385
471] ON THE DETERMINATION OF A PLANETS ORBIT.
1
(observe that n + n" — n' is = twice the triangle CC'G"), for neighbouring positions of
0"
the planet, the values of P and Q are approximately = -w and 6 6" respectively: the
solution consists in the determination of an orbit for which P and Q have these
approximate values; then, by means of such approximate orbit, the values of P and Q
are more accurately determined, and by means of these new values of P and Q, a
new determination is effected of the orbit: and so on, to the requisite accuracy of
approximation.
The foregoing approximate values of P and Q respectively are deduced from the
accurate values
ec 1
7)7)' r'r" COS jf cos f cos f" ’
where r, r, r" are the radius vectors SC, SC', SC" \ 2f 2f, 2f" are the angular
distances CSC", CSC", CSC' (/' = f+f") and tj, tj', 7)" are the ratios of the sectorial
areas CSC", CSC", CSC", to the triangular areas represented by the same letters
respectively: the doubles of the sectorial areas are thus ny, ii’t)', and n"rj", and if the
half latus rectum be denoted by p, then we have
d' v
/- _ U7] _ n'7)' _ n"7)"
Vp = -Q-~Q r - -Q,T A
and it thus at once appears that the accurate value of P is =gr>>> as a ^ove. To
obtain the expression for Q, taking <£, <£', cf>" for the true anomalies (and, for greater
symmetry, writing for the moment v, —v, v", g, —g, g" in place of n, n, n", f f', f"
respectively), we have
whence identically
or writing
this is
v =
V =
p
1 + e cos </>
V
1 + e cos </>
, 2g = cj>" - cj)',
> , V =<*>
P
1 + e cos (j)'
, 2g” = V
(9 + 9 + 9" = 0) ;
sin 2g sin 2jq , sin 2g" 4 sin g sin g sin g"
4. — j _ - ,
r r V p
v = r'r" sin 2g, v = r"r sin 2g', v" = rr' sin 2g",
4rr'r" sin g sin g sin g"
V + V + V = —
p
(rr'r") 2 sin 2g sin 2g' sin 2g"
2prr'r" cos g cos g' cos g"
f //
VV V
2prr'r" cos g cos g' cos g" ‘
C. VII.
49