)<!9,-tan,ysinE<J;f. (19') 



If the planet's place in the corrected orbit be referred to the plane of the uncor- 

 rected, it wUl not be in this plane, but above or below it, if the node and inclination 

 do not remain unchanged. The coordinate 8 Z, perpendicular to the plane (A), which, 

 as before stated, is that of the approximate orbit, can be ascertained by the usual 

 method. 



Thus, let (counted from the ascending node upon the ecliptic) be the ascending 

 node of the corrected orbit upon the plane (A), and 8 I its inclination to that plane. 

 We shall then have 5 Z = r sin 5 I sin [v -\- a — 0), where o = n — II. 



But to determine © and 5 I we shall have, by spherical trigonometry (omitting 

 infinitesimals of the second order), 



sin ^ I sin =: sin i sin 5 Ji , 

 sin 8 I cos = sin ^ i, 



or, (5 Z = r sin 5 I sin (v -j- to) — r sin i sin 8 SI cos (v -\- w). (20) 



14. So far, we have briefly shown how to resolve the eff"ect upon the planet's place in 

 its orbit of slight variations of the first order in the elements, into three portions : the 

 first, which, divided by its velocity at the time, we have called 8 z, we have considered 

 as the variation of a function, z, of the time t; difiering from t only by variations of 

 the order of the changes in the elements. This is, of course, in the direction of the 

 tangent to the orbit. The second portion we denote by r 8 w, and is in the direction 

 of the radius-vector ; the third, perpendicular to the plane of the orbit. 



The first practical use we shall make of this will be to compute the variations of any 

 geocentric coordinates with respect to the elements ; and afterwards study the same 

 relations with respect to certain new geocentric coordinates which we shall adopt. 



Goetze, Ergdnzungsheft zu den Astronomische Nachrichten, Altona, 1849, has shown 

 that if the planet's geocentric place be defined by its position with regard to any 

 plane (II.), and if any other plane (I.) (with regard to which cjo, i2o, Jq, and;^o denote 

 what the distance of the perihelion from the ascending node upon the eclijitic, the 

 longitude of that latter point, the inclination, and, lastly, the angle x-, used above, do 



