J. C. Watson on the Elements of the Orbit of a Comet. 221 
For the longitude and distance of the perihelion we put 
4f,a 
tang (45°-+-w)— J : : 
and then we shall have 
Dales tang 20 
Mat Sis 2 oe ry ? , 
Jd sin$ (0 —8) 4 rr" (14) 
1 r. sec 20 
Jy pees 4 (0'—B) rr” 
Where 2F=4 (0-+-.6") — 7, q denoting the perihelion distance, and 7 the 
— of the perihelion. 
tv and vw” be the true anomalies at the times ¢ and ¢”, and we have 
v0 — x, v6" — 1, - for direct motion, and 
v1 - 0, vlan — 6, for retrogade motion. 
Then for the time of perihelion passage T, we have 
T= tT Vv" (25 tang? 4v-+ 75 tang 4v), (15) 
which should agree with the value of T found by using the values of 
t,o, instead of ¢ and 2, 
V/2 
log —— = 0°0398723. 
°8 75k 
The preceding formule are all that are required for finding the elements 
of the orbit from two observations, when one of the geocentric distances 
'S given. To solve the problem proposed, we assume, in the rst ple 
#0 approximate value of J, and compute the elements of the orbit from 
the first and third observations, by means of these formule. With the’ 
elements thus obtained we compute the place of the comet for the time ¢’, 
compare it with the corresponding observation, and if we denote the 
oe longitude and latitude by 4/9, and 9, respectively, we shall 
ve 
w=, and B’+4-w’=)",, | 
oe w’ and w’ are the differences between computation and observation. 
ext, assume a second value of the distance of the comet from the earth 
at the time ¢, which we represent by J-+04, and compute the correspond- 
IDg system of elements, and we shall have as before 
Meu i, and B’--w"=F',. 
We also compute a third system of elements from 4-44, (04 being the 
Same 48 before,) and denote the differences between computation and ob- 
“etvation by « and w, then we shall have’ 
u=f(d-—d4), u’==f (4), uf (d+04), 
d values of A—3A, A, and A-++3A, should be so taken that the ag 
a The assume f ee 
/2€ Of A—viz, that for which the differences « and w are a minimum 
Within the limits 4-34 and a++3A, which may always be effect 
Xt Jour. Sc1.—Seconp SERrEs, VoL. XXXV, No. 104—Maxcu, 1563. 
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