Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 7)

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
	        
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