﻿462 A NEW METHOD FOR CORRECTING A PLANET S ORBIT. 



respect to the elements, we shall introduce certain intermediate linear functions of the 

 variations 5M, &c., which last may be considered as themselves variations of conceiv- 

 able finite functions of M, &c. 



4. There are, however, two distinct kinds of changes to which the elements are 

 subject; we may (1) vary the magnitude of the orbit and its position in its own plane, 

 or (2) the position of the plane of the orbit. If the latter change alone takes place, it 

 will affect i2 and i, and. through S2 the value of tt ; so that in this case Srt ^ 3 £2 

 (1 — cos ^). But n is also subject to other changes, those of the kind (1); it Avill 

 therefore be 'better if we make 



5^ = 5 71 — (1 — cos i) d SI. (1) 



Putting, for convenience, x° = ^"5 a-^d substituting ^ = ^° -\- S ^, 71 ^ n'' -\- S n, 



;f = 71 — (1 — cos i) d SI. (2) 



We shall, instead of the longitude n of the perihelion, substitute the distance x of 

 the perihelion, counted from that point. A, fixed in the orbit, which is as far back, in a 

 dkection contrary to the motion, from the ascending node, as the equinox is ; the 

 former distance being counted upon the orbit and the latter upon the ecliptic. 



5. Let now ?*, v denote the planet's actual radius-vector and true anomaly, at the 

 time t ; and (as before stated, see § 2) r°, i;° the same, calculated with the approximate 

 elements ; and 8 r, 8 v the variations r — }*°, v — v° ; we shall then get these quantities, 

 as also the planet's heliocentric longitude. A, counted from the point A, just alluded 

 to, in the following way : — 



E° — e° sin E° = ;u° < + W, 

 r° cos v° = a" (cos E" — e°), 



r° sin v° = a" cos 9° sin E°, 



jf = v° + n". 



(3) 



E — e sin E = (it -\-M, • 

 r cos V = a (cos E — e), 



r sia ?.' r= a cos qp sin E, 



(4) 



6. Let now she a. new function of the elements and their variations, and of the time 



t, such that 



<B — e°sm(£ = n°z-\- M°, 



r cos n = a° (cos (E — e°), 



r sin J) =: a° cos <p° sin (ft, 



(5) 



We have thus obtained A, the direction in the orbit-plane as seen from the sun, as a 

 function of the approximate elements and z, precisely as it is of the corrected ones and 

 of t ; but in order to determine the position in space which the varied elements would 

 give us, we must superpose other variations. 



