220 J. C. Watson on the Elements of the Orbit of a Comet: 
These equations (8) together with Lambert’s equation, 
(rer! x)? — (npr! — x)= M(t" — 2), (9) 
where log. M= 9:0137327, will enable us = determine J” by successive 
approximations, when the value of 4 is giv 
e may therefore assume a value of 4 ae eans of the approximate 
elements of the orbit, and then find the value of 4” for which the corres- 
ponding values of 7” and * will satisfy equation (9). It will be observed 
that the value of 4” must be shes by trial; and, if we assume also an 
approximate value of 4”, we may find r” from the last of equations (8) 
and then papers * from seaation (9). Then we obtain a second value 
of 4” from the equa 
Mat 22 — 52 
With the value of 4” thus obtained we coe r” and x as before, and 
n a similar manner find a still nearer approximation to 4”, A few trials 
will generally give the correct result. 
When 4 thus been determined we find the heliocentric places of 
the comet by the following: 
4 cos 8 cos (A—@)—R=r cos 6 cos (1—O), 
4cosfsin(A—©) =r cosb sin(/—O), (10) 
4 sin 8 =r sin }, 
1" cos B" cos (1 — 2. Sigh 2: ” cos b" cos (1”/—@"), 
4" cos 8" sin (i! -O "y os b” sin (1"—©"), (1 1) 
dl sin fl main | 
where 4, b”, and 1,1”, arer pectively the heliocentric latitudes and longi- 
tudes of the comet at the tines t end S e values of r and r” should 
agree with those obtained from equations (8). 
The elements of the orbit are then chad from the heliocentric Lagos 
Ee means of the well known formule. For the node and inclination, Wé 
ve 
tang é sin (3(/-L1)— rere sec b sec ”, 
bb 
. tangicos(F(l40)—Q jaan oe =p sec b sec b”, 
the upper sign being used when the motion is direct and the lower siga 
when the motion is retrograde, corresponding respectively to the case of 
where 2” >/ and l"<7. In these equa’ ations, Q denotes the longitude 
ecliptic. 
The longitudes i in the orbit are given by the equations: 
tang (9 — Q )=tang (7 — Q) ase (13) 
tang (0 — §3 )=tang (2"— $3) sec 4 
where 6 and 6” are the longitudes in the orbit. 
on the accuracy of the computation we have 
x2 }r—r" cos (6”—8) { 2-4+-r//2 sin? (6” —6). 
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