THE DISTANCES OF A COMET FROM THE EARTH. 103 



the middle place of the comet, at nearly right angles with its path. 

 A change of twenty or thirty degrees in the position of C having (40) 

 seldom much influence, it may be taken of a convenient value for 

 computation in (41), (42). Substituting in (13) A= C, £>' = 0, 

 A = a, and D =: &, it becomes, 



0=:[sin. (C— «) tan.^'— sin. (C — «')tan. 5] |^' ? cos. 5 + [sin. (C — a") tan. (9' (41) 



— sin. ( C — «') tan. (9"] p" cos. 6"— [sin. (C— ©) tan. fl'— sin. (C—a') tan.0] 

 £[^ Bcos. + [sin. (C- ©') tan. fl' — sin. (C — «') tan. 0'] ^ B'cos. o' 



— [sin. (C— O") tan. ^' — sin. {C — a') tan. 0"] R' cos. 0". 



And in a similar manner is found, 



0=[sin. (C — a) tan. (9" — sin. (C — «") tan. ^] ^ 9 cos. ^— [sin. (C — «') (42) 

 tan. fi"— sin. (C — «") tan. 5'] '^ p' cos. 6' — [sin. (C — ©) tan. 5" — 

 sin. (C — «") tan. 0]^' J? cos. 0+ [sin. (C— ©') tan. 6" — sm.{C—a") 

 tan- e>'J '^] li' COS. 0'— [sin. (C— Q") tan. 5" — sin. (C — «") tan.V] 



fi" COS. 0". 



As C in these equations is an arbitrary quantity, they may be made (43) 

 to satisfy other conditions, as C = 0; this value gives from (41), 

 = [sin. „ _ ;-^ tan. 6] ^ g cos. 6 + [sin. «" - ^1 tan. 6"] 9" cos. 6" - (44) 

 [-"•O- ^; '-• ^VS^ cos. 0+ [sin. 0'_ fi^; tan. 0'] [ij B'cos. 0' 



— [sin. ©"— ^' tan. 0"] K" cos. 0". 



Neglecting the sum of the last three terms of {SQ), it becomes, 

 using (9), — 7 = 7;,iirTr,. In which d and d" represent, nearly, (45) 

 the apparent motion of the comet during the intervals t" and t. 

 Hence, in small intervals of time, the distance of a comet from 

 the earth varies nearly inversely as its apparent motion. When, 

 therefore, its motion is mostly in right ascension or declination, this 

 relation gives a rough approximation to the ratio of the distances 

 g, g', and g". 



In all the equations here given, except those which contain the 

 quantities J and J', allowance for parallax may be made at the (46) 



