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



the distance (in the direction of the motion) of the perihelion from this node, and the in- 

 clination to the equator. Let a, d denote the geocentric right-ascension and declination. 



Then will if, 6 be the same functions of a — -fii, S, ii, that the geocentric longitude 

 and latitude are of a, S, and the obliquity. 



We shall thus have ( Theoria Motus, Art. 68, p. 64), 



sin (45° — ^6) sin J (E — ,;) = cos [45° -|- ^ („ — J2,)] sin [45° — | {{, + 5)'] ; 

 sin (45° — ^6) cos ^ (E — ,,) = sin [45° -(- x („ — />,)] cos [45° — 4- (i, — 5)] ; 

 cos (45° — ^ 5) sin J (E + ,,) = sin [45° + J. (« — J2,)] sin [45° — ^ (i, — 5)] ; 

 cos (45° — ^6) cos ^ (E + Tj) = cos [45° -f ^ (« _ j>,)] cos [45° — ^ (j", + ^)]. 



It will not be necessary in general to compute t/, 6 with extreme accuracy, as what 

 we most wish are their variations. These will be (see same article in Theoria Motus, 

 adfinem, where, as in (27), E is not the eccentric anomaly), 



<? ij cos 5 = sin E (£ a cos 3 -j- cos 'E d S; 



(27) 



... —3; ) 



(25') \ (26) 



(27') 

 d 6 = — cos E rf o cos 3 -f- sin E rf ^. ) 



In applying now the formulae (24) to (26), we shall have 1 = 0. J2', indefinite, 

 but Jl' -\- a = ca , as the origin of if is at the ascending node, on the equator. The 

 letters if and d are substituted, as the things which they denote are here the same, for 

 / and b, Goetze's notation. 



Then, ^D = 90°; ■. 



M = — , + Jl' ; ( • (24') 



M' = QJi -|- y — r] ; ) 



WE — d; > WF = 90° — 6; 



> (25') 



]Sf' = 270° + to, 4-f — 7;;5 P' = 90° + 



From (23), (24') we have the expression 



Dz7i=0; (27) 



and as the variations s, w do not contain SI or i, and are thus (to terms of the first 

 order with respect to the changes of the elements) independent of small changes in 

 the orbit-plane, so also is rf. 



That is, so soon as we know the positions of the orbit-plane correct within limits 

 of error of the first order, we can obtain one geocentric coordinate which depends only 

 upon four elements. 



16. If a set of rectangular coordinates be employed, the axes oi x,y in which are in 

 the approximate orbit-plane, and the former passes through the nodes upon the equator, 

 the positive direction of x being towards the ascending node, and if S, T, 'P denote 

 the sun's geocentric coordinates, we shall have, if the elements are supposed unvaried 

 (J denoting, as usual, geocentric distance). 



