﻿A NEW METHOD FOR CORRECTING A PLANETS ORBIT. 469 



r" cos (o),° + v°) A^ S=/f cos j," cos 6° ; ^ 



r" sill (0),° + 1'°) ■\.T—A° sin ,;" sin ^^ ; \ (28) 



!// =: ^° sin 6". ; 



The elements being supposed changed, we shall have h {oy -|- tj) = 5 ^, provided we 

 consider 5 cj, as the variation arising from a change in the position of the perihelion, 

 and do not permit it to be affected by the change in the orbit-plane. 



So that Q being any angle whatever, and S, T, 1' being invariable, we shall have 



<5 [r sin (co, + f + Q)] = ,5 [j sin (, + Q) cos ^]. (28') 



17. We will now apply the formulae we have obtained to a practical problem of very 

 frequent occurrence, — the case in which a set of elements is desired which shall satisfy 

 two observed or normal positions precisely, and certain others nearly. 



In our case we will, with regard to those positions which are not to be exactly 

 represented, content ourselves with satisfying the observed value of our coordinate, tf ; 

 the plane to which it refers being that of an approximate orbit satisfying the first- 

 mentioned positions. 



We will, therefore, assume values of J corresponding to times ti, %, for which we 

 have observed right-ascensions. From this the orbit is easily found, corresponding to 

 these distances. 



A variation of the distance for the time t, which we may denote by J, will produce 

 the following change in equation (28) : 



5 [r sin (wi + i; -}- Q)] =:= 3 ^ . sin (7; + Q) cos 6. (29) 



But we put 5 L = 5 (cji -J- v) under the form 8z.'D^v=^Sz.'DiA, as in (9) ; 

 whence by (18) 



5 log r = Dt log r . Sz -\- r 8 w, 



and (29) becomes 



S [r sin (w^ -\- v -\- Q)] = Dj [r sin (a,, -f- t, -f- Q)] 5 c 4- r sin (wj + y -f Q) 3 w. 

 Making now D, r =:: — tan 9 sin E = c cos i^, ^ 



r U» t; = — = c sm li , 1 



J. T 1 J 



we shall have, if 3 denote the effect of changing J only, 



3 ^ sin (7; + Q) cos e = c sin (wi + »' + V + Q) i^ - + '^ *in (w, + i' + Q) 5 w. (31) 



If Q = — «! — V — 1/;, (31) becomes 



h A sin (?) — 0)1 — V — y) cos 6 :^ — r sin y/ . J w ; 



D^ «; = giigL+! ' + ^-'') COS 6. (32) 



r sin y ^ ' 



VOL. VI. NEW SERIES. 70 



