204- Prof. Encke on Olbers's Method of 



f> cos (a. — ©) — R = r cos /3 cos (X — ©) 

 (III) psm(a-O) = rcosjS sin (X — ©) 



p tan 8 = r sin /3 



p"cos(a"- 0") - R" = r"cos /3" cos (x" - 0") 

 |C sin (a"- 0") = r" cos j3" sin (V - 0") 



p" tan 8" = r" sin |3", 



by which the heliocentric places are given. The accordance of 

 the r and r" with those obtained by the trials is the first proof 

 of correct calculation. The comet is director retrograde, ac- 

 cording as X" is greater or less than X, 



Next comes the determination of ft and i by these formulae : 



, TV x ± tan |3 = tan i sin (X — ft) 



tanj^ tan j cos ^ - x) ^ . _ 



— sin (X" — X) v 



where the upper signs refer to direct, the lower ones to retro- 

 grade, motion of the comets. Instead of these, the following 

 formulas may be applied : 



tan i sin (* (* + h") - ft) ffl ££$»-?) sec P sec ^ 



tan J cos G (A + V) - ft) = j^S""?) «« 6 ^c |f, 



The longitude on the orbit is obtained by these formulae : 



tan(„-a)= ton < ,l -. a) 

 (V) cos ' 



1 y COS I 



where v — ft and w" — ft must be taken in the same qua- 

 drants in which X — ft and X" — ft are respectively situated. 

 If a further check on the calculation should be deemed ne- 

 cessary, one may calculate 



k = 4/ {(r" - rcos (v" - i>)) 2 + r'sin (v" - i;) 2 }, 



where the value of k must perfectly agree with the former. 



For the longitude of the perihelion and the distance we have 

 these formulae : 



1 . , , x cotani(u" — v) 1 

 ~t- . sin \ (v— o>)= *-} ' : — r-rr, > yu 



' 1 1 



-r-.cosi (u— w) =— r-, 

 V q z v ' V r 



