THE DISTANCES OF A COMET FROM THE EARTH. 101 



The sum of the last two terms is by (6) and (8), when r and 

 x" are nearly equal, of the third order in t. If they are neglected, 

 (20) gives 



5"cos. «" ■T_ sin. (0' — a) tan. ^< — sin. (Q' — g') tan. 6 -f sin, (g — a') tan. 0' , 



g COS. d t" sin. (©' — a') tan-^"— sin- (©' — «") tan. «'-|- sin. («'—«") tan. 0' "T ('!) 



terms of the third order in t when t = x". 



If we put 0'=O, that is, supposing a, Q, &c., to represent 

 longitudes and latitudes, (21) becomes identical with Olber's equa- 

 tion for determining the ratio of the curtate distances of a comet 

 from the earth, at the first and last observations; (21) is therefore (22) 

 this equation adapted to direct computation from right ascensions 

 and declinations. The same values neglected give, from (18), 

 ^=: — ^, 1^, Q and Q" having the signification stated in (19). (93) 

 It follows from (23), that in practice the accuracy of (17), (20), (24) 

 and (21) will be proportional to the sine of the angle which the 

 direction of the comet's apparent motion makes with the great 

 circle joining the places of the sun and comet at the second ob- 

 servation. 



When this angle is small, the terms neglected in (21) and (23), (25) 

 and errors of observation, acquire an important influence. The best 

 position is when the comet is near the ecliptic at the middle ob- 

 servation, and its motion is mostly in latitude. The equation (20), 

 on which the method of Olbers depends, has the peculiar advan- 

 tage of eliminating '[^\ of which the approximate value (9) is less (26) 

 accurate than that of ^!^\ which is retained. 



If the motion of the comet is mostly in right ascension, the (27) 

 following equations may be employed, which result from substi- 

 tuting in (13) J' = J = a', D' = 0, and A' = A — a:\ D' = 0: 



0=.sin. («' — a) '^ g COS. 6 -{- sin. («' — «") g" cos. 5" — sin. («'—©) ^j^ (28) 

 R COS. -1- sin. («' — ©') ^j R' COS. ©' — sin. («' — ©") R" cos. ©". 



