Olbers's Essai/ on Comets, 139 



1 11 1 ^/ 1 ^ / T^ tan? x"' 



be called C and C . If now we put cot u = r-- — 77^, — t^s. 



tang A sin (U — L>) 



— cot (C" — C), we shall have u the distance of the comet in 

 longitude, at the time of the first observation, from the ascend- 

 ing node ; and consequently C — co the longitude of the node. 

 The inclination of the orbit is obtained from the formula tang 



tan"" x' 



i t= — ^ For the two heliocentric distances of the comet 



sm a 



from its node in the plane of its orbit, u', ?/', we have cos u' = 



cos a' cos a, and cos ?«'"=: cos a'" cos (C" — C'+ij) ; and li" — u' 



is the difference of the two true anomalies. Now if we make (p 



the true anomaly, in the first observation (p, we have, from the 



known properties of the parabola, tang ^ <p = cot — coscc 



• \/ -—, whence we have the longitude of the perihe- 



Hum ; and the distance w at the perihelium is found = r' cos^ 

 § <p ; the time may also readily be found either by direct com- 

 putation, or by a table. [Or since we must previously have 

 determined the values of r, r" and k\ we may obtain that of 



TT from the equation A, Note 3 ; — i. 



* ?•' + r' - ^(r' + r'Y —k'')' 



and then cos ffl ™ ■ — ^~, 1. The longitude of the 



r r' ° 



node may also be deduced from its tangent — — '^-— 4; — , 



X X z" — X " z' 



Note 2. Tr.] Note communicated by the Author. [It may be 



somewhat more convenient to employ the formula tang (w + 



C"-C' sin (a'" + x') /C— C'^ 



tanr 



sin (>. '" — a') 



^ 43 



(^>] 



It is natural, as soon as g' is found, to have the curiosity to 

 calculate all the elements of the orbit, although there is no 

 actual necessity to do -so, until further corrections have been in- 

 troduced ; and, as Laplace has justly remarked, in so long a 

 computation, we ought to spare ourselves every unnecessary 



