﻿470 A NEW METHOD FOR CORRECTING A PLANETS ORBIT. 



So also, making Q = — a, — v, we find 



D^ , ^ 'hSl^ "■ - ') cos =: ''''°^'',~ "^^''^ cos 5. (33) 



' c sin ^ K y^ 



If ^, and consequently the orbit, is changed, so that the plane to which the coor- 

 dinates are referred is no longer the actual plane of the orbit, the last equation of the 

 group (27) becomes 



S Z-\- ■4< = Jsme. (34) 



And thus D^ Z = sin e. 



An arbitrary variation in either assumed geocentric distance is thus separated into 

 three components ; one, cS z, in the direction of the tangent to the orbit, at the planet's 

 place for the time to which the varied distance refers, c being the tangential velocity ; 

 another, r 8 w, in the direction of the radius vector, and so not making an angle = 90°, 

 but one = xp, with the former ; and a third S Z, perpendicular to the orbit-plane, which 

 is the plane in which the other two lie. 



A new orbit computed with the distance Ji, varied by the amount 8 z/j, and the dis- 

 tance Jo unvaried, but still using a,, a.., 5,, 8^, will be connected with the former in the 

 following way. 



Ify; = D:^, ^1,^1 = D^, iVt, then there will be four equations, of which the known 

 terms will be / , D^, Zj = 0, ^i , D^, u:^ = 0, and the unknown the differential coeffi- 

 cients of these elements which lie in the orbit-plane ; the coefficients of these unknown 

 quantities being the coefficients of the variations of the elements in (11) and (19'). 



But instead of fi as an element, we shall use p, whereby 



d log fi = — i d]ogp -^3 dlogcoa (f; or, 



rf log fi = — ^dlogp — 3 tan (f d <f. (13) 



Our four equations will be, then, 



= lD^,L,-§^.D.,logp+(^^^smE,-3tany.^.)D.,^+^(^,;;--l)D^,;r; 



ri cos Ml 

 a cos (f 



a, = Da log » Da 9 — tan qp sin Ji>i D^ x 5 



_ Da logo — r-a cos v^ ^ — tan qp sin Ej Da ;ir■ 



l ° -^ a cos cp ' ' 



(35) 



By subtracting the first of these equations from the second, we can at once reduce 

 our equations to the number of three, with three unknown quantities ; determining 

 Da L, afterwards from the second equation explicitly, or, what is perhaps better, leav- 

 ino- i . Da L expressed in terms of the variations Da, log p , &c., in which form it can 

 be easily used. 



